Avoiding the Maratos Effect in Non-Convex Optimization Through Piecewise Convexity: A Case Study in Optimal PMU Placement Problem

Abstract

In constrained nonlinear optimization, we aim to achieve two goals: one is to minimize the objective function, and the other is to satisfy the constraints. A common way to balance these competing targets is to use penalty functions. Suppose that an algorithm generates a descent direction and produces a step that decreases the objective function value but increases the constraint violation—a phenomenon known as the Maratos effect. This leads to the rejection of the full step by the non-smooth penalty function; therefore, superlinear convergence is not preserved. This work leverages a piecewise convexity model to solve the optimal PMU placement. A quadratic objective function is minimized subject to a non-convex equality constraint within box constraints [0, 1] × [0, 1] ⊂ R². The initial non-convex region is reconsidered as a union of piecewise line segments. This decomposition enables algorithms to converge to a local optimum while preserving superlinear convergence near the solution. An analytical solution is presented using the Karush–Kuhn–Tucker conditions. First-and-second-order optimality conditions are applied to find the local minimum. We show how the Maratos effect is avoided by adopting the piecewise convexity without needing a non-smooth penalty function, second-order corrections or employing the watchdog methods. Simulations demonstrate that the algorithms partially search the space along the line segments—avoiding zig-zag trajectories—and reach (0, 1) or (1, 0), where both feasibility and optimality are satisfied at once.

1. Introduction

Solving nonlinear programming models with Sequential Quadratic Programming (SQP) and Interior Point Methods (IPMs) remains an active area of research [1,2,3,4,5,6,7,8]. These methods often encounter the Maratos effect as they approach local optima upon convergence. The appearance of the Maratos effect can slow convergence speed and lead to step length rejection imposed by non-smooth penalty functions to correct the infeasibility [1,2,3,4,5,6,7,8]. During the iterative process, even if the objective function improves, the line search may reject the unit step that could otherwise lead to a solution due to constraint violations with infeasible iterates [1,2,3,4,5,6,7,8]. Consequently, the convergence rate slows down [1,2,3,4,5,6,7,8]. Second-Order Correction (SOC) steps and the watchdog method are strategies used to mitigate the Maratos effect when a non-smooth penalty function is employed [3].

Alternatively, an exact penalty function can be used to overcome the negative issues of the Maratos effect [4]. The watchdog technique is a type of non-monotone line search that accepts a step if the next step successfully decreases the penalty function, even if it initially causes a temporary increase [4]. SOC involves solving an additional, smaller-scale problem to calculate a correction term that improves the search direction [4,5].

Unconstrained optimization is also the principal domain in both theoretical and practical implementations. It is widely used to solve nonlinear, constrained, and multivariable optimization problems [1,2,3,4,5,6,7,8]. In addition, many recurrent neural networks designed for constrained optimization problems adopt a penalty function to transform the original problem into an unconstrained one [9,10,11,12,13]. A novel multi-agent-based approach for solving the non-convex Economic Dispatch (ED) problem in smart grids is presented in [14]. The authors leverage derivative-free optimization (DFO) methods to enhance performance and facilitate the integration of distributed energy sources through dynamic communication topologies. Novel contributions are presented in nonlinear programming in recent publications [15,16,17,18,19,20,21,22,23,24,25,26,27,28]. Within this broader optimization framework, the Optimal PMU Placement (OPP) problem is a real-world engineering optimization problem, expressed in either the combinatorial or continuous domain [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44].

Although the OPP is an NP-complete problem [29,30,31,32], branch-and-bound algorithms can be utilized to solve it [33]. These algorithms provide a global solution within zero-gap optimality [33]. The OPP problem is analogous to the Trade Hub Location and Allocation problem, as both involve optimizing the placement and distribution of resources subject to constraints [38]. Deterministic and heuristic algorithms for single- and multi-channel PMU allocation have been proposed [39], allowing branch-and-bound methods to find global solutions efficiently, with the key aspect of maximum observability covered.

The objective is to minimize the PMU numbers satisfying complete observability along with maximum redundancy [33,34,39,40]. Generalized Pattern Search [40], Genetic Algorithm–based heuristic for comprehensive PMU placement under practical constraints [41], Recursive Tabu Search (RTS) [42], Genetic Algorithms [43], Binary Gravitational Search [44], and Binary Swarm Optimization [45] are heuristic algorithms being applied to the OPP problem solving [46,47]. A semi-definite programming (SDP) model with a linear objective under Linear Matrix Inequality (LMI) constraints and binary decision variables is proposed in [48,49,50,51,52,53].

Integer linear programming (ILP) and cutting planes transform the conic constraints into polytopes [33]. With this way, the problem is solvable with branch-and-bound algorithms adopted to solve the zero-one SDP model, returning global solutions [33].

Weighted Least Squares (WLS) algorithms [54], Sequential Quadratic Programming (SQP) [55,56,57,58,59] and Interior-Point Methods (IPMs) [59] (both approaches discussed in [59]) are used to solve nonlinear OPP models giving a least number of PMUs along with PMU locations around the grid [33]. Groebner bases [60] are also applied to OPP problem-solving. A strategic PMU placement using a CNN, improving line outage detection is proposed in [61]. Some novel approaches integrated the OPP solution with a state estimation and power flow analysis in [62,63,64,65,66,67,68,69,70,71,72]. A Wide-Area Monitoring System is constituted by PMUs and Phasor Data Concentrators (PDCs). PMUs provide high-accuracy, time-synchronized measurements, while PDCs aggregate data centrally or at the edge, enhancing latency and bandwidth efficiency [62,63,64,65,66,67,68,69,70,71,72].

Building on distributed economic operations, the integration of PMUs is crucial for real-time monitoring and control. PMUs provide accurate, time-synchronized data, enhancing the effectiveness of distributed optimization algorithms, as discussed in [73]. By incorporating PMU data, the proposed model-based and model-free approaches can better handle system dynamics, improve coordination of distributed energy sources, and optimize electric vehicle charging/discharging schedules under uncertainty.

To adopt a nonlinear model for solving the OPP problem, we often face the challenge of avoiding the Maratos effect [74], in order to ensure optimality while maintaining superlinear convergence near the solution [59]. SOC steps and the watchdog method are remedies employed to avoid the Maratos effect when a non-differentiable penalty function is used [4]. Our approach offers a remedy based on a geometry-driven optimization method, rather than relying solely on an algorithmic approach [74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

Building on the idea proposed in our previous work [59], this study refines the piecewise convexity and demonstrates its effectiveness in achieving global-like solutions for the OPP problem while avoiding the Maratos effect [74]. As a result, there is no need to employ non-smooth penalty functions, and the negative impact on superlinear convergence is avoided. The union of piecewise line segments ensures that feasibility and optimality are simultaneously satisfied through the iterations. Consequently, there is no need for trust-region calculations, SOC adjustments, or watchdog methods [74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

In this context, we address a partially convex programming problem, in which the non-convex structure is reconsidered through a piecewise convexity approach [39,58,59].

This novel approach enables the optimizer to guarantee both feasibility and optimality at convergence. By locally exploring line segments in ${\mathrm{R}}^2$ or convex subsets in ${\mathrm{R}}^{\mathrm{n}}$, solutions are achieved [59]. Accordingly, we studied the OPP model using a nonlinear formulation in ${\mathrm{R}}^2$, reflecting a two-bus electric transmission network [89,90,91,92,93]. The nonlinear model is solved using iterative schemes such as SQP and IPM methods [59].

The SQP methods and IPM algorithms are calling via the Fmincon and Ipopt optimizers in MATLAB 2016a [89,90,91]. The algorithms locate local solutions that exhibit global-like behavior, reinforcing the suitability of the proposed nonlinear formulation and aligning with recent research developments [94,95,96,97,98,99,100].

2. Mathematical Nonlinear Programming Formulation

Nonlinear programming methods are based on penalty functions to balance feasibility and optimality, which can distort the descent direction and degrade convergence, especially near the solution. The proposed piecewise convexity model decomposes the feasible region into a union of line segments or convex subregions, which is a geometry-driven optimization, as our foundational designs demonstrated in [33,39,55,58,59].

Consequently, the piecewise convex feasible region permits SQP and IPMs to avoid using penalized terms or correction steps to ensure convergence at a superlinear rate. Therefore, feasibility and optimality are attained simultaneously [39,55,58,59,96].

Within this framework, the feasible region can be considered as a local convexification of the feasible region, bridging the gap between convex optimization and nonlinear programming [3,4]. Let us consider the n-dimension vector $\mathrm{x}={({\mathrm{x}}_1, {\mathrm{x}}_2, \dots {\mathrm{x}}_{\mathrm{n}})}^{\mathrm{T}}\in {\mathrm{R}}^{\mathrm{n}}$. The nonlinear model is stated in ${\mathrm{R}}^{\mathrm{n}}$ as [1,2,3,4,5,6,7,8,39,58,59,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,96]:

$${\mathrm{m}\mathrm{i}\mathrm{n}}_{\mathrm{x}\in {\mathrm{R}}^{\mathrm{n}}}\mathrm{J}(\mathrm{x})$$

(1)

$$\mathrm{s}.\mathrm{t}.: \{\begin{array}{c}{\mathrm{h}}_{\mathrm{i}}(\mathrm{x})=0, \mathrm{i}\in \mathrm{E}, \mathrm{i}=\{1, 2, \dots \mathrm{m}\}\\ {\mathrm{g}}_{\mathrm{j}}(\mathrm{x})\ge 0, \mathrm{i}\in \mathrm{I}, \mathrm{j}=\{1, 2, \dots \mathrm{\rho }\}\\ x \in X\\ \end{array}$$

(2)

where $\mathrm{X}$ is a vector of entries $\mathrm{x}={[{\mathrm{x}}_1, {\mathrm{x}}_2, \dots , {\mathrm{x}}_{\mathrm{n}}]}^{\mathsf{{\rm T}}}$. ${\mathrm{h}}_{\mathrm{i}}(\mathrm{x})$: ${\mathrm{R}}^{\mathrm{n}}\to {\mathrm{R}}^{\mathrm{m}}$, $\mathrm{i}=\{1, 2, \dots \mathrm{m}\}$, $\mathrm{m}\le \mathrm{n}$ with inequality constraints ${\mathrm{g}}_{\mathrm{j}}(\mathrm{x})$: ${\mathrm{R}}^{\mathrm{n}}\to {\mathrm{R}}^{\mathrm{\rho }}\mathrm{j}=\{1, 2, \dots \mathrm{\rho }\}$ [1,2,3,4]. Here, I and E are sets of indices for equality and inequality constraints, respectively [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,58,59,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,96].

We assume that the functions $\mathrm{J}(\mathrm{x})$ and ${\mathrm{h}}_{\mathrm{i}}(\mathrm{x})$, ${\mathrm{g}}_{\mathrm{i}}(\mathrm{x})$ are real-valued and twice continuously differentiable [1,2,3,4,5]. A vector x ∈ X satisfying all the constraints is a feasible solution to the optimization problem; the collection of all such points builds the feasible region [3]. The NLP problem has a local minimum [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Quasi-Newton methods have been widely utilized for solving such nonlinear programming problems in the framework of SQP methods [1,2,3,4,5,6,7,8,9,10,11,12]. Global convergence can be induced through a line search requiring at each step the reduction of a certain penalty function [74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

However, superlinear convergence requires taking a full step $(\mathrm{\alpha }=1)$ near the solution of the nonlinear programming model [3]. During the iterative process, it is possible for these algorithms to face the Maratos effect, leading to an undesired truncation of the step due to the failure of the penalty functions to decrease when a full step is calculated.

The Maratos effect arises in SQP methods when the line search rejects the full step near a solution [74]. The linearized constraints fail to capture the relevant second-order curvature of the true constraints [74]. As a consequence, even though the quadratic problem (QP) subproblem step may be essentially correct, the trial point yields insufficient improvement in the objective for a minimization and induces a constraint violation. Therefore, the merit penalty function rejects the unity step [1,2,3,4,5,6,7,8,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]. This abnormal behavior can slow convergence and prevent the attainment of superlinear convergence [74]. To face this abnormal situation [74], second-order corrections to the steps are adopted, or adjustments made to the line search direction using the watchdog method [75,76,77,78,79,80,81,82,83,84,85,86,87,88].

In this context, while the OPP problem is NP-complete in theory [33,39,58,59], we address a nonlinear programming model with multiple-choice observability constraints, expressed as products of $(1-{\mathrm{x}}_{\mathrm{i}})(1-{\mathrm{x}}_{\mathrm{i}+1})(1-{\mathrm{x}}_{\mathrm{i}+2})$, previously presented in [33,39,55,58,59]. This formulation is decomposed into a piecewise-convex feasible region, which enables the optimizer to converge to a binary solution (0 or 1) without requiring penalty functions, as outlined in our previous work [59]. By decomposing the original feasible region into piecewise-convex segments, the optimizer can achieve a local solution while maintaining superlinear convergence through the iterations. In this way, the solution is obtained smoothly, avoiding the Maratos effect and its negative effects on superlinear convergence near the solution, also proven in our previous works [33,39,55,58,59].

Our paper does not aim to address all non-convex optimization problems. Instead, it focuses on a specific subclass where the feasible region can be decomposed into piecewise convex segments. This decomposition allows nonlinear optimization algorithms to take full steps and avoid the Maratos effect, which constitutes the main methodological contribution of this work. In particular, the polynomial constraint [33,39,40,55,58,59]:

$$(1-{\mathrm{x}}_{\mathrm{i}})(1-{\mathrm{x}}_{\mathrm{k}})(1-{\mathrm{x}}_{\mathrm{j}})=0$$

(3)

Equation (3) represents a “multiple choice” condition [3]. To satisfy it, at least one of the variables must be activated: ${\mathrm{x}}_{\mathrm{i}}=1$, ${\mathrm{x}}_{\mathrm{k}}=1$, ${\mathrm{x}}_{\mathrm{j}}=1$ [33]. This reflects the underlying structure of the problem and ensures that the system is observable at this point [58,59,96].

One reason why the algorithms easily identify a feasible direction and reduce the constraint violation to zero is the piecewise convex structure of the feasible region. Once a line segment is captured, the local geometry guides the iterates directly toward the endpoints. Consequently, the objective function decreases smoothly along this segment, full steps are accepted in the line search, and the Maratos effect is naturally avoided. The constraint is expressed as follows [33,39,40,55,58,59,96]:

$$(1-{\mathrm{x}}_{\mathrm{i}})\cdot {\prod }_{\mathrm{j}\in \mathrm{N}(\mathrm{i})}(1-{\mathrm{x}}_{\mathrm{j}})=0, \forall \mathrm{i}\in \{1, 2, \dots , \mathrm{n}\}$$

(4)

This polynomial constraint imposes that either ${\mathrm{x}}_{\mathrm{i}}=1$ or at least one of its neighbors ${\mathrm{x}}_{\mathrm{j}}=1.$ It represents a logical “OR” condition across a node and its neighborhood. This structure naturally arises in sensor activation, graph-based resource allocation, and coverage optimization problems. It also explains why the algorithms can easily drive the feasibility to zero while the objective function smoothly decreases, avoiding the Maratos effect, and progressing efficiently toward the local and global optima [39,40,55,59,60].

Building on this framework, the present study formulates the optimal PMU placement problem as a constrained nonlinear program that leverages the piecewise-convex structure of the feasible region (see Equation (4)).

This formulation allows iterative algorithms (SQP and IPM) to converge smoothly to non-strict global minima, simultaneously satisfying feasibility and optimality. The following sections detail the mathematical formulation and optimization model, providing both analytical and computational foundations for the proposed approach [96].

2.1. Connection Between Line Search, Full Step, and Penalty Function

In constrained optimization problems, especially in SQP methods, a merit (penalty) function is used to combine the objective and constraint violations into a single scalar function [1,2,3,4,5,6,7,8]. This function determines whether a trial step is accepted or rejected during line search computations. A non-smooth penalty function can be expressed as follows [4]:

$$\mathrm{\varphi }(\mathrm{x})=\mathrm{J}(\mathrm{x})+\mathrm{\rho }\cdot \parallel \mathrm{c}(\mathrm{x})\parallel$$

(5)

where:

• $\mathrm{J}(\mathrm{x})$ = objective function
• $\mathrm{c}(\mathrm{x})$ = constraint violation vector
• $\mathrm{\rho }>0$ = penalty parameter
• $\parallel \mathrm{c}(\mathrm{x})\parallel$ = norm of the constraint violations

This function balances improving the objective and reducing constraint violations. If a full step ${\mathrm{x}}_{\mathrm{k}}+{\mathrm{p}}_{\mathrm{k}}$ improves the objective function $\mathrm{J}(\mathrm{x})$, and all constraints are satisfied (i.e.,$\parallel \mathrm{c}({\mathrm{x}}_{\mathrm{k}}+{\mathrm{p}}_{\mathrm{k}})\parallel = 0$), then the penalty function decreases [1,2,3,4,5,6,7,8,16,17,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]:

$$\mathrm{\varphi }({\mathrm{x}}_{\mathrm{k}}+{\mathrm{p}}_{\mathrm{k}})<\mathrm{\phi }({\mathrm{x}}_{\mathrm{k}})$$

(6)

Therefore, the full step is accepted because both components of the penalty function—the objective function and the constraints—are improved. However, during the iterative process, a step may improve the objective function $\mathrm{J}(\mathrm{x})$ while some constraints remain violated [4,74]. In this case, the improvement in $\mathrm{J}(\mathrm{x})$ is insufficient to offset the constraint violation, causing the penalty term to dominate [1,2,3,4,5,6,7,8,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

As a result, the penalty function fails to decrease, and the step is rejected [68]. The full step is rejected and prevents the achievement of superlinear convergence near the solution as it is well known from optimization theory [2]. This means that in certain problems, the penalty function may increase even though the new trial point is computed by the line-search [4,74]:

$$\mathrm{\varphi }({\mathrm{x}}_{\mathrm{k}}+{\mathrm{p}}_{\mathrm{k}})>\phi ({\mathrm{x}}_{\mathrm{k}})$$

(7)

This increase in the penalty function value may occur irrespective of how close ${\mathrm{x}}_{\mathrm{k}}$ is to ${\mathrm{x}}^{*}$. Thereby, the algorithm is unable to take the full step (unity) toward the solution ${\mathrm{x}}^{*}$, even though ${\mathrm{x}}_{\mathrm{k}}$ is arbitrarily close to ${\mathrm{x}}^{*}$—the so-called Maratos effect [74,75]. This leads to the occasion where the step is truncated and a smaller step is taken by the line search [75,76,77,78,79]. When the Maratos effect occurs near a local solution, a step may improve the objective; however, it is rejected by the penalty function due to constraint violations.

As a result, a full step might increase the constraint violation, even if it decreases the objective function value [1,2,3,4,5,6,7,8,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]. This causes the step to be rejected by the non-smooth penalty function [74,75]. The constraints are satisfied only to first-order accuracy at the solution of the quadratic subproblem [1,2,3,4,5,6,7,8,33,39,55,58,59,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,96].

Consequently, the line search procedure may hinder progress by rejecting the full step. As a result, the penalty function provides small steps $(\ll 1)$ to balance improvements in the objective function with constraint violations [15,16,17,76,77,78,79,80,81,82,83,84,85,86,87,88].

Accordingly, in the specific formulation of the constrained nonlinear model shown in Equations (1) and (2), the penalty function is defined as [15,16,17,18,76,77,78,79,80,81,82,83,84,85,86,87,88]:

$$\mathrm{\varphi }(\mathrm{x}, \mathrm{\rho })=\mathrm{J}(\mathrm{x})+{\sum }_{\mathrm{i}\in \mathrm{E}}{\mathrm{\rho }}_{\mathrm{\iota }}\left|{\mathrm{h}}_{\mathrm{i}}(\mathrm{x})\right|+{\sum }_{\mathrm{i}\in \mathsf{{\rm I}}}{\mathrm{\rho }}_{\mathrm{m}+\mathrm{i}}\mathrm{m}\mathrm{a}\mathrm{x}(0, {\mathrm{g}}_{\mathrm{i}}(\mathrm{x}))$$

(8)

where:

• ${\mathrm{\rho }}_{\mathrm{\iota }}$, ${\mathrm{\rho }}_{\mathrm{m}+\mathrm{i}}$: Are penalty parameters associated with the equality and inequality constraints, respectively.
• $\left|{\mathrm{h}}_{\mathrm{i}}(\mathrm{x})\right|$: Represents the absolute value of the equality constraint violation.
• $\mathrm{m}\mathrm{a}\mathrm{x}(0, {\mathrm{g}}_{\mathrm{i}}(\mathrm{x}))$: Measures the violation of the inequality constraints defined as ${\mathrm{g}}_{\mathrm{i}}(\mathrm{x})\ge 0$.

The penalty function defined in Equation (8) is non-smooth due to the presence of absolute value and max operators [1,2,3,4,5,6,7,8]. Consequently, it is not continuously differentiable, which may pose challenges for optimization methods that rely on gradient-based information. The penalty function defined in Equation (8) is an instance of an exact penalty function, commonly referred to as the ${\mathcal{l}}_1$ penalty [75].

It is non-smooth and non-differentiable and corrects infeasibility, rejecting the full step and taking small steps $(\ll 1)$ [1,2,3,4,5,6,39,55,58,59]. The penalty function has a critical property: for sufficiently large penalty parameters, any local minimum of the penalized problem also satisfies the KKT optimality conditions of the constrained problem [1,2,3,4,5,6,7,8].

In the SQP method, the search direction is obtained by solving a Quadratic Programming (QP) subproblem at each iteration [17,74]. In practical implementations, the resulting step is used to minimize a penalty function (as in Equation (8)), typically through either a line search or a trust-region strategy [39,40,55,59,60,76,77,78,79,80,81,82,83,84,85,86,87,88].

In constrained optimization problems using SQP methods, a search direction ${\mathrm{d}}_{\mathrm{k}}$ is calculated at each iteration, and the next trial-point is estimated as [1,2,3,4,5,6,74,75]:

$${\mathrm{x}}_{\mathrm{k}+1}={\mathrm{x}}_{\mathrm{k}}+\mathrm{a}{\mathrm{d}}_{\mathrm{k}}$$

(9)

Here, $\mathrm{a}\in (0, 1]$ is a step size parameter chosen to guarantee sufficient reduction in the penalty function, such that the following inequality holds (as in Equation (6)) or alternatively [1,2,3,4,39,55,58,59]:

$$\mathrm{\varphi }({\mathrm{x}}_{\mathrm{k}}+\mathrm{a}{\mathrm{d}}_{\mathrm{k}})<\mathrm{\varphi }({\mathrm{x}}_{\mathrm{k}})$$

(10)

The step length is computed by a line search strategy [39,55,58,59]. If the full step $(\mathrm{\alpha }=1)$ does not reduce the penalty function, the step is rejected [1,2,3,4,5,6,7,8,15,16,17,18,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

Then the step is reduced until the penalty function decreases, improving the objective function while avoiding constraint violations [1,2,3,4,5,6,7,8,15,16,17,18,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

As a result, the convergence speed near the local solution is slowed down [74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]. Second-order step adjustments or the watchdog method provide a way to improve step accuracy in the line-search direction and avoid the appearance of the Maratos effect [71].

Based on the above analysis, we declare the mathematical programming problem as:

$$\min \mathrm{J}(\mathrm{x}):\mathrm{x}\in \mathrm{\Omega }$$

(11)

${\mathrm{J}}^{*}=\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{J}(\mathrm{x}):\mathrm{x}\in \mathrm{\Omega }$ is solved either with SQP or IPMs [39,96]. The objective function is stated in a quadratic format involving a minimization problem [6,39,55,58,59]:

$$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{J}(\mathrm{x})={\sum }_{\mathrm{i}}^{\mathrm{n}}{\mathrm{x}}_{\mathrm{i}}^2$$

(12)

$$\mathrm{s}.\mathrm{t}. \mathrm{c}(\mathrm{x})=0$$

(13)

$$x\ge 0$$

(14)

Following the SQP formulation, the IPMs find a solution by approximately solving a sequence of barrier problems of the following formulation such as [6,39,55,58,59]:

$${\mathrm{m}\mathrm{i}\mathrm{n}}_{\mathrm{\mu }}\mathrm{\phi }(\mathrm{x})=\mathrm{J}(\mathrm{x})-\mathrm{\mu }{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{l}}_{\mathrm{n}}({\mathrm{x}}_{\mathrm{i}})$$

(15)

$$\mathrm{s}.\mathrm{t}. \mathrm{c}(\mathrm{x})=0$$

(16)

where $\mathrm{\mu }$ is a barrier parameter that is reduced and converges to zero. For this solution, Newton or quasi-Newton methods are used to achieve convergence, while some algorithms employ SQP or trust-region methods [1,2,3,4,5,6,7,8,33,39,55,58,59,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

The Newton methods use a line search direction from solving a Newton equation of the perturbed optimality system [2]. To obtain local convergence efficiently, the barrier parameter requires careful updating [39,55,58,59,96]. A pseudocode for the SQP/IPM implementation and for the iterative scheme can be as outlined in [39,55,58,59,96].

Under this unified optimization framework, three pseudocodes describe the procedures for solving nonlinear problems using a general iterative method and SQP, IPMs. Algorithm 1 in nonlinear programming is presented, which is a general pseudocode. Algorithm 1 focuses on constrained optimization using SQP/IPMs, while Algorithm 2 outlines a general iterative process that can be utilized for different nonlinear algorithms [1,2,3,4,5,6]. Algorithm 2 outlines the SQP Iterative Scheme [1,2,3,4,58,59,96].

Algorithm 1 (SQP/IPM) 3: This algorithm minimizes an objective function $\mathrm{J}(\mathrm{x})$. It starts with an initial guess ${\mathrm{x}}_{\mathrm{o}}$ and iterates until the gradient of the objective function $\nabla \mathrm{J}({\mathrm{x}}_{\mathrm{k}})$ is smaller than a specified tolerance. At each iteration, a search direction ${\mathrm{p}}_{\mathrm{k}}$ is calculated using the SQP or IPMs, followed by a line search to determine the step length ${\mathrm{a}}_{\mathrm{k}}$. The current estimate ${\mathrm{x}}_{\mathrm{k}}$ is then updated by taking a step along ${\mathrm{d}}_{\mathrm{k}}$ [1,2,3,4,5,6].

Algorithm 2 (General Iterative Scheme): This algorithm is a general framework for iterative optimization. It begins by selecting an initial guess ${\mathrm{x}}_{\mathrm{o}}$ and an iteration counter. In each iteration, a search direction ${\mathrm{h}}_{\mathrm{i}}$ is calculated relying on the current estimate $x_k$ which involves gradient, Hessian, or KKT optimality conditions computations. The step length ${\mathrm{\lambda }}_{\mathrm{i}}$ is computed to guarantee that the updated point ${\mathrm{x}}_{\mathrm{k}}+{\mathrm{a}}_{\mathrm{k}}\times {\mathrm{d}}_{\mathrm{k}}$ both satisfies the constraints (if any) and minimizes the objective function. The current estimate is updated, and a stopping criterion based on the change in $\mathrm{x}$ is checked [3]. If the change is smaller than a set threshold, the process terminates; otherwise, it continues with the next iteration [1,2,3,4,5,6,33,39,40,58,59,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,96].

Algorithm 3 (SQP Iterative Scheme): This iterative scheme is a general pseudocode for the SQP in solving constrained nonlinear programming problems. If the objective improves but small constraint violations occur, the penalty function may reject the full step. This disrupts superlinear convergence and causes the Maratos effect [74]. To mitigate this effect, the SOC step compensates for the second-order contributions of the nonlinear constraints, or watchdog methods may be employed [74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,96].

Algorithm 1: Steps of SQP/IPMs implementation

$\mathrm{Initialize} {\mathrm{x}}_{\mathrm{o}}$

$\mathrm{while} \Vert \nabla \mathrm{J}({\mathrm{x}}_{\mathrm{k}})\Vert >\mathrm{t}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$

$\mathrm{Compute} \mathrm{the} \mathrm{search} \mathrm{direction} {\mathrm{d}}_{\mathrm{k}}$ using the SQP/IPM method

Determine step length ${\mathrm{a}}_{\mathrm{k}}$ using a line search

${\mathrm{x}}_{\mathrm{k}+1}={\mathrm{x}}_{\mathrm{k}}+{\mathrm{a}}_{\mathrm{k}}\times {\mathrm{d}}_{\mathrm{k}}$

end

Algorithm 2: Steps of a general iterative scheme

Step 0: Initialization

Choose initial guess ${\mathrm{x}}_{\mathrm{o}}$ $\in \mathrm{\Omega }$

Set iteration counter $\mathrm{k}\leftarrow 0$

Step 1: Compute search direction ${\mathrm{d}}_{\mathrm{k}}$ based on current estimate ${\mathrm{x}}_{\mathrm{k}}$

(e.g., gradient, Hessian approximation, or KKT system)

Step 2: Determine step length ${\mathrm{a}}_{\mathrm{k}}$ such that:

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{a}}_{\mathrm{k}}=\mathrm{a}\mathrm{r}\mathrm{g}{\mathrm{m}\mathrm{i}\mathrm{n}}_{\mathrm{a}\ge 0}\mathrm{f}({\mathrm{x}}_{\mathrm{k}}+\mathrm{a}\times {\mathrm{d}}_{\mathrm{k}})$

Step 3: Update current estimate:

${\mathrm{x}}_{\mathrm{k}+1}={\mathrm{x}}_{\mathrm{k}}+{\mathrm{a}}_{\mathrm{k}}\times {\mathrm{d}}_{\mathrm{k}}$

Step 4: Check stopping criterion:

if $\Vert {\mathrm{x}}_{\mathrm{\kappa }+1}-{\mathrm{x}}_{\mathrm{k}}\Vert \le \mathrm{\epsilon }$:

terminate

else:

increment $\mathrm{k}\leftarrow \mathrm{k}+1$ and repeat from Step 1

Algorithm 3: Sequential Quadratic Programming (SQP)

Step 0: Initialization

Choose initial guess ${\mathrm{x}}_{\mathrm{o}}$ $\in \mathrm{\Omega }$

Set initial Lagrange multipliers ${\mathrm{\lambda }}_{\mathrm{o}}$ $\in {\mathrm{R}}^{\mathrm{m}}$

Set iteration counter $\mathrm{k}\leftarrow 0$

Set tolerance $\mathrm{\epsilon }$ for convergence

Set maximum iterations $\mathrm{m}\mathrm{a}\mathrm{x}\_\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}$

Step 1: Compute Gradient and Hessian

Compute gradient $\nabla \mathrm{J}({\mathrm{x}}_{\mathrm{k}})$

Compute Jacobian of the constraints $\nabla \mathrm{G}({\mathrm{x}}_{\mathrm{k}})$

Compute Hessian of Lagrangian ${\nabla }_{\mathrm{x}\mathrm{x}}^2\mathrm{L}({\mathrm{x}}_{\mathrm{k}}, {\mathrm{\lambda }\mathrm{x}}_{\mathrm{k}})$

Step 2: Solve Quadratic Subproblem

Solve ${\mathrm{m}\mathrm{i}\mathrm{n}}_{\mathrm{d}}{\displaystyle \frac{1}{2}}{\mathrm{d}}^{\mathrm{T}}{\mathrm{H}}_{\mathrm{L}}\mathrm{d}+{\nabla \mathrm{J}({\mathrm{x}}_{\mathrm{k}})}^{\mathrm{T}}\mathrm{d}$

Subject to:

${\mathrm{h}}_{\mathrm{i}}({\mathrm{x}}_{\mathrm{k}})+{\nabla {\mathrm{h}}_{\mathrm{i}}({\mathrm{x}}_{\mathrm{k}})}^{\mathsf{{\rm T}}}\mathrm{d}=0 \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{i}=\{1, 2, \dots \mathrm{m}\}$

${\mathrm{g}}_{\mathrm{j}}({\mathrm{x}}_{\mathrm{k}})+{\nabla {\mathrm{g}}_{\mathrm{j}}({\mathrm{x}}_{\mathrm{k}})}^{\mathsf{{\rm T}}}\mathrm{d}\le 0 \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\{1, 2, \dots \mathrm{\rho }\}$

Step 3: Line Search

Determine ${\mathrm{a}}_{\mathrm{k}}$

such that: ${\mathrm{x}}_{\mathrm{k}}+{\mathrm{a}}_{\mathrm{k}}\times {\mathrm{d}}_{\mathrm{k}}$ satisfies constraints

Step 4: Update Current Iterate

${{\mathrm{x}}_{\mathrm{k}+1}=\mathrm{x}}_{\mathrm{k}}+{\mathrm{a}}_{\mathrm{k}}\times {\mathrm{d}}_{\mathrm{k}}$

Step 5: Check Convergence

if $\Vert {\mathrm{x}}_{\mathrm{\kappa }+1}-{\mathrm{x}}_{\mathrm{k}}\Vert \le \mathrm{\epsilon }$:

Terminate

else:

Increment $\mathrm{k}\leftarrow \mathrm{k}+1$ and repeat from Step 1

2.2. Piecewise Convexity Model to Avoid the Maratos Effect

Building upon the previous discussion of penalty functions and convergence behavior, nonlinear programming problems can be smooth, and multiple local minima may be found using gradient-based algorithms or derivative-free optimization methods [14,39,40]. In this context, this research introduces a novel algorithmic framework designed to effectively address the challenges posed by non-convex feasible regions in nonlinear programming [1,2,3,4,5,6,7,8]. We utilize the OPP model as an illustrative instance to show the piecewise convexity [33,39,40,55,58,59,96]. Initially, we provide geometric insight in ${\mathrm{R}}^2$, and then generalize the method in ${\mathrm{R}}^{\mathrm{n}}$. This generalization offers an innovative contribution to solving nonlinear programming problems by splitting the feasible region into locally convex regions [59,96]. The piecewise convexity approach is adopted to address nonlinear programming problems with non-convex multiple-choice constraints [58]. The feasible region is interpreted as a union of piecewise line segments in ${\mathrm{R}}^2$, or as piecewise convex subsets in ${\mathrm{R}}^{\mathrm{n}}$. We provide a novel way to manage non-convex feasibility. This transformation permits SQP and IPM methods to preserve both feasibility and optimality simultaneously, as evidenced by numerical results in [39,55,58,59,96].

Feasibility and optimality are achieved without the need for non-smooth penalty functions or additional corrective steps avoiding the Maratos effect [74]. Each step improves the objective function, and the constraints are fully satisfied upon convergence, with full steps accepted through the feasible iterates [58,59,96]. The resulting iterates demonstrate local solution quality with global behavior, while superlinear convergence is retained near the solution, as evidenced by numerical results in [33,39,55,58,59].

2.2.1. Geometric Analysis and Solution of OPP Problem in ${\mathrm{R}}^2$

Instead of using SOC, filters, or the watchdog method [1,2,3,4,5,6,7,8], we reconsidered the feasible region as a union of piecewise line segments in ${\mathrm{R}}^2$, or as subsets of convex subsets in $\mathrm{R}\mathrm{ⁿ}$. Based on this reformulation, the non-convex feasible region can be represented as a union of piecewise line segments, which remains non-convex. To make this algorithmic idea concrete, we provide a geometric plot of the optimization model, offering insight into a non-convex nonlinear model in ${\mathrm{R}}^2$. Let us define the vector as ${\mathrm{x}=[{\mathrm{x}}_1,\mathrm{x}{}_2]}^{\mathrm{T}}$. The constrained minimization problem is stated as [33,39,55,58,59].

$${\mathrm{m}\mathrm{i}\mathrm{n}}_{\mathrm{x}1, \mathrm{x}2\in {\mathrm{R}}^2}\mathrm{J}(\mathrm{x})={\sum }_{\mathrm{i}}^2{\mathrm{x}}_{\mathrm{i}}^2={\mathrm{x}}_1^2+{\mathrm{x}}_2^2$$

(17)

Subject to:

$$\mathrm{c}(\mathrm{x})=(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)\le 0$$

(18)

$$x_1, x_2\in [0, 1]$$

(19)

where $\mathrm{J}(\mathrm{x})$, $\mathrm{c}(\mathrm{x})$ are scalar functions of the design vector $\mathbf{x}$. Here, the entries ${\mathrm{x}}_{\mathrm{i}}$ of the vector $\mathrm{x}={[{\mathrm{x}}_1, {\mathrm{x}}_2]}^{\mathrm{T}}$ are the decision variables, and they are continuous in ${\mathrm{R}}^2$.

It is non-convex due to the product term in the constraint function, which is defined on a closed and bounded decision domain $[0, 1]\times [0, 1]$. The optimization problem involves a convex objective function [33,39,55,58,59].

$$\mathrm{J}(\mathrm{x})={\mathrm{x}}_1^2+{\mathrm{x}}_2^2$$

(20)

The objective has concentric elliptical level curves, with a global minimum at the origin $(0, 0)$. Geometrically, this defines a convex paraboloid in ${\mathrm{R}}^2$ surface context. To better understand the implications of this model, we note that the problem is well-defined and structured in ${\mathrm{R}}^2$, as established in optimization theory (Figure 1 and Figure 2).

Figure 1. Plot of contour lines of the objective function $\mathrm{J}(\mathrm{x})={\sum _{\mathrm{i}}^2{\mathrm{x}}_{\mathrm{i}}^2=\mathrm{x}}_1^2+{\mathrm{x}}_2^2$.

Figure 2. Geometry and Boundary Frontier of $\mathrm{c}({\mathrm{x}}_1, {\mathrm{x}}_2)=1-{\mathrm{x}}_1-{\mathrm{x}}_2+{\mathrm{x}}_1{\mathrm{x}}_2\le 0$.

Figure 1 displays the quadratic objective function, while Figure 2 depicts the constraint function in ${\mathrm{R}}^2$. As depicted in Figure 1, the objective function level sets are circles in ${\mathrm{R}}^2$.

The constraint function is defined as [33,39,55,58,59,96]:

$$\begin{gathered} \mathrm{c}(\mathrm{x})=(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)\le 0 \\ \Rightarrow \mathrm{c}(\mathrm{x})=1-{\mathrm{x}}_1-{\mathrm{x}}_2+{\mathrm{x}}_1{\mathrm{x}}_2\le 0 \end{gathered}$$

(21)

In the constrained minimization problem, $\mathrm{c}(\mathrm{x})>0$ defines an infeasible region while $c(x)\le 0$ defines the feasible region. The equality constraint $\mathrm{c}(\mathrm{x})=0$ forms the frontier of the feasible region $(\mathrm{c}(\mathrm{x})\le 0)$. Due to the nonlinear term ${\mathrm{x}}_1{\mathrm{x}}_2$, the boundary is not a straight line but rather a parabolic-like curve, introducing curvature into the constraint geometry. We reconsider the original non-convex feasible set as a union of piecewise segments and nonlinear boundary curves due to the product ${\mathrm{x}}_1{\mathrm{x}}_2$ as displayed in Figure 2.

This frontier separates the non-feasible region $(\mathrm{c}(\mathrm{x})>0)$ from the feasible region $(\mathrm{c}(\mathrm{x})\le 0)$. The feasible region corresponds to the union of line segments in the square space ${[0, 1]}^2$, where the union of line segments $x_1=1$ and $x_2=1$ holds, as shown in Figure 2 This union represents the frontier where the active constraint ($\mathrm{c}(\mathrm{x})=0)$ is satisfied.

The original feasible region is partitioning into distinct line segments in ${\mathrm{R}}^2$ or into convex subsets in ${\mathrm{R}}^{\mathrm{n}}$. Although the non-convex feasible frontier is separated into a union of piecewise line segments, the union is still a non-convex structure, as is well known from optimization [1,2,3,4,5,6,7,8]. This feasible region is non-convex because, for example, the midpoint between $(1, 0)$ and $(0, 1)$ is $(0.5, 0.5)$, which is not feasible since $(1-0.5)(1-0.5)=0.25\ne 0$. Despite the non-convexity presented in the mathematical programming model by the bilinear product ${\mathrm{x}}_1{\mathrm{x}}_2$, this feasible set can be approached via a piecewise-convex model [59]. Each piecewise line segment is a convex region, but the union of those line segments is not convex [1,2,3,4,5,6,33,39,40,58,59,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,96].

$$\{x_1=1\}\cup \{x_2=1\}\cap {[0, 1]}^2$$

(22)

Because $0\le x_1\le 1$ and $0\le x_2\le 1$, we have $1-x_1\ge 0$ and $1-x_2\ge 0$. Therefore $(1-x_1)(1-x_2)\ge 0$ is nonnegative for any point in the box. The constraint $(1-x_1)(1-x_2)\le 0$ forces the product to be zero, also shown in [33,39,55,58,59].

The equality constraint, that is, $(1-x_1)(1-x_2)=0$ implies that either $x_1=1$ or $x_2=1$ or both. Therefore, the constraint function implies logic of the form that at least one decision variable is equal to 1, also demonstrated in [33,39,40,55,58,59].

The feasible region consists of the union of the vertical line $x_1=1$ and the horizontal line $x_2=1$, intersecting at the point (1, 1), as depicted in Figure 2. The line segments are as:

Line segment:

$${\mathrm{L}}_1=\{({\mathrm{x}}_1, {\mathrm{x}}_2)|{\mathrm{x}}_1=1, \hspace{0.25em} 0\le {\mathrm{x}}_2\le 1\}$$

(23)

Line segment:

$${\mathrm{L}}_2=\{({\mathrm{x}}_1, {\mathrm{x}}_2)|0\le {\mathrm{x}}_1\le 1, {\mathrm{x}}_2=1\}$$

(24)

The union of the line segments implies either $1-{\mathrm{x}}_1=0$ or $1-{\mathrm{x}}_2=0$, which means:

$$(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)=0\Rightarrow {\mathrm{x}}_1=1 \mathrm{or} {\mathrm{x}}_2=1$$

(25)

Hence, the feasible region is as follows:

$$\mathit{F}=\{({\mathrm{x}}_1, {\mathrm{x}}_2)\in {[0, 1]}^2:{\mathrm{x}}_1=1\}\cup \{({\mathrm{x}}_1, {\mathrm{x}}_2)\in {[0, 1]}^2:{\mathrm{x}}_2=1\}$$

(26)

Each line segment is convex, but the union itself is non-convex:

$${\mathrm{F}}_{\mathrm{k}=1, 2}\subseteq {\mathrm{R}}^2$$

(27)

Using this geometric decomposition, we consider the geometric implementation. This connects the convex approximation to SQP and IPMs, avoiding the Maratos effect.

The quadratic convex objective is minimized near one of the line segments, avoiding, the Maratos effect [59,74]. This effect is a phenomenon that occurred in SQP methods. Even near the solution, the step produced by the QP subproblem might lead to the reduction in the non-smooth function, but a constraint violation might be noticed [4].

This affects the solution speed, thereby hindering superlinear convergence. The decomposition of the feasible region into such a piecewise feasible region enables the optimizer to act locally on each line segment in ${\mathrm{R}}^2$ or on convex subregions in ${\mathrm{R}}^{\mathrm{n}}$. SQP and IPMs guide their trajectory toward one of the line segments of the union: $\{x_1=1\}\cup \{x_2=1\}\cap {[0, 1]}^2$. SQP and IPMs avoid zig-zag behavior and the Maratos effect by moving along a piecewise line segment in ${\mathrm{R}}^2$. The objective function monotonically decreases from a certain iteration onward toward its optimal value, while feasibility remains zero, indicating no constraint violation. Consequently, feasibility and optimality are attained simultaneously without relying on a non-smooth penalty function in the iterative process.

2.2.2. Geometric Insight into Optimal PMU Placement in ${\mathrm{R}}^2$, $\mathrm{R}\mathrm{ⁿ}$

The OPP model is declared as a constrained minimization problem [39,55,58,59,96]:

$$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{J}(\mathrm{x})={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{i}}\cdot {\mathrm{x}}_{\mathrm{i}}^2, {\mathrm{w}}_{\mathrm{i}}={(\mathrm{1,1}, \dots , 1)}^{\mathsf{{\rm T}}}$$

(28)

Subject to the observability constraints [33,39,54,55,58,59]:

$${\mathrm{f}}_{\mathrm{i}}(\mathrm{x})=(1-{\mathrm{x}}_{\mathrm{i}})\cdot {\prod }_{\mathrm{j}\in \mathrm{N}\mathrm{(}\mathrm{i})}(1-{\mathrm{x}}_{\mathrm{j}})=0, \forall \mathrm{i} \in \{1, 2, \dots , \mathrm{n}\}$$

(29)

with bounds on the decision variable:

$$0\le \mathrm{x}\le 1$$

(30)

This is a non-convex problem due to the product terms in the constraints. The equality constraints ${\mathrm{f}}_{\mathrm{i}}(\mathrm{x})=0$ imply that for each $i$, either ${\mathrm{x}}_{\mathrm{i}}=1$ or at least one ${\mathrm{x}}_{\mathrm{j}}=1$ for $\mathrm{j}\in \mathrm{N}(\mathrm{i})$ [59]. This formulation implies a combinatorial aspect to the problem, as the decision variables ${\mathrm{x}}_{\mathrm{i}}$ often represent binary choices (e.g., placing a PMU or not [33,39,55,58,59]), even though they are continuous in the formulation [33,39,40,58,59,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,96].

Since the problem is non-convex, gradient-based or derivative-free optimization (DFO) algorithms can be used to solve it, as evidenced by the empirical results in [39,40].

The original feasible region is decomposed into piecewise convex subregions. SQP considers the optimization locally within a convex region. SQP methods make more straightforward progress toward a local optimum without becoming trapped in non-convex regions. This decomposition of the feasible region into convex subsets allows us to avoid the need for a penalty function to enforce feasibility [59]. When the SQP moves within a feasible region, erratic jumps and slow convergence are avoided, which leads to avoid the Maratos effect [39,40,55,58,59,96]. Therefore, there is no need to use a penalty function in the case of infeasibility issues [59,96]. The SQP optimizer demonstrates that the piecewise convexity model converges to a local solution with binary nature at the endpoints [39,40,55,58,59].

A local optimum is achieved by satisfying the multiple-choice constraints [33,39,58,59]. Running with different initial points, a global optimum can be reached with an increased probability, as evidenced by empirical results in [33,39,40,55,58,59,96].

Then, we present the geometric insight of the nonlinear OPP model in ${\mathrm{R}}^{\mathrm{n}}$. The feasible region is stated as [33,39,40,55,58,59,96].

$${\mathrm{S}}_{\mathrm{i}}=\{{\mathrm{x}\in [0, 1]}^{\mathrm{n}}:{\mathrm{f}}_{\mathrm{i}}(\mathrm{x})=0\}$$

(31)

The feasible region is piecewise convex, that is, a union of convex subsets. The equality constraint ${\mathrm{f}}_{\mathrm{i}}(\mathrm{x})=0$ holds, as also presented in [39,40,55,58,59,96].

$${\mathrm{f}}_{\mathrm{i}}(\mathrm{x})=(1-{\mathrm{x}}_{\mathrm{i}})\cdot {\prod }_{\mathrm{j}\in \mathrm{N}(\mathrm{i})}\left(1-{\mathrm{x}}_{\mathrm{j}}\right)=0, \forall \mathrm{i} \in \{1, 2, \dots , \mathrm{n}\}$$

(32)

Therefore, at least one product must be zero:

$$(1-{\mathrm{x}}_{\mathrm{i}})=0\Rightarrow {\mathrm{x}}_{\mathrm{i}}=1\mathrm{or} \exists \mathrm{j}\in \mathrm{N}(\mathrm{i}):(1-{\mathrm{x}}_{\mathrm{j}})=0\Rightarrow {\mathrm{x}}_{\mathrm{j}}=1$$

(33)

That means:

$${\mathrm{x}}_{\mathrm{i}}=1 \mathrm{or} \exists \mathrm{j}\in \mathrm{N}(\mathrm{i}):{\mathrm{x}}_{\mathrm{j}}=1$$

(34)

The representation of the Set ${\mathrm{S}}_{\mathrm{i}}$ is as follows:

$${\mathrm{S}}_{\mathrm{i}}={\bigcup }_{\mathrm{k}\in \{\mathrm{i}\}\bigcup \mathrm{N}(\mathrm{i})}\{\mathrm{x}\in [{0, 1]}^{\mathrm{n}}:{\mathrm{x}}_{\mathrm{k}}=1\}$$

(35)

Set ${\mathrm{S}}_{\mathrm{i}}$ is a union of affine subsets of the hypercube, where for every $\mathrm{k}$:

$${\mathrm{H}}_{\mathrm{k}}=\{\mathrm{x}\in {[0, 1]}^{\mathrm{n}}:{\mathrm{x}}_{\mathrm{k}}=1\}$$

(36)

Each set ${\mathrm{H}}_{\mathrm{k}}$ is a convex affine subset of the hypercube. Set ${\mathrm{S}}_{\mathrm{i}}$ is the union of such affine faces:

$${\mathrm{S}}_{\mathrm{i}}={\bigcup }_{\mathrm{k}}{\mathrm{H}}_{\mathrm{k}}$$

(37)

Therefore, ${\mathrm{S}}_{\mathrm{i}}$ is a non-convex set, but it consists of a finite union of convex subsets, i.e., it is piecewise convex. The entire feasible set is composed by all constraints. All constraints ${\mathrm{f}}_{\mathrm{i}}(\mathrm{x})=0$ must hold simultaneously [39,55,58,59], so the overall feasible region is as:

$$\mathrm{S}={\bigcap }_{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}$$

(38)

We substitute the expression for each ${\mathrm{S}}_{\mathrm{i}}$ and the overall feasible region is given by:

$$\mathrm{S}={\bigcap }_{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}={\bigcap }_{\mathrm{i}}({\bigcup }_{\mathrm{k}\in \{\mathrm{i}\}\bigcup \mathrm{N}(\mathrm{i})}{\mathrm{H}}_{\mathrm{k}})$$

(39)

The S is the set of all vectors ${\mathrm{x}\in [0, 1]}^{\mathrm{n}}$. Such that for every vertex $\mathrm{i}$, there exists at least one element in its neighborhood (including itself) that is equal to 1. This means that the solution set $S$ is the set of all $\mathrm{x}$ satisfying, also presented in [33,39,40,55,58,59]:

$$\mathrm{x}\in \mathrm{S}⟺\forall \mathrm{i}, \exists \mathrm{k}\in \{\mathrm{i}\}\bigcup \mathrm{N}(\mathrm{i}) \mathrm{Such} \mathrm{that} {\mathrm{x}}_{\mathrm{k}}=1$$

(40)

Therefore, $\mathrm{S}$ is not convex but is composed of a finite union of convex components, i.e., convex subsets in ${\mathrm{R}}^{\mathrm{n}}$. Therefore, the feasible region is piecewise convex. S is geometrically formed by intersecting unions of affine faces of the hypercube. It is non-convex because these affine faces form possibly overlapping convex pieces [39,59].

Each convex subset corresponds to one particular combination of nodes being active. Each sub-region is treated as a convex pathway, enabling smooth transitions along piecewise segments, as shown by empirical results in our previous works [39,58,59,96].

Together, these convex subsets make up the piecewise convex feasible region [33,39,55,58,59]. Then $\mathrm{S}$ represents all configurations where every node in the network is either observable itself or connected to at least one observable node [33,39,40,55,58,59,96].

The original feasible region is decomposed into piecewise convex subsets in ${\mathrm{R}}^{\mathrm{n}}$ such as line segments in ${\mathrm{R}}^2$. In this context, the equality constraint function ${\mathrm{f}}_{\mathrm{i}}(\mathrm{x})=0$ features the nonlinear constraint $\mathrm{c}(\mathrm{x})=(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)\le 0$ (Equation (19)) in ${\mathrm{R}}^2$.

The constraint function $\mathrm{c}(\mathrm{x})=(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)=0$ creates two piecewise line segments within the box-constraint $[0, 1]\times [0, 1]$ as shown in Figure 3. As a result, the feasible region consists of the union of the vertical line $x_1=1$ and the horizontal line $x_2=1$, intersecting at the point $(\mathrm{1,1})$.

Figure 3. Constraint Piecewise Line Segments $x_1=1$ and $x_2=1$.

As shown in Figure 3, the active constraint $\mathrm{c}(\mathrm{x})=(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)=0$ represents the union of the two line segments, and it is the frontier between the infeasible region ($\mathrm{c}(\mathrm{x})>0$) and feasible region ($\mathrm{c}(\mathrm{x})\le 0$) within a box-constraint, that is, $\mathrm{x}\in [0, 1]$ [39,55,58,59,96].

In this context, the feasible region decomposes into two convex subsets bounded by the lines ${\mathrm{x}}^1=1 \mathrm{and} {\mathrm{x}}_2=1$, as shown in Figure 4 The feasible set is a union of piecewise line segments, although the union is non-convex itself, as shown in Figure 4. The piecewise line segments ${\mathrm{x}}_1=1$ and ${\mathrm{x}}_2=1$, with the decision variable $({\mathrm{x}}_1,{\mathrm{x}}_2)$ ranging in the interval $[0, 1]$ forms the feasible set, where the solutions are found at the endpoints.

Figure 4. Piecewise Line Segments and Union of these two segments is non-convex.

Thereby, the feasible set is declared as:

$$\mathrm{F}=\{({\mathrm{x}}_1,{\mathrm{x}}_2)\in {[0, 1]}^2\mid {\mathrm{x}}_1=1 \mathrm{or} {\mathrm{x}}_2=1\}$$

(41)

Or equivalent as follows:

$$\mathrm{F}=\{({\mathrm{x}}_1, {\mathrm{x}}_2)|{\mathrm{x}}_1=1, 0\le {\mathrm{x}}_2\le 1\}\cup \{({\mathrm{x}}_1, {\mathrm{x}}_2)\mid 0\le {\mathrm{x}}_1\le 1, {\mathrm{x}}_2=1\}$$

(42)

The line segments act as boundaries and provide natural paths for the iterates, decreasing the need for excessively small step lengths. The geometry-driven optimization enables the iterative algorithm’s trajectory to move partially along one of the line segments (e.g., ${\mathrm{x}}_1=1$ or ${\mathrm{x}}_2=1$). Convex partitioning prevents curvature mismatches and instabilities near constraint intersections, allowing the SQP algorithm to take full steps without violating feasibility. The method avoids the Maratos effect in SQP by ensuring that the local quadratic model accurately represents the curvature of both the objective and constraint functions within each sub-region, as evidenced in [39,58,59,96]. The binary solutions lie at the end-points of the union of piecewise segments in ${\mathrm{R}}^2$, as shown in Figure 4.

2.2.3. Approach for Solving Piecewise Convex Regions

In this context, we exploit the geometric structure of the feasible region to guide the optimization iterates along convex subregions, naturally avoiding the Maratos effect without requiring algorithmic corrections or penalty functions [59,96]. The non-convex feasible region consists of an infeasible region ($\mathrm{c}(\mathrm{x})> 0$) and a feasible region ($\mathrm{c}(\mathrm{x})\le 0$) (Figure 5). The constraint becomes active ($\mathrm{c}(\mathrm{x})=0$), representing a frontier between these regions and giving rise to a piecewise-linear structure, as shown in Figure 5.

Figure 5. Feasible and non-feasible region.

The feasible set includes only the points where either ${\mathrm{x}}_1=1$ or ${\mathrm{x}}_2=1$; i.e., the union of two line segments on the boundary of the square ${[0, 1]}^2$. The geometry includes:

• The objective function forms a convex quadratic surface with elliptical level sets.
• An infeasible interior region within the square.
• A feasible set consisting of two straight boundary lines (${\mathrm{x}}_1=1 \mathrm{and} {\mathrm{x}}_2=1$).
• A nonlinear boundary curve that resembles a parabola due to the product term $x_1{x}_2$.

In this context, this decomposition enables the optimizers to explore locally convex regions and obtain local solutions. They are found at the endpoints of the union of piecewise line segments.

By recognizing these locally convex line segments, SQP and IPM algorithms navigate their iterates along them, as evidenced by the results in [39,58,59]. Leveraging this piecewise-convex formulation allows feasible iterates to remain on one of the line segments, effectively avoiding zig-zag behavior in the iteration process [39,40,55,58,59].

This algorithmic insight allows SQP and IPMs to move along feasible partitions while maintaining local convexity. Therefore, feasibility is satisfied once the iterative algorithm captures the line segment and remains zero, avoiding the Maratos effect [39,40,58,59].

SQP generates feasible iterates without the need for additional penalty terms or filter-based corrections. As a result, the full step is taken, and SQP reaches a solution without zig-zag behavior in the search for optimality. Solutions are found at the endpoints of the union of the line piecewise line segments.

The obtained local solutions are also globally optimal points due to the piecewise convexity of the problem [33,38,40,55,56,57]. The geometric intuition of feasible region is presented in Figure 6.

Figure 6. Constraint Function $\mathrm{c}(\mathrm{x})\le 0$ with active constraint $\mathrm{c}(\mathrm{x})=(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)=0$.

2.2.4. Behavior of the Objective Function in the Minimization Model

The minimization problem is to locate a point in the feasible region with the smallest possible value of $\mathrm{J}(\mathrm{x})={\mathrm{x}}_1^2+{\mathrm{x}}_2^2$. Note that points $({\mathrm{x}}_1, {\mathrm{x}}_2)$ with $\mathrm{J}(\mathrm{x})={\mathrm{x}}_1^2+{\mathrm{x}}_2^2$ form circles centered at $(0, 0)$. These circles are the contours of the objective function. To minimize the objective function, we must find the circle with the smallest radius that intersects the frontier of the union of piecewise line segments $\{({\mathrm{x}}_1,{\mathrm{x}}_2)\in {[0, 1]}^2\mid {\mathrm{x}}_1=1 \mathrm{or} {\mathrm{x}}_2=1\}$. The graphical approach finds an optimal solution by computing the objective contour with the smallest objective value that intersects the frontier of the union of piecewise line segments. The function $\mathrm{J}(\mathrm{x})={\mathrm{x}}_1^2+{\mathrm{x}}_2^2$ is convex and increases with distance from the origin. Since the feasible set consists of segments along the boundary of $[0, 1]\times [0, 1]$, the minimum of the function on this set will occur at the endpoints of these segments [58,59].

The objective function is minimized under the equality constraint $\mathrm{c}(\mathrm{x})=(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)=0$ within the box constraint ${[0, 1]}^2$. The local solutions at ${\mathrm{x}}_{*}^1=(1, 0)$ and ${\mathrm{x}}_{*}^2=(0, 1)$ minimize the quadratic objective subject to a constraint function and decision bound constraints. The point $\mathrm{x}=(1, 1)$ also satisfies the equality constraint but is not a local minimum, since the objective function value is larger at that point. These solutions are proven via KKT optimality conditions and MATLAB validation via Fmincon and Ipopt solvers [89,90,91]. The entire constrained minimization problem is displayed in Figure 7.

Figure 7. Objective function Contours and Feasible and Infeasible Region.

Figure 8 and Figure 9 illustrate two instances where a minimum exists at ${\mathrm{x}}_{*}^1=(1, 0)$ and ${\mathrm{x}}_{*}^2=(0, 1)$, where the objective contour $\mathrm{J}(\mathrm{x})=1$ intersects the union of the piecewise line segments: $\{x_1=1\}\cup \{x_2=1\}\cap {[0, 1]}^2$. By splitting the original feasible set into a union of piecewise line segments, the SQP algorithm recognizes these segments as distinct feasible manifolds and guides the search toward one of them, avoiding oscillations.

Figure 8. Optimization with SQP and contour Objective lines and a local solution at ${\mathrm{x}}_{*}^1=(1, 0)$.

Figure 9. Optimization with SQP and contour objective lines and a local solution at ${\mathrm{x}}_{*}^2=(0, 1)$.

As shown in Figure 8 and Figure 9, by exploiting the geometric structure, feasible iterates are produced along line segments toward the endpoints, which are both local and global minima. Zig-zag behavior and the Maratos effect are avoided as a consequence of the problem geometry [59], rather than through the use of SOC, watchdog methods, or penalty functions. The entire approach is supported both analytically and numerically [59].

By maintaining the full step, preventing non-monotone backtracking, and preserving superlinear convergence, the SQP operates efficiently without relying on non-smooth penalty functions. The sequence of trial points leads to a solution lying at one of the endpoints of the union, as shown in Figure 8 and Figure 9. SQP locates local solutions while simultaneously preserving feasibility and optimality without needing a penalty function or additional corrective steps [59]. KKT optimality conditions confirm that the endpoints, namely, ${\mathrm{x}}_{*}^1=(1, 0)$ or ${\mathrm{x}}_{*}^2=(0, 1)$ satisfy the first-order optimality condition [39,58,59].

MATLAB SQP and IPM solver logs demonstrate the optimality metrics, namely, feasibility = 0, full step lengths ($\mathrm{\alpha }=1$), while both first-order optimality and norm of the step are equal to zero. Consequently, the optimization process naturally converges to one of the feasible endpoints, i.e., ${\mathrm{x}}_{*}^1=(1, 0)$ or ${\mathrm{x}}_{*}^2=(0, 1)$. Both zero first-order optimality and the norm of the step are achieved within a zero constraint violation. The existence of the minima when minimizing the quadratic objective function on the given region consists of two non-strict local minima. This establishes that a minimum point exists on the given set within the decision bound. These optimality metrics indicate that a local minimum has been reached, as shown by the empirical results in [39,40,55,56,57,96].

The objective function is minimized toward 1.0 without constraint violation once the iterative algorithm captures one of the line segments, as displayed in Figure 8 and Figure 9. The iterative algorithms find a local solution by exploiting the specific geometric feasible region without oscillations, backtracking, or full-step rejections, as shown by the empirical results in [39,40,55,56,57,96]. Therefore, there is no need for the non-smooth penalty function to correct steps or enforce feasibility. The full step length is computed in the SQP line-search and is valid even without using the non-smooth penalty function. The quadratic convergence speed of the Newton method is achieved [33,39,40,55,58,59].

Moreover, the same geometric principles enable both SQP and IPMs to solve larger-scale optimization problems efficiently. This is possible even when direct visualization of the feasible region is not available, which would otherwise help in interpreting the numerical results. For the optimal placement of PMUs, these principles are crucial in identifying the power-dominating set in the graph [33,39,40,55,56,57,96].

3. Piecewise-Convex Feasible Regions for Avoiding the Maratos Effect

When solving nonlinear programming problems, infeasibility issues may trigger the appearance of the Maratos effect, which destroys the superlinear convergence near the solution [74]. A step that improves the objective function while keeping the constraint violation small can still cause a non-smooth penalty function to reject the full step length [74]. Near the solution, the penalty function may correct for infeasibilities by accepting only a very small step $(\ll 1)$, thereby preventing superlinear convergence [74,75].

The Maratos effect not only prevents superlinear convergence but can also degrade the quality of the computed local solution [74]. The norm of the step might not converge to zero, implying that SQP methods may fail to satisfy the KKT optimality conditions with the desired accuracy [74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]. Such behavior indicates that convergence to a true local minimum may not be achieved, or that the final iterate may violate the KKT conditions beyond acceptable tolerances [33,39,40,55,58,59,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

We adopted second-order corrections or watchdog techniques to preserve the superlinear convergence, ensure the acceptance of full steps, and guarantee that the displacement norm approaches zero ($\Vert {\mathrm{x}}_{\mathrm{k}+1}-{\mathrm{x}}_{\mathrm{k}}\Vert \to 0$) [74,75]. Consequently, a true local solution is achieved upon convergence. Second-order correction steps adjust the trial step to better satisfy the constraints, especially accounting for their nonlinear curvature [74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

In a neighborhood of a local solution, it may occur that the step produced by the SQP method does not reduce a non-smooth penalty function [74]. To reduce the penalty function while preserving superlinear convergence, a very small step may be required [4]. After calculating the initial line search direction, a correction term is applied in the next step to reduce constraint violations. It is ensured that the combined step (initial direction plus correction) is acceptable to the penalty function [39,58,59,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

This correction is essential for avoiding the Maratos effect and its negative impact on the preservation of the superlinear convergence rate near the solution [74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]. In this context, the watchdog technique complements this approach by allowing the algorithm to accept a small number of steps that do not fully satisfy the penalty function criterion.

It monitors the iterative process and can reject or reverse a step if the chosen direction proves unproductive, further helping to avoid the Maratos effect [59,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

Instead of relying solely on these algorithmic tools, we propose reconsidering the feasible region structure and viewing it as a union of piecewise line segments in R² or as a union of convex subregions in $\mathrm{R}\mathrm{ⁿ}$, as initially shown in our foundational design [57].

Both SQP and IPM methods are guided through feasible iterates along these convex subregions, ultimately capturing one of the line segments and locating solutions at the endpoints of the union: $\{({\mathrm{x}}_1, {\mathrm{x}}_2)\in [0, 1{]}^2|{\mathrm{x}}_1=1 \mathrm{or} {\mathrm{x}}_2=1\}$ [33,39,40,55,58,59].

4. Piecewise-Convex Decomposition in Nonlinear Programming for Feasibility, Optimality and Superlinear Convergence

In nonlinear constrained optimization, the convergence behavior of SQP methods is often affected by the Maratos effect [74]. When the objective function improves while small constraint violations exist, superlinear convergence may be destroyed near the solution—the Maratos effect [74]. A non-smooth penalty function may then reject the full step to fix the infeasibility issue, which can slow down the superlinear convergence [75,76,77,78,79,80,81,82,83,84,85,86,87,88]. To overcome the Maratos effect [74], second-order corrections, filter methods, or watchdog strategies have been proposed in [75,76,77,78,79,80,81,82,83,84,85,86,87,88]. We present an illustrative nonlinear programming problem in ${\mathrm{R}}^2$. The feasible region is the union of boundary line segments defined by a bilinear (polynomial) constraint [59]. The solvers handle these piecewise feasible regions to face infeasibility issues without the requirement for non-smooth penalty functions and finally reach an optimal solution [59]. The SQP and IPMs show monotonic convergence in the objective function, as the objective value decreases consistently from one iteration onward. It naturally guides the sequence of iterates toward feasibility and optimality simultaneously [39,58,59]. When the starting point lies within the infeasible region, the SQP initially increases the objective function values. Its trajectory then moves toward one of the line segments in order to satisfy feasibility from that iteration onward. Once SQP finds a point lying on a line segment ${\mathrm{x}}_1=1$ or ${\mathrm{x}}_2=1$, the objective value is decreased toward 1.0 while both feasibility and optimality are preserved.

Solver logs in the appendices confirm this. A local solution point is achieved, where no constraint violation is measured, and both first-order optimality and norm of the step are computed equal to zero [39,58,59]. Similarly, IPMs solve the nonlinear model and exhibit the same behavior. The monotonic convergence ensures that the penalty function decreases, generating feasible iterates throughout the iterative process toward optimality. Both first-order optimality and norm of step approaches zero while a local minimum is attained [39,58,59]. Therefore, there is no need for penalty functions to act on infeasibility issues that would trigger the appearance of the Maratos effect [74].

Piecewise-Convex Model Solutions: Log Analysis and Plots

In this context, the proposed approach introduces a geometry-driven decomposition of the feasible region in constrained nonlinear optimization problems.

By representing the non-convex feasible set as a union of locally convex segments—line segments in ${\mathrm{R}}^2$ and convex subsets in ${\mathrm{R}}^{\mathrm{n}}$—the method enables gradient-based SQP and IPMs to operate as if solving convex subproblems.

This ensures superlinear convergence, full step acceptance, and feasibility preservation without the need for non-smooth penalty functions [96].

Furthermore, the structure of the multiple-choice constraints naturally leads to binary solutions ($x_i = 0$ or 1) at the endpoints of these segments or vertices of the hypercube in higher dimensions. This combination of piecewise convexity and binary enforcement allows for efficient and reliable optimization of a specific subclass of non-convex problems, aligning continuous NLP solvers with combinatorial outcomes [96].

Simulation results are produced in MATLAB, using SQP and IPMs within the FMINCON and Ipopt solvers, respectively [89,90,91]. It is well known that using a non-differentiable penalty function can prevent acceptance of a full step near the solution. This hinders superlinear convergence [75]. Both SQP and IPMs produce feasible iterates, meaning that, at each step no constraint violation is measured. The optimization pathway solves the non-convex model using the following optimality criteria. If the step decreases the objective function and satisfies the constraints, the penalty function has nothing to correct [59]. In this case, the full step is accepted at convergence [33,39,40,55,58,59].

The SQP and IPMs handle the active constraint $(\mathrm{c}(\mathrm{x})=(1-\mathrm{x}{}_1)(1-\mathrm{x}{}_2)=0)$ and push the iterates toward the boundaries of the piecewise convex frontier, locating the solutions at ${\mathrm{x}}_{*}^1=(1, 0)$ and ${\mathrm{x}}_{*}^2=(0, 1)$. SQP and IPMs find solutions at the endpoints of the line segments. At convergence, the step norm is zero. The solutions are found at the endpoints of the line segments, as shown in Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. As shown in these figures, SQP and IPM trajectories naturally align along feasible line segments, avoiding zig-zag trajectories. The iterative algorithm’s trajectory toward optimality is shown in Figure 10. Starting from an initial point ${\mathrm{x}}^0=(0.5161, 0.019)$, the iterative algorithm moves toward the piecewise segment (${\mathrm{x}}_1=1$). From iteration 2 onward, it moves along the segment and reaches the local solution ${\mathrm{x}}_{*}^1=(1, 0)$. The full step is taken during the iterative process, also demonstrated in [39,58,59]. Convergence is monotonic along the piecewise line segment without oscillations or step rejections. Thus, no penalty function, SOC, or watchdog is needed, as shown in [39,58,59]. The solutions are found at the endpoints of the segments. This explains why an SQP solver can take full steps along the segments and naturally avoid the Maratos effect, also demonstrated in [39,40,55,58,59].

Figure 10. Iterative Algorithm, Contour Objective Function Lines, and Trajectory toward a Local solution at the end-point.

Figure 11. SQP Trajectory on Piecewise Feasible Segment toward local optimality.

Figure 12. IPMs Trajectory on Piecewise Feasible Segment toward Local Optimality.

Figure 13. SQP Trajectory on Piecewise Feasible Segment toward local optimality.

Figure 14. IPMs Trajectory on Piecewise toward local optimality.

As a result, local solutions are found while maintaining both optimality and feasibility, ensuring superlinear convergence. As shown in Figure 10, the piecewise convexity approach exploits a union of line segments in two dimensions. From iteration 2 onward, continuing forward along with the line segment $x_1=1$. SQP guides its trajectory that piecewise convex line segments and the solution stand at $x_{*}^1=(1, 0)$. SQP satisfies both feasibility and optimality simultaneously, also proven in [39,55,58,59].

Starting from the initial point (0.561, 0.019), the SQP method begins by moving toward the piecewise segment defined by $x_1=1$. Initially, the objective function value is approximately 0.009. Feasibility improves rapidly, with the constraint violation decreasing from 0.0653 to 0 by iteration 3. From iteration 3 onward, the algorithm moves along the piecewise segment and monotonically converges to the local solution $(1, 0)$, at which point the objective function reaches its optimal value of 1.0.

From iteration 3 onward, no constraint violation occurs. The algorithm accepts the full step length under the line search conditions. Full steps are taken along feasible directions, and feasibility is preserved from iteration 2 onward. Specifically, the line search holds, where is the solution to the QP problem at iterations [1,2,3,4,5,6,7,8,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]. The SQP method calculates the trial point from the initial point to the final solution either (1.0) or (0.1), converging to a local solution without any constraint violation [33,39,40,55,58,59].

As iterations progress, both the norm of the step and the first-order optimality approach zero, confirming that a local solution has been reached. By iteration 7, both the first-order optimality measure and the step norm reach zero, indicating convergence to a local solution while preserving superlinear convergence, also proven in [39,40,58,59].

At this point, the algorithm continues to accept the full step length under the line-search conditions, avoiding backtracking and ensuring smooth convergence [59].

The quadratic approximations used in the SQP method prevent reduced step lengths (i.e., steps smaller than 1), ensuring that the Maratos effect is avoided [74].

As a result, no non-smooth penalty function is needed to handle constraint violations [75]. These calculations allow the SQP method to converge to a local solution without compromising superlinear convergence near the optimal solution [39,40,58,59,96].

Table 1 presents all optimality metrics, including the objective function values, feasibility, step length, first-order optimality, and step norm at each iteration. During the line search toward optimality, no corrective measures, such as step adjustments or penalty modifications are needed. By the final iteration, the norm of the step vector reaches zero, indicating that no further improvement is possible [33,39,40,55,58,59,96].

Table 1. SQP Log Analyses in solving the Nonlinear Programming Model.

Iter	F-Count	f(x)	Feasibility	Step Length	Norm of Step	First-Order Optimality
0	3	8.998584 × 10−2	6.230 × 10−1		4.776 × 10−1
1	7	4.744174 × 10−1	2.631 × 10−1	7.000 × 10−1	3.924 × 10−1	6.148 × 10−1
2	10	1.105716 × 100	6.533 × 10−2	1.000 × 100	3.630 × 10−1	6.821 × 10−1
3	13	1.514371 × 100	0.000 × 100	1.000 × 100	2.310 × 10−1	1.434 × 100
4	16	1.459627 × 100	0.000 × 100	1.000 × 100	3.924 × 10−2	1.356 × 100
5	19	1.242558 × 100	0.000 × 100	1.000 × 100	1.855 × 10−1	9.850 × 10−1
6	22	1.000000 × 100	0.000 × 100	1.000 × 100	4.925 × 10−1	5.720 × 10−1
7	25	1.000000 × 100	0.000 × 100	1.000 × 100	0.000 × 100	0.000 × 100
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the selected value of the optimality tolerance,
and constraints are satisfied to within the selected value of the constraint tolerance.
<stopping criteria details>
x1	x2	fval
_______	______	________
0.2388	0.18155	0.089986
0.4962	0.47771	0.47442
0.76899	0.7172	1.1057
1	0.7172	1.5144
1	0.67796	1.4596
1	0.4925	1.2426
1	0	1
1	0	1
Optimal Solution:
1
0
Objective Function Value at Optimum:
1
Start: (0.239, 0.182) --> Opt: (1.0000, 0.0000), f = 1.0000

The first-order optimality measure becomes exactly zero, confirming that the KKT conditions are satisfied. The maximum constraint violation is also zero, ensuring that both feasibility and optimality are achieved [33,39,40,55,58,59,96].

As the algorithm smoothly converges to a local minimum, the objective function is minimized monotonically from an iteration onward, until the optimal solution is reached, as illustrated in Figure 11.

The SQP algorithm exhibits smooth convergence from the initial point (0.239, 0.182) to the optimal solution (1.0000, 0.0000), with an objective value of 1.

Feasibility and optimality improve at every iteration, and no constraint violations occur. From iteration 2 onward, the solver consistently takes the unit step, illustrating smooth progress along the search direction. Feasibility is reduced to zero by iteration 3, indicating that the constraints are fully satisfied early in the process [39,40,55,58,59,96].

Both the first-order optimality and the norm of the step ($\Vert {\mathrm{x}}_{\mathrm{k}+1}-{\mathrm{x}}_{\mathrm{k}}\Vert \to 0)$ approach zero, confirming that the solution satisfies the necessary conditions for optimality. The iterative algorithm moves along the piecewise-linear segments of the feasible region, with feasible iterates converging efficiently toward the optimal solution [39,40,55,58,59,96].

In this context, the structure of the feasible region naturally improves the objective function without producing constraint violations. Hence, no additional corrections are required. Therefore, no non-smooth penalty function is needed to correct infeasibility or enforce small step lengths. The full step is accepted at each iteration, and feasibility is preserved. The local solution obtained satisfies both optimality and feasibility, without the need for penalty functions to correct infeasibility or step reductions [59]. The iterates naturally follow the boundary line segments of the feasible set and converge to a local optimum [59]. Consistent with this feasible progress, the penalty function decreases with each iteration, indicating that the full step is not rejected, as feasibility and optimality are satisfied from a certain iteration onward. SQP progresses smoothly toward the optimal solution ${\mathrm{x}}_{*}^1=(1, 0)$ with the objective function value ${\mathrm{J}(\mathrm{x}}^{*})=1$.

A monotonic decrease in the objective value was observed, from 1.5144 at iteration 3 to 1.0000 at iteration 7. Constraint violations decrease steadily and reach zero by iteration 3, while the line search consistently computes unit steps, also proven in [39,55,58,59]. Table 2 reports the iteration history of IPMs solving the nonlinear model in ${\mathrm{R}}^2$.

Table 2. IPMs Log Analyses toward optimality.

Iter	F-Count	f(x)	Feasibility	First-Order Optimality	Norm of Step
0	3	3.381258 × 10−1	3.471 × 10−1	1.824 × 10−2
1	7	6.227646 × 10−1	1.952 × 10−1	3.980 × 10−1	2.082 × 10−1
2	10	1.214015 × 100	4.330 × 10−2	7.788 × 10−1	3.184 × 10−1
3	14	1.283015 × 100	2.791 × 10−3	1.057 × 100	1.945 × 10−1
4	17	1.121541 × 100	1.787 × 10−3	6.878 × 10−1	2.028 × 10−1
5	20	9.972296 × 10−1	1.386 × 10−3	2.000 × 10−1	3.380 × 10−1
6	23	1.000007 × 100	3.681 × 10−6	2.921 × 10−4	2.987 × 10−3
7	26	1.000000 × 100	4.447 × 10−10	4.693 × 10−5	1.208 × 10−4
8	29	1.000000 × 100	5.551 × 10−15	4.146 × 10−8	1.236 × 10−5
9	32	1.000000 × 100	0.000 × 100	2.980 × 10−8	1.488 × 10−8
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the selected value of the optimality tolerance,
and constraints are satisfied to within the selected value of the constraint tolerance.
<stopping criteria details>
x1	x2	fval
_______	___________	_______
0.4388	0.38155	0.33813
0.58441	0.53031	0.62276
0.86199	0.68629	1.214
0.99389	0.54333	1.283
1.0027	0.34075	1.1215
0.99861	0.0027746	0.99723
1	0.00013311	1
1	1.2355 × 10−5	1
1	−4.1762 × 10−9	1
1	1.0705 × 10−8	1
Optimal Solution:
1.0000
0.0000
Objective Function Value at Optimum:
1.0000
Start: (0.439, 0.382) --> Opt: (1.0000, 0.0000), f = 1.0000

The table shows the objective function, feasibility, first-order optimality, and step norm at each iteration. The logs confirm that a local minimum is found, as both the norm of the step and first-order optimality converge to zero, terminating at a local minimum.

The IPM efficiently solves the nonlinear minimization model, converging from a starting point inside the feasible region, as observed from the data in Table 2. Starting from the initial point $x_o=(0.439, 0.382),$ the IPM improves the objective function value while maintaining feasibility. IPMs inherently stay within the feasible region and adjust the barrier parameter inherently, rather than using penalty terms [39,40,55,58,59,96].

IPMs calculate accurate steps early on and then gradually refine the solution as it approaches the endpoint. The step sizes and first-order norms show this transition from larger corrective steps at the start to very small refinements near convergence [3].

IPMs smoothly drive the solution toward feasibility and optimality simultaneously, ultimately reaching the same global minimum found by SQP, but through a different and more interior-oriented search path. Constraint satisfaction is achieved quickly, while the optimality measure decreases to a negligible level, also demonstrated in [39,58,59].

The acceptance of the full step indicates that the piecewise local model, including the Hessian and constraint Jacobians, results in a local optimum. From iteration 6 onward, the iterative process proceeds along a line segment, along with local convexity [59]. By iteration 9, both feasibility and optimality are satisfied at once, and the IPMs avoid unnecessary step rejections or penalty corrections [33,39,40,55,58,59,96].

The algorithm navigates the feasible region efficiently, reaches the constraint boundary, and terminates upon satisfying the optimality and feasibility criteria 3. The optimization successfully found a local solution, which is also the global minimum of the objective function, with the given constraint fully satisfied and predefined tolerances.

This result was consistently obtained using optimizer routines invoking SQP and IPM algorithms implemented via Fmincon and IPOPT [39,40,55,58,59,89,90,96].

Finally, an optimal solution is achieved, namely, $x_{*}^1=(1, 0)$ with an objective value of ${\mathrm{J}(\mathrm{x}}^{*})=1$, with the first-order optimality and norm of the step approaching zero. This remarkable convergence achievement is also evidenced by empirical results in our prior works in this domain [33,39,40,55,58,59,96].

Figure 12 shows the trajectory of the IPM iterates: the method moves from the interior of the feasible region toward a boundary line segment. Once it reaches the horizontal line $x_1=1$, feasibility remains zero, and the iterates proceed along this boundary until reaching the endpoint: $x_{*}^1=(1, 0)$.

As shown in Figure 13, the SQP methods efficiently converged from ${\mathrm{x}}_{\mathrm{o}}=(0.632, 0.098)$ to the boundary solution ${\mathrm{x}}^{*}=(1.000, 0.000)$. It adapts the search direction in a way that progressively moves toward a solution satisfying the polynomial equality constraint function. The objective function is minimized within the feasible region. The trajectory moves along the segment $x_1=1$ and converges to the local point ${\mathrm{x}}_{*}^1=(1, 0)$, which lies at the endpoint of the union of line segments [59].

Feasibility is preserved from a certain iteration onward. Feasible iterates are generated, and the SQP trajectory navigates toward one of the line segments, moving locally on convex subregions. The full step is taken, and the exploitation of local convexity is achieved [39]. The penalty function is decreased without rejecting the unit step, and the need for step truncation is eliminated. Additionally, a monotonic behavior of the objective function appears once the feasibility measure reaches zero [39,40,55,58,59,96]. The SQP methods converge to a local solution, satisfying both feasibility and optimality simultaneously. Therefore, the Maratos effect does not occur [39,55,58,59,96].

Figure 14 illustrates that the IPMs move from the interior of the feasible set toward one of the piecewise line segments. Starting from an initial point, the IPMs decrease the objective function to reach a boundary solution at ${\mathrm{x}}_{*}^1=(1, 0)$, where the minimum value is 1.0 within the feasible frontier. The frontier is the union of the line segments $x_1=1\cup x_2=1$. The IPMs terminate at a local solution, with no constraint violation, while both the first-order optimality conditions and the norm of the step approach zero.

This remarkable convergence achievement is also evidenced by empirical results in our prior works in this domain [33,39,40,55,58,59,96].

5. Implementation Part and Simulation Results

In optimization methods, the objective function may improve even when minor constraint violations occur [74]. However, the penalty term may not decrease sufficiently.

This can lead to step rejection and the loss of superlinear convergence [39,58,59]. The Maratos effect can cause a superlinear convergence rate to fail near a local solution.

This happens because search steps might lead to infeasible points. It appears when the step calculated by the line-search procedure fails to simultaneously improve the objective function and satisfy the constraint functions. MATLAB implementations are presented for producing the numerical results. The rejection of full steps due to temporary constraint violations is avoided. This occurs even when the objective improves at convergence [59]. Non-smooth penalty function is not needed [33,39,40,55,58,59,96].

The proposed piecewise convexity model enables both SQP and IPMs to deliver a local solution [39,58,59]. By guiding SQP and IPMs along these segments, full-step acceptance is maintained [59]. Feasible iterates are produced which successively satisfy the constraints. All subproblems remain feasible without any infeasibility [33,39,40,55,58,59,96]. The step improves the objective function value without any constraint violation, also shown in [58,59]. The full step is accepted at convergence, eliminating the need for a penalty function. By leveraging piecewise convexity, both SQP and IPM algorithms move from a starting point toward optimality [59]. At some iteration, they reach the feasible region and proceed along one of the boundaries of the feasible set (e.g., ${\mathrm{x}}_1=1$ or ${\mathrm{x}}_2=1$) corresponding to the line segments. Thereby, the algorithm always performs within the feasible region and naturally avoids infeasible steps that may trigger the Maratos effect, rejecting the full step length [74]. The step improves the objective function in the case of minimization and does not violate any constraints.

As a result, there is no need for the penalty function to correct infeasibility as our previous works also demonstrated. In this case, the full step is accepted, and the Maratos effect does not occur. SQP and IPMs satisfy both optimality and feasibility.

In this context, SQP and IPM trajectories move toward one of the line segments. They locate a solution at the endpoints such as $x_{*}^1=(1, 0)$ and $x_{*}^2=(1, 0)$. These points are also non-strict global optima laying at the endpoints of the union. The optimization logs indicate that the Maratos effect did not occur. Full steps (step-length = 1) were taken.

Feasibility is achieved once the SQP methods captured one of the line segments defined by ${\mathrm{x}}_1=1 \mathrm{or} {\mathrm{x}}_2=1$. A smooth and monotonic decrease of the objective function toward its optimal value $\mathrm{J}({\mathrm{x}}^{*})=1$ is noticed through the iterations in the logs.

The SQP is invoked via Fmincon and the optimization log is presented in Table 3, as demonstrated by empirical results in [39,55,58,59]. Local minimum points are found at the endpoints ${\mathrm{x}}_{*}^1=(1, 0)$ and ${\mathrm{x}}_{*}^2=(0, 1)$.

Table 3. SQP optimization log analysis toward optimality.

Iteration	x1	x2	Objective Function	Feasibility	Step Length	Norm of Step	First-Order Optimality
0	0.4961	0.1511	0.2689	0.4278	−	0.9921	−
1	0.6688	0.4544	0.6537	0.1807	0.7	0.3490	0.7802
2	1.0000	0.4544	1.2064	0	1	0.3312	0.9087
3	1.0000	0.3485	1.1215	0	1	0.1058	0.6970
4	1.0000	0.0000	1.0000	0	1	0.3485	0.4990
5	1.0000	0.0000	1.0000	0	1	0	0

As the SQP moves along a local convex segment, the trajectory is guided consistently toward an optimal solution along that segment. As a result, SQP performs the iterative process well while preserving the descent direction and allowing a full step length to be taken [39,58,59]. Initially, the objective function increases monotonically as the SQP trajectory approaches the vertical line ${\mathrm{x}}_1=1$. Once the trajectory reaches ${\mathrm{x}}_1=1$, feasibility is satisfied, thus there is no need for penalty function to fix infeasibility [59].

This behavior confirms that the non-smooth penalty function did not reject the full Newton step near the solution. The step-acceptance mechanism did not reject the full-step [39,55,58,59]. Consequently, superlinear convergence is achieved.

The log shows that the objective function decreases monotonically toward its optimal value of $\mathrm{J}({\mathrm{x}}^{*})=1.0$. As observed in the optimization logs in Table 3, from iteration 2 onward, the SQP produces iterates that satisfy the equality constraint, also proven in [39,58,59]. The SQP accepts full steps, and the norm of step and first-order optimality approach zero by iteration 5, and feasibility is preserved. This behavior confirms that the SQP successfully maintains both feasibility and optimality without requiring a penalty function. The superlinear convergence is preserved near the solution, as evidenced by empirical results in our previous works [33,39,40,58,59,96].

Figure 15 illustrates the iterative trajectory of the SQP algorithm toward feasibility and local optimality. The plot shows the progression of the iterates starting from the initial point (0.4961, 0.1511) and moving along the feasible piecewise line segments toward the final solution at (1, 0). Arrows indicate the search direction at each iteration, while the trajectory reflects the step lengths and descent paths. The SQP iterates successfully satisfy the equality constraint from iteration 2 onward, taking full steps and gradually reducing both the norm of step and the first-order optimality measures to zero, as summarized in Table 3. This demonstrates that the algorithm converges smoothly without oscillations or violations of feasibility, and reaches the endpoints of the piecewise feasible set where the objective function is minimized [33,39,40,55,58,59,96].

Figure 15. Fmincon iterative trajectory toward feasibility and optimality.

The following convergence metrics in the iterative scheme leads to optimality.

• Iteration 0–1: The iterative scheme starts from an infeasible point (Feasibility > 0, i.e., constraint violation). The initial point (0.4961, 0.1511) does not satisfy the constraints.
• Iteration 2 and onwards: Feasibility becomes zero, indicating that the solution is now within the piecewise line segment where feasible iterates are produced.
• Step-length: The steps are relatively large, with the full step is taken that is, step-length = 1 from iteration 2 onwards, demonstrating smooth convergence.
• Oscillations, full-step rejections, and zig-zag behavior of the iterates between line segments are avoided.
• Norm of step and Optimality norm: These quantities reduce gradually, indicating that the SQP steadily approaches the local solution ${\mathrm{x}}_{*}^1=(1, 0)$.

The optimization process using Fmincon with SQP shows steady convergence toward the optimal solution at $x*=(1, 0)$. From iteration 5 onward, the objective function decreases smoothly to its optimal value of 1, with zero constraint violation as shown in the optimization logs in Figure 4. By iteration 9, the solution is reached with no further change in the objective function, indicating that the algorithm converges to a local minimum. Moreover, feasibility is preserved from that iteration onward, while the objective function smoothly decreases until reaching its optimal value, as also demonstrated in our prior work [59]. The step norm and first-order optimality decrease gradually, confirming stable convergence without oscillatory behavior. The robustness of this convergence result is also demonstrated by empirical evidence from our earlier research in this mathematical programming area [33,39,40,55,58,59,96].

The nonlinear model is solved using SQP methods, solving the nonlinear problem in the entire decision domain as shown in Table 4 and Table 5. The optimization logs displayed in these tables satisfy both feasibility and optimality tolerances without any oscillations, with step sizes gradually decreasing to zero. Unnecessary backtracking is avoided, indicating that the Maratos effect has been successfully avoided, as evidenced by the norm of the step reaching zero. By decomposing the feasible region into line segments or convex subsets, the algorithm exploits local convexity, preserves the full step in the SQP line search strategy. This ensures that its trajectory remains smooth while moving toward optimality and maintaining superlinear convergence, thus avoiding the Maratos effect (Figure 16 and Figure 17). The minimization problem becomes convex on that piece [39,58,59]. Τhe iterative algorithm moves on that piece, and the local minima found are also global, allowing full (unit) steps to be taken near the solution without unnecessary step length reductions. Principal characteristics of this minimization problem instance being solved by SQP and IPMs are as follows:

Table 4. SQP embedded in Fmincon Solver Iteration Logs toward optimality.

Iter	F-Count	f(x)	Feasibility	Step Length	Norm of Step	First-Order Optimality
0	3	2.023812 × 10−1	4.902 × 10−1		8.500 × 10−1
1	7	5.927443 × 10−1	2.068 × 10−1	7.000 × 10−1	3.707 × 10−1	7.351 × 10−1
2	10	1.187826 × 100	2.545 × 10−2	1.000 × 100	3.415 × 10−1	8.862 × 10−1
3	13	1.296461 × 100	0.000 × 100	1.000 × 100	5.588 × 10−2	1.089 × 100
4	16	1.291746 × 100	0.000 × 100	1.000 × 100	4.347 × 10−3	1.080 × 100
5	19	1.268920 × 100	0.000 × 100	1.000 × 100	2.156 × 10−2	1.037 × 100
6	22	1.172290 × 100	0.000 × 100	1.000 × 100	1.035 × 10−1	8.302 × 10−1
7	25	1.000001 × 100	0.000 × 100	1.000 × 100	4.142 × 10−1	8.231 × 10−1
8	28	1.000000 × 100	0.000 × 100	1.000 × 100	8.723 × 10−4	7.177 × 10−4
9	31	1.000000 × 100	0.000 × 100	1.000 × 100	0.000 × 100	0.000 × 100
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, within the selected optimality tolerance, and constraints
are satisfied within the selected constraint tolerance.
<stopping criteria details>
--- Optimal Solution ---
Var1	Var2	fval	exitflag
1.0000	0.0000	1.0000	1
--- Lagrange Multipliers ---
eqlin		[0 × 1 double]
eqnonlin		2.0000
ineqlin		[0 × 1 double]
lower		[2 × 1 double]
upper		[2 × 1 double]
ineqnonlin		[0 × 1 double]
The extreme point is a local minimum.

Table 5. SQP embedded in Fmincon with Iteration Logs.

Iter	F-Count	f(x)	Feasibility	Step Length	Step	Optimality
0	3	3.765469 × 10−1	3.270 × 10−1		1.094 × 100
1	7	7.745387 × 10−1	1.369 × 10−1	7.000 × 10−1	2.795 × 10−1	7.892 × 10−1
2	10	1.236784 × 100	0.000 × 100	1.000 × 100	2.667 × 10−1	9.732 × 10−1
3	13	1.169973 × 100	0.000 × 100	1.000 × 100	7.433 × 10−2	8.246 × 10−1
4	16	1.009489 × 100	0.000 × 100	1.000 × 100	3.149 × 10−1	5.189 × 10−1
5	19	1.000000 × 100	0.000 × 100	1.000 × 100	9.741 × 10−2	3.876 × 10−2
6	22	1.000000 × 100	0.000 × 100	1.000 × 100	0.000 × 100	0.000 × 100
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, within the selected optimality tolerance, and constraints
are satisfied within the selected constraint tolerance.
<stopping criteria details>
--- Optimal Solution ---
Var1	Var2	fval	exitflag
1.0000	0.0000	1.0000	1
--- Lagrange Multipliers ---
eqlin		[0 × 1 double]
eqnonlin		2.0000
ineqlin		[0 × 1 double]
lower		[2 × 1 double]
The extreme point is a local minimum.

Figure 16. SQP convergence and trajectory toward optimality.

Figure 17. SQP convergence and trajectory toward optimality.

• The solvability of the problem is instance-dependent.
• Piecewise convexity ensures that local solutions are also global.
• Both SQP and IPMs behave globally because of this structure in the specific instance.

Rather than allowing the SQP to wander between competing feasible line segments (${\mathrm{x}}_1=1$ or ${\mathrm{x}}_2=1$) and then correcting it, we reshaped the feasible space so it moves along a single segment. This prevents zig-zag behavior and avoids the Maratos effect [74].

The objective consistently decreases toward its optimal value, ${\mathrm{J}}^{*}(\mathrm{x})=1$ as shown in Table 4 and Table 5. Full steps are taken along the SQP line search direction for all feasible iterates, and feasibility is preserved. The feasibility is satisfied, the first-order optimality measure reaches zero while the step length is taken. The associated Lagrange multipliers further verify that the equality constraint is active at the boundary point namely $(1, 0) \mathrm{or} (0, 1)$. The extreme points (1, 0) and (0, 1) are constrained local minima [59].

Any local minimum at the endpoints of the union is also a global minimum [59]. Consequently, the algorithms reach non-strict global minima lying at (1, 0) and (0, 1). Those solutions are global minima since no other points can be produced by feasible iterates that lead to a smaller objective value. These remarkable optimality metrics achievement is further supported by empirical findings presented in [33,39,40,55,58,59,96]. The optimal solutions are non-strict local optima as well as non-strict global minima due to the piecewise nature of the feasible region: $\{(\mathrm{x}{}_1,\mathrm{x}{}_2)\in [0, 1{]}^2:{\mathrm{x}}_1=1\}\cup \{\mathrm{x}\in {[0, 1]}^2:{\mathrm{x}}^2=1\}$. These optimality findings indicate that local convexity and the effective step-size strategy prevent the appearance of the Maratos effect [33,39,40,55,58,59,96].

This allows for the computation of full steps, that is, unit steps without unnecessary step length reductions near the local solution. This notable convergence achievement is further supported by empirical results reported in our prior works in this domain [33,39,40,55,58,59,96].

The challenge in understanding the novelty is that, to a reader, it may appear that the SQP or IPM algorithms simply converge to an endpoint, namely $x_{*}^1=(1, 0)$ or $x_{*}^2=(1, 0)$, as shown in Table 4 and Table 5. However, the proper interpretation of the logs—and the iterative trajectory shown in Figure 16 and Figure 17—is as follows:

• The iterative algorithm naturally navigates to one of the line segments. Then, it follows the line segment: $x{}_1=1 \mathrm{or} x{}_2=1$, without zigzagging between them.
• This iterative trajectory shows in practice that there is no need for correction mechanisms (SOC, watchdog, or penalty functions) to avoid step rejection.
• A partial search of the space is performed, restricted to convex segments, so the Maratos effect never occurs. The convergence of the algorithm is rapid and does not exhibit any zig-zagging behavior. Thereby, superlinear convergence is preserved naturally, because the iterates never “jump” between lines.

As shown in Figure 16 and Figure 17, the SQP’s trajectory moves along the boundary line segments, locating local solutions at the endpoints. This smooth behavior aligns with expectations for piecewise convex regions. The local convexity helps the optimizer avoid oscillations. By effectively selecting search directions and step sizes, the algorithm progresses smoothly and achieves superlinear convergence, as proven in [39,58,59,96].

No penalty term is required to handle infeasibility. There is no need to reduce the step length, thereby avoiding the Maratos effect [74]. The optimality and convergence criteria are completely satisfied at the boundaries of the feasible region [39,58,59].

6. Conclusions

In this paper we solve a nonlinear programming model by reinterpreting the feasible region as a union of piecewise feasible subsets. A quadratic objective function is minimized under polynomial multiple-choice constraints with box constraints $[0, 1]\times [0, 1]$.

The nonlinear model is solved either using SQP methods or via IPMs solving the nonlinear problem in the entire decision domain. The nonlinear model is analytically solved in ${\mathrm{R}}^2$ using the KKT optimality conditions. Instead of relying on penalty functions or step-size reductions to handle the Maratos effect, our algorithmic approach leverages the intrinsic geometry of the feasible region.

By decomposing the non-convex region into piecewise convex subsets, the optimizer naturally follows convex pathways, ensuring convergence without violating constraints. This geometry-driven strategy provides an innovative approach instead of using traditional algorithmic interventions to mitigate or avoid the Maratos effect.

The non-convex feasible region is represented as a union of piecewise line segments in ${\mathrm{R}}^2$, or as a subset of convex subsets in $\mathrm{R}\mathrm{ⁿ}$. Such a decomposition of the feasible region into a piecewise convexity model enables iterative algorithms to preserve feasibility and optimality, without needing a non-smooth penalty function.

The Fmincon optimizer decomposes the original feasible region into a union of piecewise line segments and locates a local solution with binary values. Both optimality and feasibility are satisfied simultaneously. Step rejections during line search and non-monotonic zig-zag trajectories between feasible line segments are avoided.

Thereby, oscillations and non-monotone behavior does not appear at all. Graphical plots demonstrate how the SQP methods guide their trajectories from the infeasible region, toward one of the line segments and reach the local solution. Similarly, IPMs move from the interior of the feasible region in a similar trajectory, reaching the same solution at the endpoints. The Fmincon and Ipopt produce local solutions located at the endpoints of the union of the piecewise line segments. The first-order optimality and the norm of step are zero for SQP, and both optimality quantities approach zero for IPMs at these local solutions. Hence, feasibility and optimality are achieved simultaneously without needing a non-smooth penalty, SOC, watchdog methods, or filter techniques to correct infeasibilities. The full step is taken in the line search direction, preventing the Maratos effect.

Consequently, local optima are attained at the endpoint of the union of piecewise line segments with superlinear convergence.

7. Future Work

Future work could explore integrating PMU placement with the decarbonization of energy-intensive loads. The optimization techniques discussed here could be adapted to manage computing power migration in data centers, as demonstrated by Du et al. (2025) [94], where load and storage migration are optimized to minimize costs and carbon footprint. Additionally, reinforcement learning (RL) could be incorporated to improve decision-making in complex environments, as shown in Kaloev and Krastev (2023) [95], where sparse updates techniques enhance computational efficiency in large action spaces.

Appendix A.1. Analytical Solution–Equality-Constrained Optimization

We want to minimize the optimization problem involving a quadratic objective function under a polynomial function defined on a closed and bounded set on the decision variables. We consider the constrained minimization problem in $({\mathrm{x}}_1, {\mathrm{x}}_2)\in {\mathrm{R}}^2$:

$${\mathrm{m}\mathrm{i}\mathrm{n}}_{({\mathrm{x}}_1, {\mathrm{x}}_2)\in {[0, 1]}^2}\mathrm{J}(\mathrm{x})={\mathrm{x}}_1^2+{\mathrm{x}}_2^2$$

Subject to inequality constraint:

$$\mathrm{c}({\mathrm{x}}_1, {\mathrm{x}}_2)=(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)\le 0$$

and within the box-constraints as:

$$0 \le x_1 \le 1, 0 \le x_2 \le 1$$

Because ${\mathrm{x}}_{\mathrm{i}}\in [0, 1]$, implies $1-{\mathrm{x}}_{\mathrm{i}}\ge 0$. Therefore, the product $(1 - x{}_1) (1 - x{}_2)$ is nonnegative on the box $[0, 1]\times [0, 1]$ such as:

$$(1-\mathrm{x}{}_1)(1-\mathrm{x}{}_2)\ge 0$$

The constraint requires: $(1-\mathrm{x}{}_1)(1-\mathrm{x}{}_2)\le 0$. It separates the entire search space ${\mathrm{R}}^{\mathrm{n}}$ in infeasible and feasible regions with $(1-\mathrm{x}{}_1)(1-\mathrm{x}{}_2)=0$ to be the frontier of the union. But since the product is always ≥0, the only way both conditions are satisfied is if:

$$(1-\mathrm{x}{}_1)(1-\mathrm{x}{}_2)=0$$

That means, either:

$$\begin{array}{c}1-\mathrm{x}{}_1=0\Rightarrow \mathrm{x}{}_1=1\\ or\\ 1-\mathrm{x}{}_2=0\Rightarrow \mathrm{x}{}_2=1\end{array}$$

Therefore, the feasible region is decomposed into a union of piecewise line segments:

$$\{(\mathrm{x}{}_1,\mathrm{x}{}_2):{\mathrm{x}}_1=1, 0\le \mathrm{x}{}_2\le 1\}\cup \{(\mathrm{x}{}_1,\mathrm{x}{}_2):0\le \mathrm{x}{}_1\le 1, {\mathrm{x}}_2=1\}$$

Geometrically, this means the feasible region consists of two boundary line segments within the square [0, 1] × [0, 1], which is the frontier between the infeasible $\mathrm{c}(\mathrm{x})>0$ and feasible $\mathrm{c}(\mathrm{x})\le 0$ regions.

• The vertical segment at ${\mathrm{x}}_1=1$, from $(1, 0)$ to $(1, 1)$
• The horizontal segment at ${\mathrm{x}}_2=1$, from $(0, 1)$ to $(1, 1)$

Therefore, the feasible set coincides with the boundary of these two sides of the square, rather than any interior region, only boundary segments. The equality constraint $c(x)=(1-x_1)(1-x_2)=0$ relates to the observability constraint in an OPP model with a case study based on a two-bus electric transmission network [33,39,54,55,58,59].

Formulation of Lagrangian Function Considering Only the Equality Constraint

We form the Lagrangian function including only the equality constraints and introduce a Lagrange multiplier $\mathrm{\lambda }$ for the equality constraint [1,2,3,4,5,6,7,8].

$$\mathrm{L}({\mathrm{x}}_1, {\mathrm{x}}_2, \mathrm{\lambda })={\mathrm{x}}_1^2+{\mathrm{x}}_2^2+\mathrm{\lambda }(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)$$

The KKT optimality conditions are explicitly declared as follows [1,2,3,4,5,6,7,8,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]:

• Stationarity:

  $${\displaystyle \frac{\partial \mathrm{L}}{\partial {\mathrm{x}}_1}}=2x_1-\mathrm{\lambda }(1-x_2)=0$$

  $${\displaystyle \frac{\partial \mathrm{L}}{\partial {\mathrm{x}}_2}}=2x_2-\mathrm{\lambda }(1-x_1)=0$$
• Primal feasibility:

  $${\displaystyle \frac{\partial \mathrm{L}}{\partial \mathrm{\lambda }}}=(1-x_1)(1-x_2)=0$$

$\mathrm{\lambda }$ is unrestricted in sign because this is an equality constraint.

3.

Finding potential solutions and langrage multiplier

Solving the System, we find the potential solutions such as:

From $(1-{\mathbf{x}}_1)(1-{\mathbf{x}}_2)=\mathbf{0}$, we see that the decision variables take $\mathbf{0} \mathbf{or}\mathbf{ }\mathbf{1}$.

$$x_1=1\mathrm{or} x_2=1$$

Case $x_1=1$:

$$2(1)-\mathrm{\lambda }(1-{\mathrm{x}}_2)=0⟹2-\mathrm{\lambda }(1-{\mathrm{x}}_2)=0\to \mathrm{\lambda }={\displaystyle \frac{2}{1-{\mathrm{x}}_2}}$$

The feasible domain is within the box-constraint relating to the decision variable ${\mathrm{x}}_2$.

$$0 \le x_2 \le 1$$

The objective function $\mathrm{J}(\mathrm{x})=1+{\mathrm{x}}_2^2$ is minimized at ${\mathrm{x}}_2=0$. Hence, we calculate the candidate solution with a $\mathrm{\lambda }$ parameter.

$$x_{*}^1=(1, 0), \mathrm{\lambda }=2$$

Case ${\mathrm{x}}_2=1$:

$$2(1)-\mathrm{\lambda }(1-{\mathrm{x}}_1)=0⟹2-\mathrm{\lambda }(1-{\mathrm{x}}_1)=0⟹\mathrm{\lambda }={\displaystyle \frac{2}{1-{\mathrm{x}}_1}}$$

The feasible domain is within the box-constraint relating to the decision variable ${\mathrm{x}}_1$.

$$0\le {\mathrm{x}}_1\le 1$$

The objective function $\mathrm{J}(\mathrm{x})=1+{\mathrm{x}}_1^2$ is minimized at ${\mathrm{x}}_1=0$. Hence, the candidate solution is found that is equal with a $\mathrm{\lambda }$ parameter.

$$x_{*}^2=(0, 1), \mathrm{\lambda }=2$$

Both candidates satisfy the multiple-choice constraint, that is, $\mathrm{c}(\mathrm{x})=(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)=0$. The potential solutions ${\mathrm{x}}_{*}^1=(1, 0)$ and ${\mathrm{x}}_{*}^2=(0, 1)$ with $\mathrm{\lambda }=2$ minimizes the product representing the observability constraint in the OPP model relating to a two-bus power network transmission system for solving the OPP [33,39,48,49,50,51,52,53,54,55,58,59].

4.

Computation of Hessian Matrix

The Hessian (matrix of second derivatives) is [1,2,3,4,5,6,7,8]:

$${\mathsf{H}=\nabla }_{\mathrm{x}}^2\mathrm{L}=\left[\begin{array}{cc}{\displaystyle \frac{{\partial }^2\mathrm{L}}{\partial {\mathrm{x}}_1^2}}& {\displaystyle \frac{{\partial }^2\mathrm{L}}{\partial {\mathrm{x}}_1\partial {\mathrm{x}}_2}}\\ {\displaystyle \frac{{\partial }^2\mathrm{L}}{\partial {\mathrm{x}}_2\partial {\mathrm{x}}_1}}& {\displaystyle \frac{{\partial }^2\mathrm{L}}{\partial {\mathrm{x}}_2^2}}\end{array}\right]=\left[\begin{array}{cc}2& \mathrm{\lambda }\\ \mathrm{\lambda }& 2\end{array}\right]$$

The eigenvalues μ of H are solutions to the characteristic equation such as [1,2,3,4,5,6,7,8]:

$$\mathrm{d}\mathrm{e}\mathrm{t}(\mathrm{H}-\mathrm{\mu }\mathrm{I})=0$$

$$\mathrm{d}\mathrm{e}\mathrm{t}\left[\begin{array}{cc}2-\mathrm{\mu }& \mathrm{\lambda }\\ \mathrm{\lambda }& 2-\mathrm{\mu }\end{array}\right]=0\to (2-\mathrm{\mu })(2-\mathrm{\mu })-\mathrm{\lambda }\mathrm{\lambda }=0$$

The equality is simplified to:

$${(2-\mathrm{\mu })}^2-{\mathrm{\lambda }}^2=0\to {(2-\mathrm{\mu })}^2={\mathrm{\lambda }}^2$$

The square root of both sides is calculated such as [1,2,3,4,5,6,7,8,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]:

$$(2-\mathrm{\mu })=\pm \mathrm{\lambda }$$

Eigenvalues of $\mathrm{H}$:

$$\mathrm{\mu }{}_1=2+\mathrm{\lambda }, \mathrm{\mu }{}_2=2-\mathrm{\lambda }$$

• $-2< \mathrm{\lambda }< 2$$\to \mathrm{\mu }{}_1,\mathrm{\mu }{}_2> 0$$\to$ H positive definite
• $\mathrm{\lambda }=2\to \mathrm{\mu }{}_1=4, \mathrm{\mu }{}_2=0\to$ H positive semi-definite (flat direction)
• $\mathrm{\lambda }> 2 \mathrm{or} \mathrm{\lambda }< -2\to$ H indefinite (one positive, one negative eigenvalue)

At the candidate solution $\mathrm{\lambda }=2$, the eigenvalues are found ${\mathrm{\mu }}^1=4, \mathrm{\mu }{}_2=0$

5.

Tangent Space Analysis (Equality Constraint Only)

Active constraint: $\mathrm{c}(\mathrm{x})=(1-\mathrm{x}{}_1)(1-\mathrm{x}{}_2)=0$

Gradient at ${\mathrm{x}}^{*}=(1, 0)$:

$$\nabla \mathrm{c}({\mathrm{x}}^{*})={[-1, 0]}^{\mathrm{T}}$$

Linearized tangent space: $\mathrm{T}=\{\mathrm{d}=({\mathrm{d}}_1, {\mathrm{d}}_2)\in {\mathrm{R}}^2:{\nabla \mathrm{c}({\mathrm{x}}^{*})}^{\mathrm{T}}\mathrm{d}=0\}$.

We compute $\nabla \mathrm{c}(\mathrm{x}{)=[-(1-{\mathrm{x}}_2), -(1-{\mathrm{x}}_1)]}^{\mathrm{T}}$. At $(1, 0)$ we get $\nabla \mathrm{c}(1, 0)={[-1, 0]}^{\mathrm{T}}$, so

$$\mathrm{T}=\{\mathrm{d}=({\mathrm{d}}_1, {\mathrm{d}}_2)\in {\mathrm{R}}^2:-1\cdot {\mathrm{d}}_1+0\cdot {\mathrm{d}}_2=0\}=\{(0, {\mathrm{d}}_2)\}\Rightarrow {\mathrm{d}}_1=0$$

i.e., tangent directions are $\mathrm{d}=(0, {\mathrm{d}}_2).$

Second-Order Sufficient Condition (SOSC)

Restricted to this tangent space ${\mathrm{x}}^{*}=(1, 0)$.

$${\mathrm{d}}^{\mathrm{T}}\mathrm{H}\mathrm{d}=[0{\mathrm{d}}_2]\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]\left[\begin{array}{c}0\\ {\mathrm{d}}_2\end{array}\right]=2{\mathrm{d}}_2^2>0, (\mathrm{for} {\mathrm{d}}_2\ne 0).$$

Some observations are as follows [1,2,3,4,5,6,7,8]:

• H is positive definite on tangent space $⟹$ SOSC satisfied $⟹$ local minimum is achieved along feasible directions.
• Flat direction is along the line segment in the feasible set (${\mathrm{d}}_2$ direction at ${\mathrm{x}}_1=1$).

Therefore H is positive definite on the tangent space, not on the full space, and the second-order sufficient condition (SOSC) for a constrained (non-strict) local minimum under the equality constraint is satisfied at (1, 0). The same argument applies at $x_{*}^2=(0, 1)$ by symmetry. We conclude that each candidate is a non-strict constrained local minimum. Since there is no feasible point with a smaller objective value, they are non-strict global minima over the feasible set. For any feasible tangent direction d, the inequality must hold [1,2,3,4,5,6,7,8,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]:

$${\mathrm{d}}^{\mathrm{T}}\mathrm{H}\mathrm{d}\ge 0$$

If only the equality constraint were active, the tangent space would be $\mathrm{T}=\{(0, {\mathrm{d}}_2)\}$. Then,

${\mathrm{d}}^{\mathrm{T}}\mathrm{H}\mathrm{d}=2{\mathrm{d}}_2^2>0$, for ${\mathrm{d}}_2\ne 0\to$ H is positive definite on the tangent space [1,2,3,4,5,6,7,8,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88].

For $\mathrm{\lambda }=2\to$ one zero eigenvalue → flat direction along boundary.

For $\mathrm{\lambda }< 2$$\to$ positive definite in interior → strict minimum if feasible interior existed.

Based on the above analytically calculations, we notice the following optimality metrics.

• Candidate solutions: $x_{*}^1=(1, 0) \mathrm{and} x_{*}^2=(0, 1).$
• Objective function value: ${\mathrm{J}}^{*}(\mathrm{x}) = 1.$
• Hessian eigenvalues: $\mathrm{\lambda }< 2$$\to$ positive definite, $\mathrm{\lambda }=2$→ positive semi-definite.
• Boundary effect: minima are non-strict due to flat directions along feasible boundary segments
• $\mathrm{\lambda }$ indicates sensitivity of objective to equality constraint; $\mathrm{\lambda }=2\to$ fully active.

Appendix A.2. Analytical Solution–Equality-Constrained Optimization and Decision Bounds

We consider the full Lagrangian Function formulation, including both the equality constraint and the box constraints directly in the Lagrangian function [1,2,3,4,5,6,7,8,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]:

$$\mathrm{L}({\mathrm{x}}_1, {\mathrm{x}}_2, \mathrm{\lambda }, {\mathrm{\mu }}_1, {\mathrm{\mu }}_2, {\mathrm{\mu }}_3, {\mathrm{\mu }}_4)={\mathrm{x}}_1^2+{\mathrm{x}}_2^2+\mathrm{\lambda }(1-{\mathrm{x}}_1)(1-{\mathrm{x}}_2)+{\mathrm{\mu }}_1(-{\mathrm{x}}_1)+{\mathrm{\mu }}_2(-{\mathrm{x}}_2)+{\mathrm{\mu }}_3({\mathrm{x}}_1-1)+{\mathrm{\mu }}_4({\mathrm{x}}_2-1)$$

where:

• ${\mathrm{\mu }}_{\mathrm{i}}\ge 0$ are the Lagrange multipliers for box constraints.
• $\mathrm{\lambda }$ is unrestricted (equality constraint multiplier).

1.

The KKT optimality conditions is as follows:

1.

Stationarity:

$$\{\begin{array}{c}2{\mathrm{x}}_1-\lambda (1-{\mathrm{x}}_2)-{\mathrm{\mu }}_1+{\mathrm{\mu }}_3=0\\ 2{\mathrm{x}}_2-\lambda (1-{\mathrm{x}}_1)-{\mathrm{\mu }}_2+{\mathrm{\mu }}_4=0\end{array}$$

2.

Primal feasibility:

$$(1-x_1)(1-{\mathrm{x}}_2)=0, -{\mathrm{x}}_{\mathrm{i}}\le 0, \hspace{1em}\hspace{1em}{\mathrm{x}}_{\mathrm{i}}-1\le 0$$

3.

Complementary slackness:

$${\mathrm{\mu }}_1{\mathrm{x}}_1=0, {\mathrm{\mu }}_2{\mathrm{x}}_2=0, {\mathrm{\mu }}_3({\mathrm{x}}_1-1)=0, {\mathrm{\mu }}_4({\mathrm{x}}_2-1)=0$$

4.

Dual Feasibility:

$${\mathrm{\mu }}_{\mathrm{i}}\ge 0, \mathrm{i}=1, \dots , 4$$

2.

The candidate solutions of the minimization problem are as follows:

• ${\mathrm{x}}_{*}^1=(1, 0)$

➢

Active constraints: $1-{\mathrm{x}}_1=0$, ${\mathrm{x}}_2=0$

➢

Objective function: $\mathrm{J}({\mathrm{x}}^{*})=1$

➢

Stationarity: $\mathrm{\lambda }=2$, ${\mathrm{\mu }}_{\mathrm{i}}$ satisfy complementary slackness

• $2(1)-\mathrm{\lambda }(1-0)-{\mathrm{\mu }}_1+{\mathrm{\mu }}_3=0$$\to 2-\mathrm{\lambda }-{\mathrm{\mu }}_1+{\mathrm{\mu }}_3=0$
• $2(0)-\lambda (1-1)-{\mathrm{\mu }}_2+{\mathrm{\mu }}_4=0$$\to -{\mathrm{\mu }}_2+{\mathrm{\mu }}_4=0$

Complementary slackness:

• ${\mathrm{\mu }}_1{\mathrm{x}}_1=0\to {\mathrm{\mu }}_1=0$
• ${\mathrm{\mu }}_2{\mathrm{x}}_2=0\to {\mathrm{\mu }}_2=0$
• ${\mathrm{\mu }}_3$(${\mathrm{x}}_1-1)=$0$\to {\mathrm{\mu }}_3\ge 0$ free,
• ${\mathrm{\mu }}_4({\mathrm{x}}_2-1)=0\to {\mathrm{\mu }}_4=0$
   
• Solve stationarity: $\mathrm{\lambda }=2$, ${\mathrm{\mu }}_3=0$
• Objective function: $\mathrm{J}({\mathrm{x}}^{*})=1$
   
• ${\mathrm{x}}_{*}^2=(0, 1)$
   

➢

Active constraints: $1-x_2 = 0$, ${\mathrm{x}}_1= 0$

➢

Objective function: $\mathrm{J}({\mathrm{x}}^{*})=1$

➢

Stationarity: $\mathrm{\lambda }=2$, ${\mathrm{\mu }}_{\mathrm{i}}$ satisfy the complementary slackness

• $2(0)-\mathrm{\lambda }(1-1)-{\mathrm{\mu }}_1+{\mathrm{\mu }}_3=0\to -{\mathrm{\mu }}_1+{\mathrm{\mu }}_3=0$
• $2(1)-\mathrm{\lambda }(1-0)-{\mathrm{\mu }}_2+{\mathrm{\mu }}_4=0\to 2-\mathrm{\lambda }-{\mathrm{\mu }}_2+{\mathrm{\mu }}_4=0$

• Complementary slackness:

• ${\mathrm{\mu }}_1{\mathrm{x}}_1=0\to {\mathrm{\mu }}_1=0$,
• ${\mathrm{\mu }}_2{\mathrm{x}}_2=0\to {\mathrm{\mu }}_2=0$,
• ${\mathrm{\mu }}_3({\mathrm{x}}_1-1)=0\to {\mathrm{\mu }}_3=0$
• ${\mathrm{\mu }}_4({\mathrm{x}}_2-1)=0\to {\mathrm{\mu }}_4\ge 0$
   

• Solve stationarity: $\mathrm{\lambda }=2$, ${\mathrm{\mu }}_4=0$
• Objective function: $\mathrm{J}({\mathrm{x}}^{*})=1$
• Both optimal solution give an objective function value equal to $\mathrm{J}({\mathrm{x}}^{*}) = 1$

3.

The Hessian and the Tangent Space are as:

The Hessian of the Lagrangian with respect to $x$ is:

$$\mathrm{H}={\nabla }_{\mathrm{x}}^2\mathrm{L}=\left[\begin{array}{cc}2& \mathrm{\lambda }\\ \mathrm{\lambda }& 2\end{array}\right], \mathrm{\lambda }=2$$

Eigenvalues: ${\mathrm{\mu }}_1=4, {\mathrm{\mu }}_2=0$

Tangent space.

• Equality constraint only ($\mathrm{c}(\mathrm{x})=0$):

  $$\nabla \mathrm{c}({\mathrm{x}}^{*})={[-(1-{\mathrm{x}}_2), -(1-{\mathrm{x}}_1)]}^{\mathrm{T}}$$

  $$\mathrm{At} {\mathrm{x}}_{*}^1=(1, 0)$$

  $$\nabla \mathrm{c}(1, 0)={[-1, 0]}^{\mathrm{T}}$$

The tangent space of feasible directions (ignoring box constraints) is:

$${\mathrm{T}}_{\mathrm{e}\mathrm{q}}=\{\mathrm{d}=({\mathrm{d}}_1, {\mathrm{d}}_2)\in {\mathrm{R}}^2:{\nabla \mathrm{c}({\mathrm{x}}^{*})}^{\mathrm{T}}\mathrm{d}=0\}=\{(0, {\mathrm{d}}_2)\}$$

Case Study-Restricted Hessian along tangent space ${\mathrm{T}}_{\mathrm{e}\mathrm{q}}$

$${\mathrm{d}}^{\mathrm{T}}\mathrm{H}\mathrm{d}=[0{\mathrm{d}}_2]\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]\left[\begin{array}{c}0\\ {\mathrm{d}}_2\end{array}\right]=2{\mathrm{d}}_2^2>0, {\mathrm{d}}_2\ne 0$$

This shows positive curvature along the tangent direction, satisfying the SOSC along equality-constrained directions. The zero eigenvalue corresponds to the flat direction along the boundary.

• Positive definite on tangent space $\to$ SOSC satisfied
• Confirms local minimum along feasible directions
• Flat direction corresponds to the zero eigenvalue along the boundary

The Hessian restricted to this tangent space is positive definite ${\mathrm{d}}^{\mathrm{T}}\mathrm{H}\mathrm{d}=2{\mathrm{d}}_2^2>0$, satisfying the SOSC and confirming a local minimum along feasible directions. The zero eigenvalue of the full Hessian corresponds to a flat direction along the boundary, making the local minimum non-strict. By symmetry, the same applies to ${\mathrm{x}}_{*}^1=(1, 0)$. By symmetry, the same calculations are applied to locate the local solution as: ${\mathrm{x}}_{*}^2=(0, 1)$.

2.

Full active constraints (including box constraints):

• The active constraint gradients at $x_{*}^1=(1, 0)$ are calculated as:

  $${\mathrm{g}}_1=\nabla \mathrm{c}(x_{*}^1)={[-(1-{\mathrm{x}}_2), -(1-{\mathrm{x}}_1)]}^{\mathrm{T}}={[-1, 0]}^{\mathrm{T}}$$

  When the box-constraints are considered active:

  $$\mathrm{Box} \mathrm{constraint} {\mathrm{x}}_2=0\Rightarrow {\mathrm{g}}_2=\nabla (-{\mathrm{x}}_2{)=[0, -1]}^{\mathrm{T}},$$

  $$\mathrm{Box} \mathrm{constraint} {\mathrm{x}}_1=1\Rightarrow {\mathrm{g}}_3=\nabla ({\mathrm{x}}_1-1)={[1, 0]}^{\mathrm{T}}$$

  The matrix of active gradients is as follows:

  $$\mathrm{G}=\left[\begin{array}{cc}-1& 0\\ 0& -1\\ 1& 0\end{array}\right]$$

  We consider only linearly independent rows:

  $${\mathrm{G}}_{\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}}=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$$

  Which leads to a Tangent space defined as follows:

  $${\mathrm{G}}_{\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}}^{\mathrm{T}}\mathrm{d}=0\to \mathrm{d}={[0, 0]}^{\mathrm{T}}$$

• Projection matrix onto tangent space:

  $$\mathrm{P}=\mathrm{I}-{\mathrm{G}}_{\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}}^{\mathrm{T}}{({\mathrm{G}}_{\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}}{\mathrm{G}}_{\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}}^{\mathrm{T}})}^{-1}{\mathrm{G}}_{\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}}=0_{2\times 2}$$

We noticed the following optimality metrics as:

• Tangent space is trivial, so SOSC is trivially satisfied.
• Hessian eigenvalues: 4, 0 → The zero eigenvalue corresponds to flat boundary direction.
• There are no feasible directions along which the objective can decrease.
• The solution is a non-strict local minimum.

The Interpretation of the Eigenvalues is as follows:

• Zero eigenvalue $({\mathrm{\mu }}_2=0)$ corresponds to flat direction along boundary.
• Positive eigenvalue $({\mathrm{\mu }}_1=4)$ corresponds to curvature along orthogonal direction.
• Solutions are local minima but not strict due to boundary-induced flatness.
• $\mathbf{\lambda }=\mathbf{2}$$\to$ The equality constraint is fully active.
• No feasible directions exist that can further reduce the objective function.
• Therefore, non-strict local minimum are achieved.

Finally, some conclusions on achieving global optimality are as follows:

• Candidate solutions: ${\mathrm{x}}_{*}^1=(1, 0).$ and ${\mathrm{x}}_{*}^2=(0, 1)$
• Objective function value: $\mathrm{J}({\mathrm{x}}^{*})=1$.
• Hessian eigenvalues: ${\mathrm{\mu }}_1=4, {\mathrm{\mu }}_2=0$.
• Tangent space: full KKT with active box constraints.
• Boundary effect: local minima are non-strict along the flat direction of the feasible boundary due to zero eigenvalue.
• Globality: Since no feasible point has a smaller objective value, the solutions are non-strict global minima over the feasible region and decision bound.

As observed, the SQP generates a step that decreases the objective function without violating the constraint, meaning that the feasibility measure is zero, as shown in Table A1.

The iteration log shows the convergence behavior of the SQP algorithm implemented via fmincon. Starting from an interior point, the objective function increases until feasibility is achieved, after which it decreases monotonically along the active boundary [39,55,58,59,89,96].

Constraint violation and first-order optimality converge to zero, while full step lengths are consistently accepted. The algorithm terminates at the endpoint (1, 0) with objective value of 1.0, which is a non-strict local minimum and also a global minimum. The computed Lagrange multiplier λ = 2 matches the analytical KKT solution, confirming optimality.

From iteration 3 onward, the objective function decreases monotonically until it reaches its optimal value of 1.0, while feasibility remains zero [39,40,55,58,59,96].

More specifically, the iterates are feasible, which means that the constraints are inherently satisfied [59]. Thereby, there is no need for a penalty function [59]. Quadratic convergence is preserved near the solution. The step direction decreases both the objective function and preserves feasibility. The Fmincon produces feasible iterates once it captures one of the line segments [91]. It moves smoothly along that segment, avoiding zig-zag oscillations between the horizontal and vertical lines, as shown in Figure A1.

The SQP method reaches a solution located at endpoints of the union: ${\mathrm{x}}_{*}^1=(1, 0)$ and ${\mathrm{x}}_{*}^2=(0, 1)$. The objective function decreases monotonically toward its optimal value of 1.0. The algorithm takes full steps. Both the first-order optimality and the norm of the step gradually approach zero [59]. This indicates that a local minimum is reached. As observed in Figure A1, both norm of the step and first-order optimality decrease smoothly and monotonically. This confirms a stable convergence without oscillatory behavior [39,55,58,59]. Zig-zag oscillations are avoided and the algorithm progresses monotonically toward optimality [59]. The quadratic form ${\mathbf{d}}^{\mathbf{T}}\mathbf{H}\mathbf{d}=0$ does not violate the SOC [39,58,59]. It just indicates a flat direction along the boundary where the objective cannot decrease further. The computed Lagrange multiplier $\mathrm{\lambda }=2$ indicates that the constraint is active. It has a precise influence on the optimal solution [96].

The numerical optimization process calculated the optimality parameter that is crucial for optimizing the nonlinear model. This is confirmed by empirical results in [39,58,59]. KKT optimality conditions are used to solve the nonlinear model analytically. The optimality quantities are found: $\mathrm{\lambda }=2$, objective $\mathrm{J}({\mathrm{x}}^{*})=1$, $\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}=1$, and zero constraint violation (feasibility = 0) [1,2,3,4,5,6,7,8,39,40,55,58,59,96].

Both candidate points satisfy the KKT conditions with λ = 2, zero constraint violation, and zero stationarity residuals [1,2,3,4]. These optimality metrics confirm they are stationary points meeting the first-order necessary optimality conditions. The second-order condition (SOC) is adopted using a quadratic form providing ${\mathrm{d}}^{\mathrm{T}}\mathrm{H}\mathrm{d}=0$ at both solution points. This is confirmed by empirical results in [39,40,55,58,59,96].

This indicates that the curvature of the Lagrangian in feasible directions is flat as shown in Figure A1 and Figure A2. The Hessian neither curves upward or downward along certain boundary directions, so no descent direction exists to further decrease the objective function [39,58,59]. This zero curvature represents a neutral, degenerate direction, typical when the minimum lies on a boundary or multiple global minima exist [55,58,59,96].

Therefore, ${\mathrm{d}}^{\mathrm{T}}\mathrm{H}\mathrm{d}=0$ is consistent with non-negative curvature (positive semidefinite), confirming that both points are local—and indeed global—minima. Both the norm of the step and the first-order optimality are zero, as shown by empirical results in our previous works [39,55,58,59,96].

The full step taken by the line search indicates no issues requiring step reduction avoiding the Maratos effect [74]. Optimality and feasibility are simultaneously satisfied, so no penalty functions are needed for infeasibility corrections.

Table A1. SQP Log Analyses toward optimality.

Iter	F-Count	f(x)	Feasibility	Step Length	Norm of Step	First-Order Optimality
0	3	1.506170 × 10−1	5.273 × 10−1		5.920 × 10−1
1	7	5.575777 × 10−1	2.228 × 10−1	7.000 × 10−1	3.602 × 10−1	6.537 × 10−1
2	10	1.167324 × 100	5.508 × 10−2	1.000 × 100	3.342 × 10−1	7.083 × 10−1
3	13	1.537776 × 100	0.000 × 100	1.000 × 100	2.066 × 10−1	1.467 × 100
4	16	1.492423 × 100	0.000 × 100	1.000 × 100	3.160 × 10−2	1.403 × 100
5	19	1.303075 × 100	0.000 × 100	1.000 × 100	1.512 × 10−1	1.101 × 100
6	22	1.000000 × 100	0.000 × 100	1.000 × 100	5.505 × 10−1	1.216 × 100
7	25	1.000000 × 100	0.000 × 100	1.000 × 100	0.000 × 100	0.000 × 100
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the default value of the optimality tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.
<stopping criteria details>
Start: (0.296, 0.251) --> Opt: (1.0000, 0.0000), f = 1.0000
Optimization Results:
Optimal x: [1; 0]
Optimal Objective Value: 1
Exit Flag: 1
Solver Information:
iterations: 7
funcCount: 25
algorithm: ‘sqp’
message: ‘Local minimum found that satisfies the constraints.…’
constrviolation: 0
stepsize: 0
lssteplength: 1
firstorderopt: 0
Constraint Violation:
0
The optimum is a local minimum point.
---multipler Lagrange (λ) from fmincon---
λ (fmincon) =
---computation λ (computed by KKT optimality conditions)---
λ (KKT) = 2.0000
Unique theoretical global minima:
x1   x2
0   1
1   0

Table A1. SQP Log Analyses toward optimality.

Iter	F-Count	f(x)	Feasibility	Step Length	Norm of Step	First-Order Optimality
0	3	1.506170 × 10−1	5.273 × 10−1		5.920 × 10−1
1	7	5.575777 × 10−1	2.228 × 10−1	7.000 × 10−1	3.602 × 10−1	6.537 × 10−1
2	10	1.167324 × 100	5.508 × 10−2	1.000 × 100	3.342 × 10−1	7.083 × 10−1
3	13	1.537776 × 100	0.000 × 100	1.000 × 100	2.066 × 10−1	1.467 × 100
4	16	1.492423 × 100	0.000 × 100	1.000 × 100	3.160 × 10−2	1.403 × 100
5	19	1.303075 × 100	0.000 × 100	1.000 × 100	1.512 × 10−1	1.101 × 100
6	22	1.000000 × 100	0.000 × 100	1.000 × 100	5.505 × 10−1	1.216 × 100
7	25	1.000000 × 100	0.000 × 100	1.000 × 100	0.000 × 100	0.000 × 100
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the default value of the optimality tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.
<stopping criteria details>
Start: (0.296, 0.251) --> Opt: (1.0000, 0.0000), f = 1.0000
Optimization Results:
Optimal x: [1; 0]
Optimal Objective Value: 1
Exit Flag: 1
Solver Information:
iterations: 7
funcCount: 25
algorithm: ‘sqp’
message: ‘Local minimum found that satisfies the constraints.…’
constrviolation: 0
stepsize: 0
lssteplength: 1
firstorderopt: 0
Constraint Violation:
0
The optimum is a local minimum point.
---multipler Lagrange (λ) from fmincon---
λ (fmincon) =
---computation λ (computed by KKT optimality conditions)---
λ (KKT) = 2.0000
Unique theoretical global minima:
x1   x2
0   1
1   0

Figure A1. Optimization with Equality Constraints $\mathrm{c}(\mathrm{x})=0$ and decision bounds.

Figure A1. Optimization with Equality Constraints $\mathrm{c}(\mathrm{x})=0$ and decision bounds.

Table A2. KKT and Checking the Second-Order Optimality Conditions.

Point #1: (0.00, 1.00)

Lambda (λ) = 2.0000

Upper bounds multipliers (μ) = [0.00 0.00]

Lower bounds multipliers (ν) = [0.00 0.00]

Stationarity residual = 0.0000 × 100

Quadratic form d^T H d = 0.0000 (Second-Order check)

Point #2: (1.00, 0.00)

Lambda (λ) = 2.0000

Upper bounds multipliers (μ) = [0.00 0.00]

Lower bounds multipliers (ν) = [0.00 0.00]

Stationarity residual = 0.0000 × 100

Quadratic form d^T H d = 0.0000 (Second-Order check)

Table A2. KKT and Checking the Second-Order Optimality Conditions.

Point #1: (0.00, 1.00)

Lambda (λ) = 2.0000

Upper bounds multipliers (μ) = [0.00 0.00]

Lower bounds multipliers (ν) = [0.00 0.00]

Stationarity residual = 0.0000 × 100

Quadratic form d^T H d = 0.0000 (Second-Order check)

Point #2: (1.00, 0.00)

Lambda (λ) = 2.0000

Upper bounds multipliers (μ) = [0.00 0.00]

Lower bounds multipliers (ν) = [0.00 0.00]

Stationarity residual = 0.0000 × 100

Quadratic form d^T H d = 0.0000 (Second-Order check)

From the optimization logs, supported by the corresponding plots, the optimality metrics were observed as follows, as also presented in [39,40,55,56,58,59,96]:

• Unique theoretical global minima: ${\mathrm{x}}_{*}^1=(1, 0)$ and ${\mathrm{x}}_{*}^2=(0, 1)$
• The KKT conditions are fully satisfied.
• Second-order conditions (SOC): ${\mathrm{d}}^{\mathrm{T}}\mathrm{H}\mathrm{d}\ge 0$$\to$ verified.
• Flat directions occur along boundaries due to zero eigenvalue.
• Visual confirmation: the two minima are unique in the feasible domain.

We noticed that the quadratic form is exactly zero. This happens because:

• One eigenvalue of the Hessian is zero, which means a flat direction.
• The tangent space at the boundary is trivial → there are no nonzero feasible directions that decrease the objective function.

The SOC is interpreted as follows:

• The zero value indicates a flat direction along the boundary.
• There is no feasible direction that can further decrease the objective.
• The SOC is technically satisfied, even though it is not strictly positive definite.

Figure A2. Feasible Set with Unique Theoretical Global Minima and Active Constraints.

Figure A2. Feasible Set with Unique Theoretical Global Minima and Active Constraints.

The connection with the KKT optimality conditions is as follows:

• Stationarity residual $= 0$ $\to$ KKT stationarity is satisfied.
• Multipliers $(\mathsf{\mu }, \mathsf{\nu })=0$$\to$ dual feasibility is satisfied.
• Primal feasibility = constraint satisfied.
• Quadratic form $\ge \mathbf{0}$$\to$ second-order optimality condition is satisfied.

Based on the KKT optimality condition calculations, the optimality metrics are as:

• We consider two scenarios with and without box constraints in Lagrangian Function.
• Both scenarios yield the solutions that is, ${\mathrm{x}}_{*}^1=(1, 0).$ and ${\mathrm{x}}_{*}^2=(0, 1)$
• The box constraints defined on the decision bounds enforce boundary solutions.
• Hessian positive semi-definite along feasible directions $\to$ flat direction present.
• Tangent space confirms $\to$ no feasible descent direction.
• The langrage multiplier $(\mathrm{\lambda }=2)$$\to$ fully activates equality constraint along boundary.