Search Results (12)

Accurate estimation of branch-level traffic flows at urban Y-intersections from limited mainline measurements remains a critical challenge in intelligent transportation systems. Y-intersections, with their symmetric geometric configuration where multiple branches converge, pose unique challenges from flow coupling, signal-induced periodicity, and merging delays. This study develops a hybrid mathematical modeling framework that integrates piecewise linear segments with periodic components for each branch flow. The model enforces physical constraints including flow conservation, non-negativity, and segment continuity, while incorporating operational features such as signal timing and merging delays. Parameter estimation employs a two-stage optimization approach combining least-squares fitting with constrained nonlinear programming, utilizing sparse mainline detector data and minimal historical priors. Experimental validation across five progressive problem formulations demonstrates robust performance, achieving RMSE values of 3.3432 and 5.4467 for complex multi-branch scenarios while accurately capturing 10-min green/8-min red signal cycles and 2-min merging delays. The method successfully reconstructs branch flow profiles at required time points (07:30 and 08:30), reducing observation requirements by 60–80% while maintaining estimation accuracy. The proposed framework provides a practical and interpretable solution for branch flow estimation under sparse sensing conditions, bridging physics-based modeling with data-driven techniques and offering transportation agencies a deployable tool for intersection monitoring without extensive instrumentation. Full article

In this paper, a nonlinear system is interpreted as an operator $\mathcal{F}$ transforming random vectors. It is assumed that the operator is unknown and the random vectors are available. It is required to find a model of the system represented by a best constructive operator $\mathcal{F}$ approximation. While the theory of operator approximation with any given accuracy has been well elaborated, the theory of best constrained constructive operator approximation is not so well developed. Despite increasing demands from various applications, this subject is minimally tractable because of intrinsic difficulties with associated approximation techniques. This paper concerns the best constrained approximation of a nonlinear operator in probability spaces. The main conceptual novelty of the proposed approach is that, unlike the known techniques, it targets a constructive optimal determination of all $3p+2$ ingredients of the approximating operator where p is a nonnegative integer. The solution to the associated problem is represented by a combination of new best approximation techniques with a special iterative procedure. The proposed approximating model of the system has several degrees of freedom to minimize the associated error. In particular, one of the specific features of the developed approximating technique is special random vectors called injections. It is shown that the desired injection is determined from the solution of a special Fredholm integral equation of the second kind. Its solution is called the optimal injection. The determination of optimal injections in this way allows us to further minimize the associated error. Full article

This paper integrates $L_1$-norm structural risk minimization with $L_1$-norm approximation error to develop a new optimization framework for solving the parameters of sparse kernel regression models, addressing the challenges posed by complex model structures, over-fitting, and limited modeling accuracy in traditional nonlinear system modeling. The first $L_1$-norm regulates the complexity of the model structure to maintain its sparsity, while another $L_1$-norm is essential for ensuring modeling accuracy. In the optimization of support vector regression (SVR), the $L_2$-norm structural risk is converted to an $L_1$-norm framework through the condition of non-negative Lagrange multipliers. Furthermore, $L_1$-norm optimization for modeling accuracy is attained by minimizing the maximum approximation error. The integrated $L_1$-norm of structural risk and approximation errors creates a new, simplified optimization problem that is solved using linear programming (LP) instead of the more complex quadratic programming (QP). The proposed sparse kernel regression model has the following notable features: (1) it is solved through relatively simple LP; (2) it effectively balances the trade-off between model complexity and modeling accuracy; and (3) the solution is globally optimal rather than just locally optimal. In our three experiments, the sparsity metrics of $SVs\%$ were 2.67%, 1.40%, and 0.8%, with test RMSE values of 0.0667, 0.0701, 0.0614 (sinusoidal signal), and 0.0431 (step signal), respectively. This demonstrates the balance between sparsity and modeling accuracy. Full article

We consider an irreducible positive-recurrent discrete-time Markov process on the state space $X={\mathbb{Z}}_{+}^M\times J$, where ${\mathbb{Z}}_{+}$ is the set of non-negative integers and $J=\{1,2,\dots ,n\}$. The number of states in $J$ may be either finite or infinite. We assume that the process is a homogeneous quasi-birth-and-death process (QBD). It means that the one-step transition probability between non-boundary states $(k,i)$ and $(n,j)$ may depend on $i,j$, and $n-k$ but not on the specific values of $k$ and $n$. It is shown that the stationary probability vector of the process is expressed through square matrices of order $n$, which are the minimal non-negative solutions to nonlinear matrix equations. Full article

In this paper, we investigate the iterative methods for the solution of different types of nonlinear matrix equations. More specifically, we consider iterative methods for the minimal nonnegative solution of a set of Riccati equations, a nonnegative solution of a quadratic matrix equation, and the maximal positive definite solution of the equation $X+A^{\ast }{X}^{-1}A=Q$. We study the recent iterative methods for computing the solution to the above specific type of equations and propose more effective modifications of these iterative methods. In addition, we make comments and comparisons of the existing methods and show the effectiveness of our methods by illustration examples. Full article

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern, too. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative (or geometric) nature of averaging. The original concept of PMA was motivated by the practice of matrix population models as a tool to assess the population viability from long-term monitoring data. The task has reduced to searching for an approximate solution to an overdetermined system of polynomial equations for unknown elements of the average matrix (G), and hence to a nonlinear constrained minimization problem for the matrix norm. Former practical solutions faced certain technical problems, which required sophisticated algorithms but returned acceptable estimates. Now, we formulate (for the first time in ecological modeling and nonnegative matrix theory) the PMA problem as an eigenvalue approximation one and reduce it to a standard problem of linear programing (LP). The basic equation of averaging also determines the exact value of λ₁(G), the dominant eigenvalue of matrix G, and the corresponding eigenvector. These are bound by the well-known linear equations, which enable an LP formulation of the former nonlinear problem. The LP approach is realized for 13 fixed-pattern matrices gained in a case study of Androsace albana, an alpine short-lived perennial, monitored on permanent plots over 14 years. A standard software routine reveals the unique exact solution, rather than an approximate one, to the PMA problem, which turns the LP approach into ‘’the best of versatile optimization tools”. The exact solution turns out to be peculiar in reaching zero bounds for certain nonnegative entries of G, which deserves modified problem formulation separating the lower bounds from zero. Full article

Malaria is a serious illness caused by a parasite, called Plasmodium, transmitted to humans through the bites of female Anopheles mosquitoes. The parasite infects and destroys the red blood cells in the human body leading to symptoms, such as fever, headache, and flu-like illness. Awareness campaigns that educate people about malaria prevention and control reduce transmission of the disease. In this research, a mathematical model is proposed to study the impact of awareness-based control measures on the transmission dynamics of malaria. Some basic properties of the proposed model, such as non-negativity and boundedness of the solutions, the existence of the equilibrium points, and their stability properties, have been studied using qualitative theory. Disease-free equilibrium is globally asymptotic when the basic reproduction number, ${\mathcal{R}}_0$, is less than the number of current cases. Finally, optimal control theory is applied to minimize the cost of disease control and solve the optimal control problem by applying Pontryagin’s minimum principle. Numerical simulations have been provided for the confirmation of the analytical results. Endemic equilibrium exists for ${\mathcal{R}}_0>1$, and a forward transcritical bifurcation occurs at ${\mathcal{R}}_0=1$. The optimal profiles of the treatment process, organizing awareness campaigns, and insecticide uses are obtained for the cost-effectiveness of malaria management. This research concludes that awareness campaigns through social media with an optimal control approach are best for cost-effective malaria management. Full article

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern as well. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative nature of averaging. The concept of PMA was motivated in a number of application fields, of which we consider the matrix population models and illustrate solving the PMA problem with several sets of model matrices calibrated in particular botanic case studies. The patterns of those matrices are typically nontrivial (they contain both zero and nonzero elements), the PMA problem thus having no exact solution for a fundamental reason (an overdetermined system of algebraic equations). Therefore, searching for the approximate solution reduces to a constrained minimization problem for the approximation error, the loss function in optimization terms. We consider two alternative types of the loss function and present a general algorithm of searching the optimal solution: basin-hopping global search, then local descents by the method of conjugate gradients or that of penalty functions. Theoretical disadvantages and practical limitations of both loss functions are discussed and illustrated with a number of practical examples. Full article

This paper puts forward a new version of the Isotropic Material Design method for the optimum design of structures made of an elasto-plastic material within the Hencky-Nadai-Ilyushin theory. This method provides the optimal layouts of the moduli of isotropy to make the overall compliance minimal. Thus, the bulk and shear moduli are the only design variables, both assumed as non-negative fields. The trace of the Hooke tensor represents the unit cost of the design. The yield condition is assumed to be independent of the design variables, to make the design process as simple as possible. By eliminating the design variables, the optimum design problem is reduced to the pair of the two mutually dual Linear Constrained Problems (LCP). The solution to the LCP stress-based problem directly determines the layout of the optimal moduli. A numerical method has been developed to construct approximate solutions, which paves the way for constructing the final layouts of the elastic moduli. Selected illustrative solutions are reported, corresponding to various data concerning the yield limit and the cost of the design. The yield condition introduced in this paper results in bounding the values of the optimal moduli in the places of possible stress concentration, such as reentrant corners. Full article

Recently, operations research, especially linear integer-programming, is used in various grids to find optimal paths and, based on that, digital distance. The 4 and higher-dimensional body-centered-cubic grids is the nD ($n\ge 4$) equivalent of the 3D body-centered cubic grid, a well-known grid from solid state physics. These grids consist of integer points such that the parity of all coordinates are the same: either all coordinates are odd or even. A popular type digital distance, the chamfer distance, is used which is based on chamfer paths. There are two types of neighbors (closest same parity and closest different parity point-pairs), and the two weights for the steps between the neighbors are fixed. Finding the minimal path between two points is equivalent to an integer-programming problem. First, we solve its linear programming relaxation. The optimal path is found if this solution is integer-valued. Otherwise, the Gomory-cut is applied to obtain the integer-programming optimum. Using the special properties of the optimization problem, an optimal solution is determined for all cases of positive weights. The geometry of the paths are described by the Hilbert basis of the non-negative part of the kernel space of matrix of steps. Full article

Unique k-SAT is the promised version of k-SAT where the given formula has 0 or 1 solution and is proved to be as difficult as the general k-SAT. For any $k\ge 3$ , $s\ge f(k,d)$ and $(s+d)/2>k-1$ , a parsimonious reduction from k-CNF to d-regular (k,s)-CNF is given. Here regular (k,s)-CNF is a subclass of CNF, where each clause of the formula has exactly k distinct variables, and each variable occurs in exactly s clauses. A d-regular (k,s)-CNF formula is a regular (k,s)-CNF formula, in which the absolute value of the difference between positive and negative occurrences of every variable is at most a nonnegative integer d. We prove that for all $k\ge 3$ , $f(k,d)\le u(k,d)+1$ and $f(k,d+1)\le u(k,d)$ . The critical function $f(k,d)$ is the maximal value of s, such that every d-regular (k,s)-CNF formula is satisfiable. In this study, $u(k,d)$ denotes the minimal value of s such that there exists a uniquely satisfiable d-regular (k,s)-CNF formula. We further show that for $s\ge f(k,d)+1$ and $(s+d)/2>k-1$ , there exists a uniquely satisfiable d-regular $(k,s+1)$ -CNF formula. Moreover, for $k\ge 7$ , we have that $u(k,d)\le f(k,d)+1$ . Full article

The present paper focuses on the development of an algorithm for safely and optimally managing the routing of aircraft on an airport surface in future airport operations. This tool is intended to support air traffic controllers’ decision-making in selecting the paths of all aircraft and the engine startup approval time for departing ones. Optimal routes are sought for minimizing the time both arriving and departing aircraft spend on an airport surface with engines on, with benefits in terms of safety, efficiency and costs. The proposed algorithm first computes a standalone, shortest path solution from runway to apron or vice versa, depending on the aircraft being inbound or outbound, respectively. For taking into account the constraints due to other traffic on an airport surface, this solution is amended by a conflict detection and resolution task that attempts to reduce and possibly nullify the number of conflicts generated in the first phase. An example application on a simple Italian airport exemplifies how the algorithm can be applied to true-world applications. Emphasis is given on how to model an airport surface as a weighted and directed graph with non-negative weights, as required for the input to the algorithm. Full article