Search Results (71)

The integration of high-penetration renewable energy sources (RESs) into transmission networks introduces profound uncertainty that challenges traditional infrastructure planning approaches. Existing transmission expansion planning (TEP) models either rely on static scenario sets or over-conservative worst-case assumptions, failing to capture the operational stress triggered by rare but structurally impactful renewable behaviors. This paper proposes a novel bi-level optimization framework for transmission planning under adversarial uncertainty, coupling a distributionally robust upper-level investment model with a lower-level operational response embedded with physics and market constraints. The uncertainty space was not exogenously fixed, but instead dynamically generated through a physics-informed spatiotemporal generative adversarial network (PI-ST-GAN), which synthesizes high-risk renewable and load scenarios designed to maximally challenge the system’s resilience. The generator was co-trained using a composite stress index—combining expected energy not served, loss-of-load probability, and marginal congestion cost—ensuring that each scenario reflects both physical plausibility and operational extremity. The resulting bi-level model was reformulated using strong duality, and it was decomposed into a tractable mixed-integer structure with embedded adversarial learning loops. The proposed framework was validated on a modified IEEE 118-bus system with high wind and solar penetration. Results demonstrate that the GAN-enhanced planner consistently outperforms deterministic and stochastic baselines, reducing renewable curtailment by up to 48.7% and load shedding by 62.4% under worst-case realization. Moreover, the stress investment frontier exhibits clear convexity, enabling planners to identify cost-efficient resilience strategies. Spatial congestion maps and scenario risk-density plots further illustrate the ability of adversarial learning to reveal latent structural bottlenecks not captured by conventional methods. This work offers a new methodological paradigm, in which optimization and generative AI co-evolve to produce robust, data-aware, and stress-responsive transmission infrastructure designs. Full article

Over-the-air federated learning (Air-FL) has emerged as a promising paradigm that integrates communication and learning, which offers significant potential to enhance model training efficiency and optimize communication resource utilization. This paper addresses the challenge of interference management in multi-cell Air-FL systems, focusing on parallel multi-task scenarios where each cell independently executes distinct training tasks. We begin by analyzing the impact of aggregation errors on local model performance within each cell, aiming to minimize the cumulative optimality gap across all cells. To this end, we formulate an optimization framework that jointly optimizes device transmit power and denoising factors. Leveraging the Pareto boundary theory, we design a centralized optimization scheme that characterizes the trade-offs in system performance. Building upon this, we propose a distributed power control optimization scheme based on interference temperature (IT). This approach decomposes the globally coupled problem into locally solvable subproblems, thereby enabling each cell to adjust its transmit power independently using only local channel state information (CSI). To tackle the non-convexity inherent in these subproblems, we first transform them into convex problems and then develop an analytical solution framework grounded in Lagrangian duality theory. Coupled with a dynamic IT update mechanism, our method iteratively approximates the Pareto optimal boundary. The simulation results demonstrate that the proposed scheme outperforms baseline methods in terms of training convergence speed, cross-cell performance balance, and test accuracy. Moreover, it achieves stable convergence within a limited number of iterations, which validates its practicality and effectiveness in multi-task edge intelligence systems. Full article

This paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameterized backward stochastic linear quadratic control problem. Then, under the surjectivity of the linear constraint, the equivalence between the original problem and the dual problem is proven by Lagrange duality theory. Subsequently, with the help of the maximum principle, an explicit solution of the optimal control for the unconstrained problem is obtained. This solution is feedback-based and determined by an adjoint stochastic differential equation, a Riccati-type ordinary differential equation, a backward stochastic differential equation, and an equality, thereby yielding the optimal control for the original problem. Finally, an optimal control for an investment portfolio problem with an expected-type equality constraint on the initial state is explicitly provided. Full article

In urban environments, the highly complex communication environment often leads to blockages in the link between ground users (GUs) and unmanned aerial vehicles (UAVs), resulting in poor communication quality. Although traditional reconfigurable intelligent surfaces (RISs) can improve wireless channel quality, they can only provide reflection services and have limited coverage. For this reason, we study a novel simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR–RIS)–assisted UAV–mobile edge computing (UAV–MEC) network, which can serve multiple users residing in the transmission area and reflection area, and switch between reflection and transmission modes according to the relative positions of the UAV, GUs, and STAR–RIS, providing users with more flexible and efficient services. The system comprehensively considers user transmit power, time slot allocation, UAV flight trajectory, STAR–RIS mode selection, and phase angle matrix, achieving long–term energy consumpution minimization while ensuring stable task backlog queue. Since the proposed problem is a long–term stochastic optimization problem, we use the Lyapunov method to transform it into three deterministic online optimization subproblems and iteratively solve them alternately. Specifically, we firstly use the Lambert function to solve for the closed-form solution of the transmit power; then, use Lagrange duality and the Karush–Kuhn–Tucker conditions to solve time slot allocation; finally, successive convex approximation is used to obtain trajectory planning for UAVs with lower complexity, and triangular inequalities are used to solve the STAR–RIS phase shift. The simulation results show that the proposed scheme has better performance than other benchmark schemes in maintaining queue stability and reducing energy consumption. Full article

In this paper, a class of controlled variational control models is studied by considering the notion of $(q,w)-\pi$-invexity. Our aim is to investigate a solution set in the considered interval-valued controlled models. To achieve this, we establish some characterization results of solutions in the controlled interval-valued variational models. More precisely, necessary and sufficient conditions of optimality are highlighted as part of a feasible solution. To prove that the optimality conditions are sufficient, we impose generalized invariant convexity hypotheses for the involved multiple integral functionals. Finally, a duality result is provided in order to better describe the problem under study. The methodology used in this paper is a combination of techniques from the Lagrange–Hamilton theory, calculus of variations, and control theory. This study could be immediately improved by including an analysis of this class of optimization problems by using curvilinear integrals instead of multiple integrals. The independence of path imposed to these functionals and their physical significance would increase the applicability and importance of the paper. Full article

Let X be a centered random vector in a finite-dimensional real inner product space $\mathcal{E}$. For a subset C of the ambient vector space V of $\mathcal{E}$ and $x,y\in V$, write $x{⪯}_{C}y$ if $y-x\in C$. If C is a closed convex cone in $\mathcal{E}$, then ${⪯}_C$ is a preorder on V, whereas if C is a proper cone in $\mathcal{E}$, then ${⪯}_C$ is actually a partial order on V. In this paper, we give sharp Cantelli-type inequalities for generalized tail probabilities such as $\mathrm{Pr}\left\{X{⪰}_{C}b\right\}$ for $b\in V$. These inequalities are obtained by “scalarizing” $X{⪰}_{C}b$ via cone duality and then by minimizing the classical univariate Cantelli’s bound over the scalarized inequalities. Three diverse applications to random matrices, tails of linear images of random vectors, and network homophily are also given. Full article

This article investigates robust optimality conditions and duality results for a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty (UNMPVC). Mathematical programming problems with vanishing constraints constitute a distinctive class of constrained optimization problems because of the presence of complementarity constraints. Moreover, uncertainties are inherent in various real-life problems. The aim of this article is to identify an optimal solution to an uncertain optimization problem with vanishing constraints that remains feasible in every possible future scenario. Stationary conditions are necessary conditions for optimality in mathematical programming problems with vanishing constraints. These conditions can be derived under various constraint qualifications. Employing the properties of convexificators, we introduce generalized standard Abadie constraint qualification (GS-ACQ) for the considered problem, UNMPVC. We introduce a generalized robust version of nonsmooth stationary conditions, namely a weakly stationary point, a Mordukhovich stationary point, and a strong stationary point (RS-stationary) for UNMPVC. By employing GS-ACQ, we establish the necessary conditions for a local weak Pareto solution of UNMPVC. Moreover, under generalized convexity assumptions, we derive sufficient optimality criteria for UNMPVC. Furthermore, we formulate the Wolfe-type and Mond–Weir-type robust dual models corresponding to the primal problem, UNMPVC. Full article

Two typical fixed-length random number generation problems in information theory are considered for general sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given approximation measure is one of our main concerns and has been characterized using two different information quantities: the information spectrum and the smooth Rényi entropy. Recently, optimum achievable rates with respect to f-divergences have been characterized using the information spectrum quantity. The f-divergence is a general non-negative measure between two probability distributions on the basis of a convex function f. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to f-divergences. However, optimum achievable rates with respect to f-divergences using the smooth Rényi entropy have not been clarified yet in both problems. In this paper, we try to analyze the optimum achievable rates using the smooth Rényi entropy and to extend the class of f-divergence. To do so, we first derive general formulas of the first-order optimum achievable rates with respect to f-divergences in both problems under the same conditions as imposed by previous studies. Next, we relax the conditions on f-divergence and generalize the obtained general formulas. Then, we particularize our general formulas to several specified functions f. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. Furthermore, a kind of duality between the resolvability and the intrinsic randomness is revealed in terms of the smooth Rényi entropy. Second-order optimum achievable rates and optimistic achievable rates are also investigated. Full article

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating the primal problem NIMSIPVC and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints. Full article

Multiobjective programming refers to a mathematical problem that requires the simultaneous optimization of multiple independent yet interrelated objective functions when solving a problem. It is widely used in various fields, such as engineering design, financial investment, environmental planning, and transportation planning. Research on the theory and application of convex functions and their generalized convexity in multiobjective programming helps us understand the essence of optimization problems, and promotes the development of optimization algorithms and theories. In this paper, we firstly introduces new classes of generalized $(F,\alpha ,\rho ,d)-I$ functions for semi-preinvariant convex multiobjective programming. Secondly, based on these generalized functions, we derive several sufficient optimality conditions for a feasible solution to be an efficient or weakly efficient solution. Finally, we prove weak duality theorems for mixed-type duality. Full article

This survey offers a succinct overview of the General Framework of Portfolio Theory (GFPT), consolidating Markowitz portfolio theory, the growth optimal portfolio theory, and the theory of risk measures. Central to this framework is the use of convex analysis and duality, reflecting the concavity of reward functions and the convexity of risk measures due to diversification effects. Furthermore, practical considerations, such as managing multiple risks in bank balance sheets, have expanded the theory to encompass vector risk analysis. The goal of this survey is to provide readers with a concise tour of the GFPT’s key concepts and practical applications without delving into excessive technicalities. Instead, it directs interested readers to the comprehensive monograph of Maier-Paape, Júdice, Platen, and Zhu (2023) for detailed proofs and further exploration. Full article

In this paper, by making use of some advanced tools from variational analysis and generalized differentiation, we establish necessary optimality conditions for a class of robust fractional minimax programming problems. Sufficient optimality conditions for the considered problem are also obtained by means of generalized convex functions. Additionally, we formulate a dual problem to the primal one and examine duality relations between them. In our results, by using the obtained results, we obtain necessary and sufficient optimality conditions for a class of robust fractional multi-objective optimization problems. Full article

This article deals with mathematical programs with vanishing constraints (MPVCs) involving lower semi-continuous functions. We introduce generalized Abadie constraint qualification (ACQ) and MPVC-ACQ in terms of directional convexificators and derive necessary KKT-type optimality conditions. We also derive sufficient conditions for global optimality for the MPVC under convexity utilizing directional convexificators. Further, we introduce a Wolfe-type dual model in terms of directional convexificators and derive duality results. The results are well illustrated by examples. Full article

This paper considers a risk-averse Markov decision process (MDP) with non-risk constraints as a dynamic optimization framework to ensure robustness against unfavorable outcomes in high-stakes sequential decision-making situations such as disaster response. In this regard, strong duality is proved while making no assumptions on the problem’s convexity. This is necessary for some real-world issues, e.g., in the case of deprivation costs in the context of disaster relief, where convexity cannot be ensured. Our theoretical results imply that the problem can be exactly solved in a dual domain where it becomes convex. Based on our duality results, an augmented Lagrangian-based constraint handling mechanism is also developed for risk-averse reinforcement learning algorithms. The mechanism is proved to be theoretically convergent. Finally, we have also empirically established the convergence of the mechanism using a multi-stage disaster response relief allocation problem while using a fixed negative reward scheme as a benchmark. Full article

In the future, a high proportion of distributed generations (DG) will be integrated into the distribution network. The existing active distribution network (ADN) planning methods have not fully considered multiple uncertainties, differentiated regulation modes or the cost of multiple types of interconnection switches. Meanwhile, it is difficult to solve large-scale problems at small granularity. Therefore, an expansion planning method of ADN considering the selection of multiple types of interconnection switches is proposed. Firstly, a probability distribution ambiguity set of DG output and electrical-load consumption based on the Wasserstein distance is established for dealing with the issue of source-load uncertainty. Secondly, a distributionally robust optimization model for collaborative planning of distribution network lines and multiple types of switches based on the previously mentioned ambiguity set is established. Then, the original model is transformed into a mixed integer second-order cone programming (SOCP) model by using the convex relaxation method, the Lagrangian duality method and the McCormick relaxation method. Finally, the effectiveness of the proposed method is systematically verified using the example of Portugal 54. The results indicate that the proposed method raises the annual net profit by nearly 5% compared with the traditional planning scheme and improves the reliability and low-carbon nature of the planning scheme. Full article