Vector Optimization Problems Research Articles

Second-order optimality conditions for locally Lipschitz vector optimization problems

In this paper, we derive primal and dual second-order necessary and sufficient optimality conditions for a vector optimization problem with equality and inequality constraints where the functions involved are locally Lipschitz. We introduce a weaker notion of second-order Abadie constraint qualification to derive second-order necessary conditions for weak local Pareto minima and strict local Pareto minima of order two in terms of Páles and Zeidan's second-order upper directional derivatives. Dual necessary conditions are derived for both types of minimal solutions in finite-dimensional spaces assuming the functions to be first-order continuously Fréchet differentiable. In the same setting, we derive dual and primal second-order sufficient optimality conditions for strict local Pareto minima of order two in terms of second-order lower directional derivatives.

Focus on sufficient optimality conditions in D.C. vector optimization

Sufficient optimality conditions for a D.C. vector optimization problem are given by Guo and Li (Journal of Optimization Theory and Applications 162, 821–844, 2014). Guo and Li's main result turned out to be false and was corrected by Shafie and Bozorgnia (Journal of Optimization Theory and Applications 182, 837–849, 2019). In this work, we have pointed out that Shafie and Bozorgnia's result is wrong. We correct the flaws and present an alternative proof for the main result.

Geometric Duality Results and Approximation Algorithms for Convex Vector Optimization Problems

We study geometric duality for convex vector optimization problems. For a primal problem with a -dimensional objective space, we formulate a dual problem with a -dimensional objective space. Consequently, different from an existing approach, the geometric dual problem does not depend on a fixed direction parameter, and the resulting dual image is a convex cone. We prove a one-to-one correspondence between certain faces of the primal and dual images. In addition, we show that a polyhedral approximation for one image gives rise to a polyhedral approximation for the other. Based on this, we propose a geometric dual algorithm which solves the primal and dual problems simultaneously and is free of direction-biasedness. We also modify an existing direction-free primal algorithm in such a way that it solves the dual problem as well. We test the performance of the algorithms for randomly generated problem instances by using the so-called primal error and hypervolume indicator as performance measures.

On Second-Order Semi-Differentiability of Index γ of Perturbation Maps in Parametric Vector Optimization Problems

In this paper, we prove that the Henig global proper efficient solution map and the Henig global proper efficient perturbation map of a parametric vector optimization problem are second-order semi-differentiable with index [Formula: see text] under some suitable qualification conditions. Several examples are given to illustrate the obtained results.

Hadamard well-posedness for a set optimization problem with an infinite number of constraints

This article aims to study the Hadamard well-posedness for a set optimization problem with an infinite number of constraints, where Kuroiwa's lower set less relation is used to compare entire images of an objective set-valued map. Based on the weak Slater constraint qualification, sufficient conditions of Hadamard well-posedness properties for efficient and weakly efficient solutions to the reference problem under functional perturbations of both constraint sets and objective maps are established. The obtained results can also be seen as an extension to the case of infinite/semi-infinite vector optimization problems.

Pareto efficiency without topology

Given a vector optimization problem over spaces endowed with a topological linear structure, existence results for optima (efficient points) are known. Relying only on the linear structure, the set of properly efficient points from a convex set is proved to be nonempty and the sets of Proper efficient points and Pareto efficient points coincide, provided that the set of internal points picked from the corresponding cone is nonempty. This result is appealing since the scalarization of the vector optimization problem is valid without topological requirements. A As an important consequence, we provide the Second Welfare Theorem in vector lattices and especially in Lebesgue spaces holds without topology.

Robust duality for robust efficient solutions in uncertain vector optimization problems

Robust duality for robust efficient solutions in uncertain vector optimization problems

On interval-valued vector variational-like inequalities and vector optimization problems with generalized approximate invexity via convexificators

On interval-valued vector variational-like inequalities and vector optimization problems with generalized approximate invexity via convexificators

On nondifferentiable semi-infinite multiobjective programming with interval-valued functions

In the paper, we investigate a class of nondifferentiable semi-infinite multiobjective programming problems such that all functions constituting them are interval-valued. We derive both Fritz John necessary optimality conditions and, under a constraint qualification, Karush-Kuhn-Tucker necessary optimality conditions for (weak) $ LU $-Pareto solutions in the considered nondifferentiable semi-infinite vector interval-valued optimization problem. Under appropriate invexity hypotheses, we also prove sufficient optimality conditions for this semi-infinite interval-valued vector optimization problem. Further, we also use the $ l_{1} $ exact function method for solving the aforesaid nondifferentiable interval-valued multicriteria optimization problem. Then, we analyze the property of exactness of the penalization for the absolute value exact penalty function method under assumption that the functions involved in the considered semi-infinite multiobjective programming problem are locally Lipschitz interval-valued invex functions. The conditions guaranteeing the equivalence of the sets of (weak) $ LU $-Pareto solutions in the original semi-infinite interval-valued multiobjective programming problem and its associated vector penalized optimization problem with the multiple interval-valued $ l_{1} $ exact penalty function are given.

New second-order necessary optimality conditions for constrained vector optimization problems

This paper is devoted to second-order optimality conditions of constrained vector optimization problems with continuously Fréchet differentiable data. Different from conventional approaches, we employ a new one, image space analysis, to establish suitable regularity conditions and then present new second-order necessary optimality conditions that depend on one single KKT multiplier for all critical directions. Moreover, we propose one more regularity condition for the multiobjective case and compare it with the previous one. Besides, we compare the main regularity conditions with the existing ones in the literature for the scalar case.

Levitin-Polyak well-posedness for vector optimization problems in linear spaces

In this paper, we deal with vector optimization problems where the images of the objective maps are given in linear spaces not endowed with any topological structure. Firstly, we propose new concepts of semicontinuity of vector-valued maps acting from metric spaces into linear ones and discuss their properties and characterizations. Next, using Zorn's Lemma and these properties, we formulate existence conditions for such problems. Then, we introduce two notions of Levitin-Polyak well-posedness for the reference problems, and study sufficient conditions for their fulfillment, as well as relationships of these notions. Finally, using the Kuratowski measure of noncompactness, several metric characterizations of such well-posedness properties for vector optimization problems are investigated. Many examples are given to analyze our results.

On minty variational principle for quasidifferentiable vector optimization problems

This paper deals with quasidifferentiable vector optimization problems involving invex functions wrt convex compact sets. We present vector variational-like inequalities of Minty type and of Stampacchia type in terms of quasidifferentials denoted by (QMVVLI) and (QSVVLI), respectively. By utilizing these variational inequalities, we infer vital and adequate optimality conditions for an efficient solution of the quasidifferentiable vector optimization problem involving invex functions wrt convex compact sets. We also establish various results for the solutions of the corresponding weak versions of the vector variational-like inequalities in terms of quasidifferentials.

Optimality and Duality of Approximate Quasi Weakly Efficient Solution for Nonsmooth Vector Optimization Problems

This paper aims at studying optimality conditions and duality theorems of an approximate quasi weakly efficient solution for a class of nonsmooth vector optimization problems (VOP). First, a necessary optimality condition to the problem (VOP) is established by using the Clarke subdifferential. Second, the concept of approximate pseudo quasi type-I function is introduced, and under its hypothesis, a sufficient optimality condition to the problem (VOP) is also obtained. Finally, the approximate Mond–Weir dual model of the problem (VOP) is presented, and then, weak, strong, and converse duality theorems are established.

Conditions for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the ideal value mapping associated to the parametric class of vector optimization problems. Within such an analysis, a refinement of a recent existence result, specific for ideal efficient solutions to unperturbed problem and enhanced by related error bounds, is discussed. Some connections with the concept of robustness in multi-objective optimization are also sketched.

Stability and Regularization of Vector Optimization Problems Under Possible Criteria Disturbances

Stability and Regularization of Vector Optimization Problems Under Possible Criteria Disturbances

A Perturbation Approach to Vector Optimization Problems: Lagrange and Fenchel–Lagrange Duality

In this paper we study the general minimization vector problem (P), concerning a perturbation mapping, defined in locally convex Hausdorff topological vector spaces where the "WInf" stands for the weak infimum with respect to an ordering generated by a convex cone $K$. Several representations of the epigraph of the conjugate mapping of the perturbation mapping are established. From these, variants vector Farkas lemmas are then proved. Armed with these basic tools, the {\it dual} and the so-called {\it loose dual problem} of (P) are defined, and then stable strong duality results between these pairs of primal-dual problems are established. The results just obtained are then applied to a general class (CCCV) of composed vector optimization problems with cone-constrained. For this classes of problems, four perturbation mappings are suggested. Each of these mappings yields several forms of vector Farkas lemmas and two forms of dual problems for (CCVP). Concretely, one of the suggested perturbation mapping give rises to well-known {\it Lagrange} and {\it loose Lagrange dual problems} for (CCVP) while each of the three others, yields two kinds of Fenchel-Lagrange dual problems for (CCVP). Stable strong duality for these pairs of primal-dual problems are proved. Several special cases of (CCVP) are also considered at the end of the paper, including: vector composite problems (without constraints), cone-constrained vector problems, and scalar composed problems. The results obtained in this papers when specified to the two concrete mentioned vector problems go some Lagrange duality results appeared recently, and also lead to new results on stable strong Fenchel-Lagrange duality results, which, to the best knowledge of the authors, appear for the first time in the literature.

A Norm Minimization-Based Convex Vector Optimization Algorithm

We propose an algorithm to generate inner and outer polyhedral approximations to the upper image of a bounded convex vector optimization problem. It is an outer approximation algorithm and is based on solving norm-minimizing scalarizations. Unlike Pascoletti–Serafini scalarization used in the literature for similar purposes, it does not involve a direction parameter. Therefore, the algorithm is free of direction-biasedness. We also propose a modification of the algorithm by introducing a suitable compact subset of the upper image, which helps in proving for the first time the finiteness of an algorithm for convex vector optimization. The computational performance of the algorithms is illustrated using some of the benchmark test problems, which shows promising results in comparison to a similar algorithm that is based on Pascoletti–Serafini scalarization.

Comparison of the effectiveness of thriple-loop and double-loop systems of active shielding of a magnetic field in a multi-storey old buildings

Aim. The issues of comparing the effectiveness of reducing the level of the magnetic field in a five-storey old buildings generated by a single-circuit overhead power transmission lines with a triangular suspension of wires using a thriple-loop and double-loop systems of active screening, which respectively contain three or two compensating windings are considered. Methodology. Spatial location coordinates of the compensating windings and the currents in the shielding windings were determined during the design of systems of active screening based on solution of the maximin vector optimization problem, in whith the vector of objective function is calculated based on Biot-Savart's law. The solution of this problem is calculated based on algorithms of multi-swarm multi-agent optimization. Results. The results of theoretical and experimental comparing the effectiveness of reducing the level of the magnetic field in a five-storey old generated by a single-circuit overhead power transmission lines with a triangular suspension of wires using a thriple-loop and double-loop systems of active screening, which respectively contain three or two compensating windings are presented. Originality. For the first time, the comparison the effectiveness of reducing the level of the magnetic field in a five-storey old using a thriple-loop and double-loop systems of active screening are considered. Practical value. From the point of view of the practical implementation it is shown the possibility to reduce the level of magnetic field induction in a five-storey old buildings to the sanitary standards of Ukraine for real overhead power transmission lines currents with the help of a synthesized double-loop systems of active screening. A double-loop active screening system is simpler in comparison with a thriple-loop active screening system when implementing.

Stability Analysis for Semi-Infinite Vector Optimization Problems under Functional Perturbations

This paper aims to study the stability of a class of semi-infinite vector optimization problems (SVOP) under functional perturbations. By using an important hypothesis a necessary and sufficient condition of Hausdorff continuity for weak efficient solution mappings and certain sufficient conditions for Painlevé-Kuratowski convergence of weak efficient solution sets for SVOP are established under the perturbations of both constraint sets and objective functions.

On relations between nonsmooth interval-valued multiobjective programming problems and generalized Stampacchia vector variational inequalities

In this article, we study nonsmooth interval-valued multiobjective programming problem and generalized Stampacchia vector variational inequality with its weak form for interval-valued functions. Using the tools of Mordukhovich subdifferential, we define some new classes of generalized approximate LU-convex functions. These functions are then employed to establish the relations between the solutions of generalized vector variational inequalities and the approximate LU-efficient solutions of the nonsmooth interval-valued multiobjective programming problem. Moreover, we identify the Kuhn–Tucker vector critical points of the considered nonsmooth interval-valued multiobjective programming problem. Under suitable constraint qualification, we establish the equivalence among local approximate LU-efficient points, Kuhn–Tucker vector critical points and the solutions of generalized Stampacchia vector variational inequalities. The results of this paper extend and sharpen the corresponding results of [Giannessi F. On Minty variational principle. In: Giannessi F, Komlósi S, Rapcśak T. editors. New trends in mathematical programming. Dordrecht: Kluwer Academic Publishers; 1997. p. 93–99], [Golestani M, Sadeghi H, Tavan Y. Nonsmooth multiobjective problems and generalized vector variational inequalities using quasi-efficiency. J Optim Theory Appl. 2018;179(3):896–916], [Lee GM, Lee KB. Vector variational inequalities for nondifferentiable convex vector optimization problems. J Global Optim. 2005;32(4):597–612], [Mishra SK, Upadhyay BB. Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency. Positivity. 2013;17(4):1071–1083], [Upadhyay BB, Mohapatra RN, Mishra SK. On relationships between vector variational inequality and nonsmooth vector optimization problems via strict minimizers. Adv Nonlinear Var Inequal. 2017;20(2):1–12], [Zhang J, Zheng Q,Ma X, Li L. Relationships between interval-valued vector optimization problems and vector variational inequalities. Fuzzy Optim Decis Mak. 2016;15(1):33–55] for nonsmooth interval-valued multiobjective programming problem by using the powerful tool of Mordukhovich subdifferential.