Set-valued Vector Research Articles

On Sub Convexlike Optimization Problems

In this paper, we show that the sub convexlikeness and subconvexlikeness defined by V. Jeyakumar are equivalent in locally convex topological spaces. We also deal with set-valued vector optimization problems and obtained vector saddle-point theorems and vector Lagrangian theorems.

Nonlinear Scalarization Characterizations of E-Super Efficient Solution of Set-valued Vector Optimization Problem

Abstract: In the study of vector optimization problems, the concept of (weakly) effective solution based on the definition of order cone and its properties play a very important role. The concept of superefficient solution is the unification of several existing concept of properly efficient solutions, while the E- super efficient solution of the vector optimization problem proposed based on the improvement set is an important extension of the classical concept of superefficient solution, which unifies the concepts of superefficient solution and ε-super efficient solution. Therefore, it is of great significance to study the E- super efficient solution and its related properties for the study of vector optimization problems. This paper mainly focuses on the relevant conditions of nonlinear scalarization characterization of E- super efficient solution: firstly, a sufficient and necessary condition for the nonlinear scalarization characterization about the E- super efficient solution of the set-valued vector optimization problem is obtained through the corresponding property of the seminorm p; secondly, the corresponding property of the distance function d is used to obtain another sufficient and necessary condition for the nonlinear scalarization characterization of the E- super efficient solution; then, according to the relationship between the E-optimal solution and the E-super efficient solution, a sufficient condition for the results of nonlinear scalarization characterization of the E- super efficient solution is derived by means of the distance-oriented function Δ-E which based on the improved set E under the condition of E=E+K0; finally, under the condition of 0∈clE, the distance-oriented function Δ-E is used to obtain a necessary condition for the nonlinear scalarization result of E- super efficient solution, and the corresponding sufficient and necessary result is deduced.

Generalized Lagrangian Duality in Set-valued Vector Optimization via Abstract Subdifferential

Generalized Lagrangian Duality in Set-valued Vector Optimization via Abstract Subdifferential

Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Processes

We present a new kind of Lagrangian duality theory for set-valued convex optimization problems whose objective and constraint maps are defined between preordered normed spaces. The theory is accomplished by introducing a new set-valued Lagrange multiplier theorem and a dual program with variables that are pointed closed convex processes. The pointed nature assumed for the processes is essential for the derivation of the main results presented in this research. We also develop a strong duality theorem that guarantees the existence of dual solutions, which are closely related to the sensitivity of the primal program. It allows extending the common methods used in the study of scalar programs to the set-valued vector case.

On the Existence of Super Efficient Solutions and Optimality Conditions for Set-Valued Vector Optimization Problems

In this paper, by using the normal subdifferential and equilibrium-like function we first obtain some properties for K-preinvex set-valued maps. Secondly, in terms of this equilibrium-like function, we establish some sufficient conditions for the existence of super minimal points of a K-preinvex set-valued map, that is, super efficient solutions of a set-valued vector optimization problem, and also attain necessity optimality terms for a general type of super efficiency.

On quasimonotone Stampacchia variational inequalities on Hadamard manifolds

In this paper, which is deeply inspired from Aussel and Hadjisavvas [On quasimonotone variational inequalities. J Optim Theory Appl. 2004;121:445–450] and Daniilidis and Hadjisavvas [Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions. J Optim Theory Appl. 1999;102(3):525–536], we study the existence of solutions of the Stampacchia variational inequality for a quasimonotone set-valued vector field on a Hadamard manifold. Moreover, the existence results are obtained under weak assumptions like quasimonotonicity and upper-sign continuity. An application of our results is also given.

Second-order efficient optimality conditions for set-valued vector optimization in terms of asymptotic contingent epiderivatives

We propose a generalized second-order asymptotic contingent epiderivative of a set-valued mapping, study its properties, as well as relations to some second-order contingent epiderivatives, and sufficient conditions for its existence. Then, using these epiderivatives, we investigate set-valued optimization problems with generalized inequality constraints. Both second-order necessary conditions and sufficient conditions for optimality of the Karush–Kuhn–Tucker type are established under the second-order constraint qualification. An application to Mond–Weir and Wolfe duality schemes is also presented. Some remarks and examples are provided to illustrate our results.

State Evaluation Model of High voltage Switchgear Based on Fuzzy Set-valued Statistical Method and Kernel Vector Space Model

The traditional high voltage switchgear (HVS) state evaluation model mostly adopts electrical test, live detection and historical data, neglecting the influence of real-time operation data of HVS composition equipment on the state evaluation results. This paper proposes a HVS operation state evaluation model based on fuzzy set-valued statistics method and kernel vector space model based on electrical test data and on-line monitoring data. First of all, according to the components of high voltage switchgear, the operation state of HVS is described and the evaluation index system is established. Secondly, the fuzzy set-valued statistics method is used to construct the mathematical model of evaluation index weight. Then, the kernel vector space model is introduced, and the Gaussian kernel function is used to map the sample to the features of the high-dimensional feature space. The indicator vector of the sample data and the ideal indicator vector of the high-voltage switchgear operation status level standard are defined in the high-dimensional feature space, and the angle-weighted cosine between the two vectors is calculated as the closeness of the sample to the standard status level, and then the high-voltage switchgear operation status level is obtained. Finally, the real data of a power supply company in western China are simulated. The results show that the greater the closeness degree is, the closer the HVS corresponding to the sample is to the normal state, on the contrary, the smaller the closeness degree is, the closer the HVS is to the fault state.

The connectedness of the solutions set for set-valued vector equilibrium problems under improvement sets

In this paper, we provide the connectedness of the sets of weak efficient solutions, Henig efficient solutions and Benson proper efficient solutions for set-valued vector equilibrium problems under improvement sets.

Error bounds for mixed set-valued vector inverse quasi-variational inequalities

The purpose of this paper is to introduce and study the mixed set-valued vector inverse quasi-variational inequality problems (MSVIQVIPs) and to obtain error bounds for this kind of MSVIQVIP in terms of the residual gap function, the regularized gap function, and the D-gap function. These bounds provide effective estimated distances between an arbitrary feasible point and the solution set of mixed set-valued vector inverse quasi-variational inequality problem. The results presented in the paper improve and generalize some recent results.

Semicontinuity of the Minimal Solution Set Mappings for Parametric Set-Valued Vector Optimization Problems

With the help of a level mapping, this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems. First, we introduce a kind of level mapping which generalizes one given in Han and Gong (Optimization 65:1337–1347, 2016). Then, we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping. Finally, in terms of the semicontinuity of the level mapping, we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric set-valued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge.

Continuity of the solution set mappings for parametric unified weak quasivariational inequalities via free-disposal set mappings

ABSTRACT In this study, we mainly study the stability for parametric unified weak quasivariational inequality (PUWQVI) under free-disposal set mappings. Firstly, by using a nonlinear scalarization method, we establish the semicontinuity of gap functions and solution mappings for (PUWQVI). Then, under the free-disposal set mapping assumption, we obtain some sufficient conditions of the outer-continuity and the outer-openness for (PUWQVI), and the inner-openness for parametric unified set-valued strong vector quasivariational inequality, respectively. Moreover, examples are given to illustrate our results, our results are new and improve the corresponding ones in the literature.

Ky Fan minimax inequalities for set-valued mappings on dense sets

We give a natural Ky Fan minimax inequality version of set-valued maps, and we deal with a type of vector equilibrium problem for set-valued mappings on a special dense set not on the whole domain. We use these results as applications to study the solutions of a generalized set-valued vector variational inequality.

Differential inclusions with mean derivatives having extreme right-hand sides and optimal control

First we prove the existence of solutions of some special stochastic differential inclusion with mean derivatives having lower semi-continuous right-hand sides that may not be convex. Then we show that among those solutions there is a solution that minimizes a certain cost criterion. After that this result is applied to investigation of controlled stochastic differential equations with feed back, whose right-hand sides take values in extreme sets of Hausdorff continuous set-valued vector field with bounded convex images. By reducing the equation to the inclusion of above-mentioned sort we prove that there exists a control that realizes the optimal solution of the inclusion as an optimal solution of the equation (an analogue of Filippov's theorem).

Proximal point algorithm for inclusion problems in Hadamard manifolds with applications

This paper deals with the proximal point algorithm for finding a singularity of sum of a single-valued vector field and a set-valued vector field in the setting of Hadamard manifolds. The convergence analysis of the proposed algorithm is discussed. Applications to composite minimization problems and variational inequality problems are also presented.

Relaxed $$\varvec{\mu }$$-quasimonotone variational inequalities in Hadamard manifolds

In this article, we introduce relaxed $$ \mu $$ -quasimonotone set-valued vector field on Hadamard manifolds and prove the existence of solutions of the Stampacchia variational inequality for such mappings. We also present the notion of relaxed $$ \mu $$ -quasiconvexity and show that the Upper Dini and Clarke–Rockafellar subdifferentials of a relaxed $$ \mu $$ -quasiconvex function is relaxed $$ \mu $$ -quasimonotone. Under relaxed $$ \mu $$ -quasiconvexity in the nondifferentiable sense, we establish the connection between the Stampacchia variational inequality problem and a nonsmooth constrained optimization problem.

Halpern- and Mann-Type Algorithms for Fixed Points and Inclusion Problems on Hadamard Manifolds

In this article, we consider an inclusion problem which is defined by means of a sum of a single-valued vector field and a set-valued vector field defined on a Hadamard manifold. We propose Halpern-type and Mann-type algorithms for finding a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. Some particular cases of our problem and algorithm are also discussed. We study the convergence of the proposed algorithm to a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. As applications of our results and algorithms, we derive the solution methods and their convergence results for the optimization problems, variational inequality problems and equilibrium problems in the setting of Hadamard manifolds.

Regularization of proximal point algorithms in Hadamard manifolds

In this paper, we consider the regularization method for exact as well as for inexact proximal point algorithms for finding the singularities of maximal monotone set-valued vector fields. We prove that the sequences generated by these algorithms converge to an element of the set of singularities of a maximal monotone set-valued vector field. A numerical example is provided to illustrate the inexact proximal point algorithm with regularization. Applications of our results to minimization problems and saddle point problems are given in the setting of Hadamard manifolds.

Existence and boundedness of solutions to inclusion problems for maximal monotone vector fields in Hadamard manifolds

In this paper, we consider the inclusion problems for maximal monotone set-valued vector fields defined on Hadamard manifolds. We discuss the equivalence between nonemptiness of solution set of the inclusion problem and the coercivity condition. The boundedness of solution set of the inclusion problem is studied. An application of our results to optimization problems in Hadamard manifolds is also presented.

Globally proper efficiency of set-valued optimization and vector variational inequality involving the generalized contingent epiderivative

In this paper, firstly, a new property of the cone subpreinvex set-valued map involving the generalized contingent epiderivative is obtained. As an application of this property, a sufficient optimality condition for constrained set-valued optimization problem in the sense of globally proper efficiency is derived. Finally, we establish the relations between the globally proper efficiency of the set-valued optimization problem and the globally proper efficiency of the vector variational inequality.