Vector Variational Inequalities Research Articles

Vector variational inequality with pseudoconvexity on Hadamard manifolds

In this paper, we give some properties for nondifferentiable pseudoconvex functions on Hadamard manifolds, and discuss the connections between pseudoconvex functions and pseudomonotone vector fields. Moreover, we study Minty and Stampacchia vector variational inequalities, which are formulated in terms of Clarke subdifferential for nonsmooth functions. Some relations between the vector variational inequalities and nonsmooth vector optimization problems are established under pseudoconvexity or pseudomonotonicity. The results presented in this paper extend some corresponding known results given in the literatures.

Network equilibrium of production, transportation and pricing for multi-product multi-market

In this paper, we consider a production, transportation and pricing problem for multi-product multi-market (PTPMM) as a system, and develop a PTPMM network equilibrium model. After allocating each product’s production cost and revenue to each path, we establish a profit network graph. An equilibrium PTPMM matrix and a \(\lambda \)-combination equilibrium are proposed based on a generalization of the well-known Wardrop’s equilibrium principle. The necessary and sufficient conditions for the \(\lambda \)-combination equilibrium are proposed using a linear scalarized profit function. We prove that solving the PTPMM network equilibrium problem can be reduced to the solving of the weak vector variational inequality problem, which proposes an algorithm for the PTPMM problem. Finally, an illustrative example is given to demonstrate an application of these theoretical results.

Expected Residual Minimization Formulation for a Class of Stochastic Vector Variational Inequalities

This paper considers a class of vector variational inequalities. First, we present an equivalent formulation, which is a scalar variational inequality, for the deterministic vector variational inequality. Then we concentrate on the stochastic circumstance. By noting that the stochastic vector variational inequality may not have a solution feasible for all realizations of the random variable in general, for tractability, we employ the expected residual minimization approach, which aims at minimizing the expected residual of the so-called regularized gap function. We investigate the properties of the expected residual minimization problem, and furthermore, we propose a sample average approximation method for solving the expected residual minimization problem. Comprehensive convergence analysis for the approximation approach is established as well.

Connectedness structure of the solution sets of vector variational inequalities

By a scalarization method and properties of semi-algebraic sets, it is proved that both the Pareto solution set and the weak Pareto solution set of a vector variational inequality, where the constraint set is polyhedral convex and the basic operators are given by polynomial functions, have finitely many connected components. Consequences of the results for vector optimization problems are discussed in details. The results of this paper solve in the affirmative some open questions for the case of general problems without requiring monotonicity of the operators involved.

Vector Equilibrium Problems on Dense Sets

In this paper, we provide sufficient conditions that ensure the existence of the solution of some vector equilibrium problems in Hausdorff topological vector spaces ordered by a cone. The conditions that we consider are imposed not on the whole domain of the operators involved, but rather on a self-segment-dense subset of it, a special type of dense subset. We apply the obtained results to vector optimization and vector variational inequalities.

An Introduction to Vector Variational Inequalities and Some New Results

This survey paper discusses some new results on vector variational inequalities. It can serve as an elementary introduction to vector variational inequalities and vector optimization problems. The focus point is made on the results about connectedness structure of the solution sets, which are obtained by a scalarization method and properties of semi-algebraic sets. The first major theorem says that both Pareto solution set and weak Pareto solution set of a vector variational inequality, where the constraint set is polyhedral convex and the basic operators are given by polynomial functions, have finitely many connected components. The second major theorem asserts that both proper Pareto solution set and weak Pareto solution set of a vector variational inequality, where the convex constraint set is given by polynomial functions and all the components of the basic operators are polynomial functions, have finitely many connected components, provided that the Mangasarian-Fromovitz Constraint Qualification is satisfied at every point of the constraint set. In addition, it has been established that if the proper Pareto solution set is dense in the Pareto solution set, then the latter also has finitely many connected components. Consequences of the results for vector optimization problems are shown.

Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints

This paper mainly intends to present some semicontinuity and convergence results for perturbed vector optimization problems with approximate equilibrium constraints. We establish the lower semicontinuity of the efficient solution mapping for the vector optimization problem with perturbations of both the objective function and the constraint set. The constraint set is the set of approximate weak efficient solutions of the vector equilibrium problem. Moreover, upper Painlevé–Kuratowski convergence results of the weak efficient solution mapping are showed. Finally, some applications to the optimization problems with approximate vector variational inequality constraints and the traffic network equilibrium problems are also given. Our main results are different from the ones in the literature.

Continuity Results and Error Bounds on Pseudomonotone Vector Variational Inequalities via Scalarization

Continuity (both lower and upper semicontinuities) results of the Pareto/efficient solution mapping for a parametric vector variational inequality with a polyhedral constraint set are established via scalarization approaches, within the framework of strict pseudomonotonicity assumptions. As a direct application, the continuity of the solution mapping to a parametric weak Minty vector variational inequality is also discussed. Furthermore, error bounds for the weak vector variational inequality in terms of two known regularized gap functions are also obtained, under strong pseudomonotonicity assumptions.

Polynomial Vector Variational Inequalities under Polynomial Constraints and Applications

By using a scalarization method and some properties of semi-algebraic sets, we prove that both the proper Pareto solution set and the weak Pareto solution set of a vector variational inequality, where the convex constraint set is given by polynomial functions and all the components of the basic operators are polynomial functions, have finitely many connected components, provided that the Mangasarian--Fromovitz constraint qualification is satisfied at every point of the constraint set. In addition, if the proper Pareto solution set is dense in the Pareto solution set, then the latter also has finitely many connected components. Consequences of the results for vector optimization problems are discussed in detail.

On Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Equilibrium Constraints

In this article, necessary conditions of Fritz John type for weak efficient solutions of a nonsmooth vector equilibrium problem involving equilibrium constraints (VEPEC) in terms of the Clarke subdifferentials are established. Under constraint qualifications which are suitable for (VEPEC), necessary conditions of Kuhn-Tucker type for efficiency are derived. Under assumptions on generalized convexity of data, sufficient conditions for efficiency are developed. Some applications to vector variational inequalities and vector optimization problems with equilibrium constraints are also given.

On the existence theorems of solutions for generalized vector variational inequalities

In this paper, we prove some existence theorems of solutions for two classes of generalized vector variational inequalities and Minty generalized vector variational inequalities.

Generalized proximal-type methods for weak vector variational inequality problems in Banach spaces

In this paper, we propose a class of generalized proximal-type method by the virtue of Bregman functions to solve weak vector variational inequality problems in Banach spaces. We carry out a convergence analysis on the method and prove the weak convergence of the generated sequence to a solution of the weak vector variational inequality problems under some mild conditions. Our results extend some known results to more general cases.

Approximate solutions for nonconvex set-valued optimization and vector variational inequality

This note deals with approximate solutions in vector optimization involving a generalized cone-invex set-valued mapping. First, a new class of generalized cone-invex set-valued maps, called cone-subinvex set-valued maps, is introduced. Then the sufficient optimality condition and two types dual theorems are established for weakly approximate minimizers under the assumption of cone-subinvexity. Finally, it also reveals the closed relationships between a weakly approximate minimizer of a cone-subinvex set-valued optimization problem and a weakly approximate solution of a kind of vector variational inequality.

Optimality Condition for Local Efficient Solutions of Vector Equilibrium Problems via Convexificators and Applications

Fritz John and Karush–Kuhn–Tucker necessary conditions for local efficient solutions of constrained vector equilibrium problems in Banach spaces in which those solutions are regular in the sense of Ioffe via convexificators are established. Under suitable assumptions on generalized convexity, sufficient conditions are derived. Some applications to constrained vector variational inequalities and constrained vector optimization problems are also given.

Scalar gap functions and error bounds for generalized mixed vector equilibrium problems with applications

It is well known that equilibrium problems are very important mathematical models and are closely related with fixed point problems, variational inequalities, and Nash equilibrium problems. Gap functions and error bounds which play a vital role in algorithms design, are two much-addressed topics of vector equilibrium problems. This paper is devoted to studying the scalar-valued gap functions and error bounds for the generalized mixed vector equilibrium problem (GMVE). First, a scalar gap function for (GMVE) is proposed without any scalarization methods, and then error bounds of (GMVE) are established in terms of the gap function. As applications, error bounds for generalized vector variational inequalities and vector variational inequalities are derived, respectively. The main results obtained are new and improve corresponding results of Charitha and Dutta (Pac. J. Optim. 6:497-510, 2010) and Sun and Chai (Optim. Lett. 8:1663-1673, 2014).

Coderivatives of gap function for Minty vector variational inequality

The purpose of this paper is to investigate coderivatives of the gap function involving the Minty vector variational inequality. First, we discuss the regular coderivative, the normal coderivative, and the mixed coderivative of a class of set-valued maps. Then, by using the relationships between the coderivatives of a set-valued map and its efficient points set-valued map, we obtain the coderivatives of the gap function for the Minty vector variational inequality.

A method of differential and sensitivity properties for weak vector variational inequalities

A method of differential and sensitivity properties for weak vector variational inequalities

On approximate starshapedness in multiobjective optimization

In this paper, we give the vector versions of the concepts of approximate starshapedness, equi-subdifferentiability and pseudo-equi-subdifferentiability and establish relationships among approximate vector starshapedness, vector-equi-subdifferentiability and vector-pseudo-equi-subdifferentiability. We extend the concept of ε-quasi-efficient solutions in the context of multiobjective optimization problems involving approximately starshaped functions and use approximate vector variational inequalities of Stampacchia and Minty type in terms of Fréchet subdifferentials to characterize approximate efficient solutions.

On Generalized Quasi-Vector Equilibrium Problems via Scalarization Method

In this paper, we consider the nonlinear scalarization function in the setting of topological vector spaces and present some properties of it. Moreover, using the nonlinear scalarization function and Fan---Glicksberg---Kakutani's fixed point theorem, we obtain an existence result of a solution for a generalized vector quasi-equilibrium problem without using any monotonicity and upper semi-continuity (or continuity) on the given maps. Our result can be considered as an improvement of the known corresponding result. After that, we introduce a system of generalized vector quasi-equilibrium problem which contains Nash equilibrium and Debreu-type equilibrium problem as well as the system of vector equilibrium problem posed previously. We provide two existence theorems for a solution of a system of generalized vector quasi-equilibrium problem. In the first one, our multi-valued maps have closed graphs and the maps are continuous, while in the second one, we do not use any continuity on the maps. Moreover, the method used for the existence theorem of a solution of a system of generalized vector quasi-equilibrium problem is not based upon a maximal element theorem. Finally, as an application, we apply the main results to study a system of vector optimization problem and vector variational inequality problem.

Optimality Conditions for Vector Equilibrium Problems with Set-Valued Mappings and Cone Constraints

Abstract The purpose of this paper is to establish necessary and sufficient conditions for a point to be solution of a vector equilibrium problem with setvalued mappings and cone constraints. Using a separation theorem which involves the quasi-relative interior of a convex set, we obtain optimality conditions for solutions of the considered vector equilibrium problem. The main theorem recovers an earlier established result. Then, the results are applied to vector optimization problems and to Stampacchia vector variational inequalities with cone constraints.