On the nonemptiness and compactness of the solution sets for vector variational inequalities

In this article, we investigate conditions for nonemptiness and compactness of the sets of solutions of pseudomonotone vector variational inequalities by using the concept of asymptotical cones. We show that a pseudomonotone vector variational inequality has a nonempty and compact solution set provided that it is strictly feasible. We also obtain some necessary conditions for the set of solutions of a pseudomonotone vector variational inequality to be nonempty and compact.     

Conditions for Error Bounds of Linear Complementarity Problems over Second-Order Cones with Pseudomonotonicity

We first discuss some properties of the solution set of a pseudomonotone second-order cone linear complementarity problem (SOCLCP), and then analyse the limiting behavior of a sequence of strictly feasible solutions within a new wide neighborhood of the central trajectory for the pseudomonotone SOCLCP under assumptions of strict complementarity. Based on this, we derive four different characterizations of an error bound for the pseudomonotone SOCLCP.     

Modified projection method for pseudomonotone variational inequalities

We consider and analyze a new projection method for solving pseudomonotone variational inequalities by modifying the extragradient method. The modified method converges for pseudomonotone Lipschitz continuous operators, which is a much weaker condition than monotonicity. The new iterative method differs from the existing projection methods. Our proof of convergence is very simple as compared with other methods.     

A projection algorithm for solving pseudomonotone equilibrium problems and it's application to a class of bilevel equilibria

Abstract

We propose a projection algorithm for solving an equilibrium problem (EP) where the bifunction is pseudomonotone with respect to its solution set. The algorithm is further combined with a cutting technique for minimizing the norm over the solution set of an EP whose bifunction is pseudomonotone with respect to the solution set.     

Existence results for equilibrium problems with applications to evolution equations

In this paper, we consider evolution equations with time-dependent pseudomonotone and quasimonotone operators by a new approach based on equilibrium problems theory. We establish new existence results for equilibrium problems associated to pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are then applied to derive existence of solutions for evolution equations. The approach is new and leads to improve and unify most of the results obtained in this direction.     

New inertial algorithm for a class of equilibrium problems

Numerical Algorithms - The article introduces a new algorithm for solving a class of equilibrium problems involving strongly pseudomonotone bifunctions with a Lipschitz-type condition. We describe...     

Solvability of Generalized Variational Inequality Problems for Unbounded Sets in Reflexive Banach Spaces

By employing the notion of exceptional family of elements, we establish some existence results for generalized variational inequality problems in reflexive Banach spaces provided that the mapping is upper sign-continuous. We show that the nonexistence of an exceptional family of elements is a necessary condition for the solvability of the dual variational inequality. For quasimonotone variational inequalities, we present some sufficient conditions for the existence of strong solutions. For the pseudomonotone case, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the problem having strong solutions. Furthermore, we establish several equivalent conditions for the solvability for the pseudomonotone case. As a byproduct, a quasimonotone generalized variational inequality is proved to have a strong solution if it is strictly feasible. Moreover, for the pseudomonotone case, the strong solution set is nonempty and bounded if it is strictly feasible.     

Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities

A coercivity condition is usually assumed in variational inequalities over noncompact domains to guarantee the existence of a solution. We derive minimal, i.e., necessary coercivity conditions for pseudomonotone and quasimonotone variational inequalities to have a nonempty, possibly unbounded solution set. Similarly, a minimal coercivity condition is derived for quasimonotone variational inequalities to have a nonempty, bounded solution set, thereby complementing recent studies for the pseudomonotone case. Finally, for quasimonotone complementarity problems, previous existence results involving so-called exceptional families of elements are strengthened by considerably weakening assumptions in the literature.     

On Inclusions with Monotone-Type Mappings in Nonreflexive Banach Spaces

Journal of Optimization Theory and Applications - We are concerned in this article with the existence of solutions to inclusions containing generalized pseudomonotone perturbations of maximal...     

Proximal Point Algorithms for Quasiconvex Pseudomonotone Equilibrium Problems

Journal of Optimization Theory and Applications - We propose a proximal point method for quasiconvex pseudomonotone equilibrium problems. The subproblems of the method are optimization problems...     

Modified golden ratio algorithms for pseudomonotone equilibrium problems and variational inequalities

Applications of Mathematics - We propose a modification of the golden ratio algorithm for solving pseudomonotone equilibrium problems with a Lipschitz-type condition in Hilbert spaces. A new...     

Pseudomonotone Operators: A Survey of the Theory and Its Applications

The notion of pseudomonotone operator in the sense of Karamardian has been studied for 35 years and has found many applications in variational inequalities and economics. The purpose of this survey paper is to present the most fundamental results in this field, starting from the earliest developments and reaching the latest results and some open questions. The exposition includes: the relation of (generally multivalued) pseudomonotone operators to pseudoconvex functions; first-order characterizations of single-valued, differentiable pseudomonotone operators; application to variational inequalities; the notion of equivalence of pseudomonotone operators and its application to maximality; a generalization of paramonotonicity and its relation to the cutting-plane method; and the relation to the revealed preference problem of mathematical economics.     

Coercivity Conditions for Equilibrium Problems

The study of the existence of solutions of equilibrium problems on unbounded domains involves usually the same sufficient assumptions as for bounded domains together with a coercivity condition. We focus on two different conditions: the first is obtained assuming the existence of a bounded set such that no elements outside is a candidate for a solution; the second allows the solution set to be unbounded. Our results exploit the generalized monotonicity properties of the function f defining the equilibrium problem. It turns out that, in both the pseudomonotone and the quasimonotone setting, an equivalence can be stated between the nonemptyness and boundedness of the solution set and these coercivity conditions. In the pseudomonotone case, we compare our coercivity conditions with various coercivity conditions that appeared in the literature.We thank an anonymous referee for valuable comments and suggestions.     

On the Range of the Pseudomonotone Second-Order Cone Linear Complementarity Problem

In this paper, we provide a complete characterization of the range of the pseudomonotone second-order cone linear complementarity problem. In particular, by answering the three questions that under what conditions the range is the whole space, convex and closed, respectively, we explicitly characterize and formulate the range of the pseudomonotone second-order cone linear complementarity problem.     

An algorithm for generalized variational inequality with pseudomonotone mapping

We suggest an algorithm for a variational inequality with multi-valued mapping. The iteration sequence generated by the algorithm is proven to converge to a solution, provided the multi-valued mapping is continuous and pseudomonotone with nonempty compact convex values.     

Some Results on Strongly Pseudomonotone Quasi-Variational Inequalities

Set-Valued and Variational Analysis - In this paper, strongly pseudomonotone quasi-variational inequalities are investigated. We provide sufficient conditions for existence and uniqueness of...     

Noncoercive mixed equilibrium problems under pseudomonotone perturbations and applications to nonlinear evolution equations with lack of coercivity

In this paper, we study the existence of solutions for noncoercive mixed equilibrium problems which are described by the sum of a maximal monotone bifunction and a pseudomonotone (or quasimonotone) bifunction in the sense of Brézis. Our approach is based on recession analysis and on recent results established by the authors for the existence of solutions of mixed equilibrium problems under pseudomonotone perturbations. As an application, we study the existence of solutions for nonlinear evolution equations associated with a noncoercive time-dependent pseudomonotone (or quasimonotone) operator.     

Generalized evolution variational inequalities and their applications to partial differential equations

We study the existence of solutions to generalized evolution vari ational inequalities by pseudomonotone mapping theory. We obtain results which generalize and extend previously known theorems. We then apply our results to parabolic partial differential equations with discontinuous nonlinearities.     

A New Extragradient Method for Strongly Pseudomonotone Variational Inequalities

This article proposes a new extragradient solution method for strongly pseudomonotone variational inequalities. A detailed analysis of the iterative sequences’ convergence and of the range of applicability of the method is provided. Moreover, an interesting class of strongly pseudomonotone infinite dimensional variational inequality problems is considered.     

New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings

Numerical Algorithms - We propose and study new projection-type algorithms for solving pseudomonotone variational inequality problems in real Hilbert spaces without assuming Lipschitz continuity of...