In this paper, with the help of the duality operator and K. Fan’s lemma, we present existence results for strong vector equilibrium problems, under pseudomonotonicity assumptions and without any pseudomonotonicity assumptions, respectively. Then, as an application, the main results allow us to state existence theorems for strong vector variational inequality problems.

In this paper we deal with the construction of gap functions for variational inequalities by using an approach which bases on the conjugate duality. Under certain assumptions we also investigate a further class of gap functions for the variational inequality problem, the so-called dual gap functions.

*Correspondence: [email protected] 1School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China 2Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China Full list of author information is available at the end of the article Abstract In this paper we consider a class of bilevel variational inequalities with hierarchical nesting structure. We first ...

We propose a variant of Korpelevich’s method for solving variational inequality problems with operators in Banach spaces. A full convergence analysis of the method is presented under reasonable assumptions on the problem data.

We propose several new models for separate and spatially distributed auction based markets where participants’ reactions are described by general price mappings and show that they admit equivalent variational inequality formulations. This approach is utilized for the derivation of existence and uniqueness results and for construction of efficient solution methods.

We present some sensitivity results for the spatial price equilibrium problem in the case of quantity formulation model and in presence of excess supply and excess demand. The equilibrium conditions that describe the above model are expressed in terms of a time dependent variational inequality. The variational inequality formulation plays a fundamental role in order to achieve the sensitivity r...

We propose several new equilibrium models for auction based markets and show that they admit equivalent variational inequality formulations. This approach simplifies essentially both the derivation of existence and uniqueness results and construction of efficient solution methods. The new models may be considered as alternative to the known perfectly and imperfectly competitive economic equilib...

Merit functions utilized to monitor the convergence of sequential quadratic programming (SQP) methods for nonlinear programs and variational inequality problems have in common that they include a penalty function for the explicit constraints, the value of the penalty parameter for which is subject to the requirement of being large enough compared to estimates of the optimal Lagrange multipliers...

Abstract The main goal of this paper is to study a class differential nonlinear system involving parabolic variational and history-dependent hemivariational inequalities in Banach spaces by using the penalty method. We first construct penalized problem for such then derive existence uniqueness its solution obtain an approximating sequence system. Moreover, we prove strong convergence obtained o...

Variational inequalities have found many applications in applied science. A partial list includes obstacles problems, fluid flow in porous media, management science, traffic network, and financial equilibrium problems. However, solving variational inequalities remain a challenging task as they are often subject to some set of complex constraints, for example the obstacle problem. Domain decompo...