Our aim is to provide a short analysis of the generalized variational inequality (GVI) problem from both theoretical and algorithmic point of view. First, we show connections among some well known existence theorems for GVI and for inclusions. Then, we recall the proximal point approach and a splitting algorithm for solving GVI. Finally, we propose a class of differentiable gap functions for GV...

By applying the concept of partially relaxed -strong monotonicity of set-valued mappings due to author and the auxiliary variational inequality technique, some new predictor–corrector iterative algorithms for solving generalized mixed quasi-variational-like inclusions are suggested and analyzed. The convergence of the algorithms only need the continuity and the partially relaxed -stronglymonoto...

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.

This paper introduces new mixed vector FQ-implicit variational inequality problems and corresponding mixed vector FQ-implicit complementarity problems for set-valued mappings, and studies the equivalence between them under certain assumptions in Banach spaces. It also derives some new existence theorems of solutions for them with examples under suitable assumptions without monotonicity. This pa...

In this paper, several classes of vector F -complementarity problems (in short, VF -CP) and vector F -variational inequalities (in short, VF -VI) with relations determined by nonconvex preferences are introduced in Banach spaces. Some characterizations of solution sets for (VF -CP) and (VF -VI) are also presented. Furthermore, the results obtained are applied to vector optimization problems.

The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space de…ned by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including ...

The critical points of the generalized complementary energy variational principles are clarified. An open problem left by Hellinger and Reissner is solved completely. A pure complementary energy (involving the Kirchhoff type stress only) is constructed. We prove that the well-known generalized Hellinger-Reissner’s energy L(u, s) is a saddle point functional if and only is the Gao-Strang gap fun...

By exploiting recent results, it is shown that the theories of Vector Optimization and of Vector Variational Inequalities can be based on the image space analysis and theorems of the alternative or separation theorems. It is shown that, starting from such a general scheme, several theoretical aspects can be developed – like optimality conditions, duality, penalization – as well as methods of so...

We introduce several types of the Levitin-Polyak well-posedness for a generalized vector quasivariational inequality problem with both abstract set constraints and functional constraints. Criteria and characterizations of these types of the Levitin-Polyak well-posednesses with or without gap functions of generalized vector quasivariational inequality problem are given. The results in this paper...