Generalized vector variational-like inequalities and vector optimization

Abstract In this paper, we consider different kinds of generalized vector variational-like inequality problems and a vector optimization problem. We establish some relationships between the solutions of generalized Minty vector variational-like inequality problem and an efficient solution of a vector optimization problem. We define a perturbed generalized Stampacchia vector variational-like inequality problem and discuss its relation with generalized weak Minty vector variational-like inequality problem. We establish some existence results for solutions of our generalized vector variational-like inequality problems

Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI

Iterative algorithms with regularization for hierarchical variational inequality problems and convex minimization problems

Abstract In this paper, we consider a variational inequality problem which is defined over the set of intersections of the set of fixed points of a ζ-strictly pseudocontractive mapping, the set of fixed points of a nonexpansive mapping and the set of solutions of a minimization problem. We propose an iterative algorithm with regularization to solve such a variational inequality problem and study the strong convergence of the sequence generated by the proposed algorithm. The results of this paper improve and extend several known results in the literature.</jats:p

Multi-step implicit iterative methods with regularization for minimization problems and fixed point problems

Abstract In this paper we introduce a multi-step implicit iterative scheme with regularization for finding a common solution of the minimization problem (MP) for a convex and continuously Fréchet differentiable functional and the common fixed point problem of an infinite family of nonexpansive mappings in the setting of Hilbert spaces. The multi-step implicit iterative method with regularization is based on three well-known methods: the extragradient method, approximate proximal method and gradient projection algorithm with regularization. We derive a weak convergence theorem for the sequences generated by the proposed scheme. On the other hand, we also establish a strong convergence result via an implicit hybrid method with regularization for solving these two problems. This implicit hybrid method with regularization is based on the CQ method, extragradient method and gradient projection algorithm with regularization. MSC:49J30, 47H09, 47J20.</jats:p

Minimax Theorems for Set-Valued Mappings under Cone-Convexities

The aim of this paper is to study the minimax theorems for set-valued mappings with or without linear structure. We define several kinds of cone-convexities for set-valued mappings, give some examples of such set-valued mappings, and study the relationships among these cone-convexities. By using our minimax theorems, we derive some existence results for saddle points of set-valued mappings. Some examples to illustrate our results are also given