Solution for the system of generalized nonlinear mixed variational inequality problems

In this paper, we study a set-valued extended generalized nonlinear mixed variational inequality problem and its generalized resolvent dynamical system. A three-step iterative algorithm is constructed for solving set-valued extended generalized nonlinear variational inequality problem. Convergence and stability analysis are also discussed. We have shown the globally exponential convergence of generalized resolvent dynamical system to a unique solution of set-valued extended generalized nonlinear mixed variational inequality problem. In support of our main result, we provide a numerical example with convergence graphs and computation tables. For illustration, a comparison of our three-step iterative algorithm with Ishikawa-type algorithm and Mann-type algorithm is shown.

A method of convex combined expectations of the least absolute deviation and least squares about the so-called regularized gap function is proposed for solving nonlinear stochastic variational inequality problems (for short, NSVIP). The NSVIP is formulated as a weighted expected residual minimization problem (in short, WERM) in this way. Moreover, we present a discrete approximation of WERM problem by applying the quasi-Monte Carlo method when the sample space is compact, and a compact approximation approach for the case that the sample space is noncompact. The limiting behaviors of optimal solutions of the discrete approximation problem and the compact approximation are also analyzed, respectively.

The generalized set-valued strongly nonlinear implicit variational inequalities

The alternating direction method is an attractive method for solving large-scale variational inequality problems whenever the subproblems can be solved efficiently. However, the subproblems are still variational inequality problems, which are as structurally difficult to solve as the original one. To overcome this disadvantage, in this paper we propose a new alternating direction method for solving a class of nonlinear monotone variational inequality problems. In each iteration the method just makes an orthogonal projection to a simple set and some function evaluations. We report some preliminary computational results to illustrate the efficiency of the method.

Algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in the case of variational inequalities. In this paper, we derive sufficient conditions for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for variational inequalities.

Solvability of semidefinite complementarity problems

In this paper, we consider nonlinear variational inequality problems with fuzzy variables. The fuzzy variables were introduced to deal with the variational inequality containing noise for which historical data is not available. The fuzzy expected residual minimization (FERM) problems were established. We discussed the S C 1 property of the FERM model. Furthermore, results of convergence analysis were obtained based on an approximation model of the FERM model. The convergence of global optimal solutions and the convergence of stationary points were analysed.

Role of twice fréchet-differentiable mappings in general auxiliary problem principle

General convergence analysis for two-step projection methods and applications to variational problems

In this paper, we introduce a new class of general mixed vector F-implicit complementarity problems and general mixed vector F-implicit variational inequality problems, and study the equivalence between of them under certain assumptions in Banach spaces. We also derive some new existence theorems of solutions for the general mixed vector F-implicit complementarity problems and the general mixed vector F-implicit variational inequality problems by using the FKKM theorem under some suitable assumptions without monotonicity. Moreover, we establish sufficient conditions for the upper semicontinuity and lower semicontinuity of the solution mapping of the general mixed vector F-implicit variational inequality problems.

This paper aims to establish the Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces. For this purpose, we firstly prove a very general existence result for generalized mixed variational inequalities, provided that the mapping involved has the so-called mixed variational inequality property and satisfies a rather weak coercivity condition. Finally, we establish the Tikhonov regularization method for generalized mixed variational inequalities. Our findings extended the results for the generalized variational inequality problem (for short, GVIP(F, K)) in \(R^n\) spaces (He in Abstr Appl Anal, 2012) to the generalized mixed variational inequality problem (for short, GMVIP\((F,\phi , K)\)) in reflexive Banach spaces. On the other hand, we generalized the corresponding results for the generalized mixed variational inequality problem (for short, GMVIP\((F,\phi ,K)\)) in \(R^n\) spaces (Fu and He in J Sichuan Norm Univ (Nat Sci) 37:12–17, 2014) to reflexive Banach spaces.

We explore the solvability of a general system of nonlinear relaxed cocoercive variational inequality (SNVI) problems based on a new projection system for the direct product of two nonempty closed and convex subsets of real Hilbert spaces.

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.

In this work we reflect a new system of generalized nonlinear variational inclusion problems in 2-uniformly smooth Banach spaces. By using resolvent operator technique, we offer an iterative algorithm for figuring out the approximate solution of the said system. The motive of this paper is to review the convergence analysis of a system of generalized nonlinear variational inclusion problems in 2-uniformly smooth Banach spaces. The proposition used in this paper can be considered as an extension of propositions for examining the existence of solution for various classes of variational inclusions considered and studied by many authors in 2-uniformly smooth Banach spaces.

We consider a new system of multivalued mixed variational inequality problem, which includes some known systems of variational inequalities as special cases. Under suitable conditions, the existence of solutions for the system of multivalued mixed variational inequality problem and the convergence of iterative sequences generated by the generalizedf-projection algorithm are proved. A perturbational algorithm for solving a special case of multivalued mixed variational inequality problem is formally constructed. The results concerned with the existence of solutions and the convergence of iterative sequences generated by the perturbational algorithm are also given. Some known results are improved and generalized.