Mixed Variational Inequality Problems Research Articles

Strong and weak convergence theorems for general mixed equilibrium problems and general variational inequality problems and fixed point problems for two nonexpansive semigroups in Hilbert spaces

In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of the general mixed equilibrium problem and the set of solutions of a general variational inequality for two cocoercive mappings and the set of common fixed points of two nonexpansive semigroups in Hilbert space. We obtain both strong and weak convergence theorems for the sequences generated by these iterative processes in Hilbert spaces. Our results improve and extend the results announced by many others.

Global Optimization from Concave Minimization to Concave Mixed Variational Inequality

We use techniques from global optimization to develop an algorithm for finding a global solution of nonconvex mixed variational inequality problems involving separable DC cost functions. In contrast to the convex mixed variational inequality, in these problems, a local solution may not be a global one. The proposed algorithm uses the convex envelope of the separable cost function over boxes to approximate a DC cost problem with a convex cost one that can be solved by available methods. To obtain better approximate solutions, the algorithm uses an adaptive rectangular bisection which is performed only in the space of concave variables. The algorithm is applied to solve the Nash-Cournot and Bertrand equilibrium models with logarithm and quadratic concave costs. Computational results on a lot number of randomly generated data show that the proposed algorithm is efficient for these models, when the number of the concave cost functions is moderate, while the size of the model may be much larger.

Stability analysis of landslides using BEM and variational inequality based contact model

Since the stability of a landslide is mainly controlled by the strength properties of the slip band rather than the sliding body, naturally the boundary element method is an excellent tool for the stability analysis of the landslide, where the slip surface is regarded as a contact boundary between the sliding body and the slip bed. By casting the contact problem into a mixed variational inequality problem (MiVIP), the convergence in dealing with contact nonlinearity has been proved significantly improved. In this study, the strength reduction technique is applied to bring the slope to the limit equilibrium state. A so-called critical element on the slip surface is always fixed to maintain the convergence during the strength reduction. Based on the fact that the strength redundancy of the critical element vanishes at the limit equilibrium state, the method of bisection is designed to find out the factor of safety. A typical example and an engineering bank slope are analyzed to demonstrate the numerical properties of the proposed procedure.

Error Bounds for Extended Generalized Mixed Variational Inequality Problems Using Gap Functions

In this paper, we set up gap function, regularized gap function and D-gap function for extended generalized mixed variational inequality. Then by utilizing these gap functions, we calculate error bounds for the solution of the problem. The results acquired in this paper generalize the familiar results in literature.

Some Results about the Isolated Calmness of a Mixed Variational Inequality Problem

It is well known that optimization problem model has many applications arising from matrix completion, image processing, statistical learning, economics, engineering sciences, and so on. And convex programming problem is closely related to variational inequality problem. The so-called alternative direction of multiplier method (ADMM) is an importance class of numerical methods for solving convex programming problem. When analyzing the rate of convergence of various ADMMs, an error bound condition is usually required. The error bound can be obtained when the isolated calmness of the inverse of the KKT mapping of the related problem holds at the given KKT point. This paper is to study the isolated calmness of the inverse KKT mapping onto the mixed variational inequality problem with nonlinear term defined by norm function and indicator function of a convex polyhedral set, respectively. We also consider the isolated calmness of the inverse KKT mapping onto classical variational inequality problem with equality and inequality constrains under strict Mangasarian-Fromovitz constraint qualification condition. The results obtained here are new and very interesting.

Two-level iterative method for non-stationary mixed variational inequalities

We consider a mixed variational inequality problem involving a set-valued nonmonotone mapping and a general convex function, where only approximation sequences are known instead of exact values of the cost mapping and function, and feasible set. We suggest to apply a two-level approach with inexact solutions of each particular problem with a descent method and partial penalization and evaluation of accuracy with the help of a gap function. Its convergence is attained without concordance of penalty, accuracy, and approximation parameters under coercivity type conditions.

A Perturbed Iterative Algorithm for Split General Mixed Variational Inequality Problem

In this paper, we introduce a split general mixed variational inequality problem, which is a natural extension of a split variational inequality problem, split general quasi-variational inequality problem in Hilbert spaces. Using the resolvent operator technique, we propose a perturbed iterative algorithm for the split general mixed variational inequality problem. Further, we discuss the convergence criteria of the iterative algorithm. The results presented in this paper extend and improve many previously known results in this area.

Some results concerning the solution mappings of mixed variational inequality problems

This paper shows some continuities of mappings between the space of mixed variational inequality problems and the graph space of their solution mappings. The space of mixed variational inequality problems is homeomorphic to the graph of a continuous mapping. These generalize the results in the corresponding references.

A projection-type algorithm for solving generalized mixed variational inequalities

We propose a projection-type algorithm for generalized mixed variational inequality problem in Euclidean space ℝn. We establish the convergence theorem for the proposed algorithm, provided the multi-valued mapping is continuous and f-pseudomonotone with nonempty compact convex values on dom(f), where f: ℝn → ℝU{+∞} is a proper function. The algorithm presented in this paper generalize and improve some known algorithms in literatures. Preliminary computational experience is also reported.

The existence results and Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces

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.

A strong convergence theorem for finding a common fixed point of a finite family of Bregman nonexpansive mappings in Banach spaces which solves a generalized mixed equilibrium problem

In this paper, we study an iterative method for finding a common fixed point of a finite family of Bregman strongly nonexpansive mappings in real reflexive Banach spaces. Moreover, we prove a strong convergence theorem for finding common fixed points which solve a generalized mixed equilibrium problem. As a special case, we solve mixed variational inequality problems.

Generalized F - Projection Algorithm for a Split Set - Valued Mixed Variational Inequality Problem

Generalized F - Projection Algorithm for a Split Set - Valued Mixed Variational Inequality Problem

Gap function and global error bounds for generalized mixed quasi variational inequalities

In this paper, we obtain some gap functions for generalized mixed quasi variational inequality problems in terms of regularized gap function and D-gap function. Further, by using these gap functions we obtain global error bounds for the solution of generalized mixed quasi variational inequality problems in Hilbert spaces. The results obtained in this paper generalize and improve some corresponding known results in literatures.

Convergence theorems for split equality mixed equilibrium problems with applications

AbstractIn this paper, we introduce a new algorithm for solving split equality mixed equilibrium problems in the framework of infinite-dimensional real Hilbert spaces. The strong and weak convergence theorems are obtained. As application, we shall utilize our results to study the split equality mixed variational inequality problem and the split equality convex minimization problem. Our results presented in this paper improve and extend some recent corresponding results.

A General Inertial Proximal Point Algorithm for Mixed Variational Inequality Problem

In this paper, we first propose a general inertial proximal point algorithm (PPA) for the mixed variational inequality (VI) problem. Based on our knowledge, without stronger assumptions, a convergence rate result is not known in the literature for inertial type PPAs. Under certain conditions, we are able to establish the global convergence and nonasymptotic $O(1/k)$ convergence rate result (under a certain measure) of the proposed general inertial PPA. We then show that both the linearized augmented Lagrangian method (ALM) and the linearized alternating direction method of multipliers (ADMM) for structured convex optimization are applications of a general PPA, provided that the algorithmic parameters are properly chosen. Consequently, global convergence and convergence rate results of the linearized ALM and ADMM follow directly from results existing in the literature. In particular, by applying the proposed inertial PPA for mixed VI to structured convex optimization, we obtain inertial versions of the linearized ALM and ADMM whose global convergence is guaranteed. We also demonstrate the effect of the inertial extrapolation step via experimental results on the compressive principal component pursuit problem.

A New System of Multivalued Mixed Variational Inequality Problem

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.

A Resolvent Algorithm for System of General Mixed Variational Inequalities

In this paper, we suggest and analyze a new resolvent algorithm for finding the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point of a nonexpansive mapping in Hilbert spaces. We also prove the convergence analysis of the proposed algorithm under some suitable mild conditions. In this respect, our results present a refinement and improvement of the previously known results.

Strong convergence theorems for a generalized mixed equilibrium problem and variational inequality problems

In this paper, a new iterative scheme based on the extragradient-like method for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a strongly positive linear bounded operator and the set of solutions of a mixed equilibrium problem is proposed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Our results extend recent results announced by many others.MSC:49J30, 49J40, 47J25, 47H09.

Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces

In this paper, we introduce two iterative algorithms for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of finite variational inequalities for inverse strongly monotone mappings and the set of common fixed points of an asymptotically k-strictly pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems by using the proposed algorithms under some suitable conditions. Our results improve and extend the corresponding results announced by many others.

Gap Functions and Algorithms for Variational Inequality Problems

We solve several kinds of variational inequality problems through gap functions, give algorithms for the corresponding problems, obtain global error bounds, and make the convergence analysis. By generalized gap functions and generalized D-gap functions, we give global bounds for the set-valued mixed variational inequality problems. And through gap function, we equivalently transform the generalized variational inequality problem into a constraint optimization problem, give the steepest descent method, and show the convergence of the method.