Implicit Variational Inequalities Research Articles

An Iterative Algorithm for the Nonlinear MC2 Model with Variational Inequality Method

The multiple criteria and multiple constraint level (MC 2 ) model is a useful tool to deal with the decision programming problems, which concern multiple decision makers and uncertain resource constraint levels. In this paper, by regarding the nonlinear MC 2 problems as a class of mixed implicit variational inequalities, we develop an iterative algorithm to solve the nonlinear MC 2 problems through the resolvent operator technique. The convergence of the generated iterative sequence is analyzed and discussed by a calculation example, and the stability of Algorithm 1 is also verified by error propagation. By comparing with two other MC 2 -algorithms, Algorithm 1 performs well in terms of number of iterations and computation complexity.

Solving implicit mathematical programs with fuzzy variational inequality constraints based on the method of centres with entropic regularization

The purpose of this paper is to consider a class of mathematical programs with fuzzy implicit variational inequality constraints in finite dimension real spaces. By using the "tolerance approach" and the fuzzy set theory, we also show that solving the fuzzy mathematical program problem with fuzzy implicit variational inequality constraints is equivalent to solving a fuzzy implicit complementarity constrained optimization problem, and the fuzzy implicit complementarity constrained optimization problem can be converted to a regular nonlinear parametric programming problem. Further, a new smoothing approach based on a version of the "method of centres" with entropic regularization for solving the resulting optimization problem and our main results are presented and a numerical example is provided to illustrate our main results applying quasi-Newton line search of MATLAB software.

Internal and subspace correction approximations of implicit variational inequalities

The aim of this paper is to study the existence of solutions and some approximations for a class of implicit evolution variational inequalities that represents a generalization of several quasistatic contact problems in elasticity. Using appropriate estimates for the incremental solutions, the existence of a continuous solution and convergence results are proved for some corresponding internal approximation and backward difference scheme. To solve the fully discrete problems, general additive subspace correction algorithms are considered, for which global convergence is proved and some error estimates are established.

Approximation results and subspace correction algorithms for implicit variational inequalities

This paper deals with the mathematical analysis and the subspace approximation of a system of variational inequalities representing a unified approach to several quasistatic contact problems in elasticity. Using an implicit time discretization scheme and some estimates, convergence properties of the incremental solutions and existence results are presented for a class of abstract implicit evolution variational inequalities involving a nonlinear operator. To solve the corresponding semi-discrete and the fully discrete problems, some general subspace correction algorithms are proposed, for which global convergence is analyzed and error estimates are established.

A dynamic unilateral contact problem with adhesion and friction in viscoelasticity

The aim of this paper is to study an interaction law coupling recoverable adhesion, friction and unilateral contact between two viscoelastic bodies of Kelvin–Voigt type. A dynamic contact problem with adhesion and nonlocal friction is considered and its variational formulation is written as the coupling between an implicit variational inequality and a parabolic variational inequality describing the evolution of the intensity of adhesion. The existence and approximation of variational solutions are analysed, based on a penalty method, some abstract results and compactness properties. Finally, some numerical examples are presented.

Existence theorem and algorithm for a general implicit variational inequality in Banach space

By using the generalized f-projection operator, the existence theorem of solutions for the general implicit variational inequality GIVI ( T - ξ , K ) is proved without assuming the monotonicity of operators in reflexive and smooth Banach space. An iterative algorithm for approximating solution of the general implicit variational inequality is suggested also, and the convergence for this iterative scheme is shown. These theorems extend the corresponding results of Wu and Huang [K.Q. Wu, N.J. Huang, Comput. Math. Appl. 54 (2007) 399–406], Wu and Huang [K.Q. Wu, N.J. Huang, Bull. Austral. Math. Soc. 73 (2006) 307–317], Zeng and Yao [L.C. Zeng, J.C. Yao, J. Optimiz. Theory Appl. 132 (2) (2007) 321–337] and Li [J. Li, J. Math. Anal. Appl. 306 (2005) 55–71].

A class of implicit variational inequalities and applications to frictional contact

AbstractThis paper deals with the mathematical and numerical analysis of a class of abstract implicit evolution variational inequalities. The results obtained here can be applied to a large variety of quasistatic contact problems in linear elasticity, including unilateral contact or normal compliance conditions with friction. In particular, a quasistatic unilateral contact problem with nonlocal friction is considered. An algorithm is derived and some numerical examples are presented. Copyright © 2009 John Wiley & Sons, Ltd.

Shape Optimization in Three-Dimensional Contact Problems with Coulomb Friction

We study the discretized problem of the shape optimization of three-dimensional (3D) elastic bodies in unilateral contact. The aim is to extend existing results to the case of contact problems obeying the Coulomb friction law. Mathematical modeling of the Coulomb friction problem leads to an implicit variational inequality. It is shown that for small coefficients of friction the discretized problem with Coulomb friction has a unique solution and that this solution is Lipschitzian as a function of a control variable describing the shape of the elastic body. The 2D case of this problem was studied by the authors in [P. Beremlijski, J. Haslinger, M. Kocvara, and J. V. Outrata, SIAM J. Optim., 13 (2002), pp. 561-587]; there we used the so-called implicit programming approach combined with the generalized differential calculus of Clarke. The extension of this technique to the 3D situation is by no means straightforward. The main source of difficulties is the nonpolyhedral character of the second-order (Lorentz) cone, arising in the 3D model. To facilitate the computation of the subgradient information, needed in the used numerical method, we exploit the substantially richer generalized differential calculus of Mordukhovich. Numerical examples illustrate the efficiency and reliability of the suggested approach.

Generalized vector implicit complementarity problems with corresponding variational inequality problems

In this work, we establish some existence theorems for solutions to a new class of generalized vector F -implicit complementarity problems and the corresponding generalized vector F -implicit variational inequality problems in topological vector spaces. No monotonicity or continuity assumption is imposed; thus we have generalized some recent results from the literature.

The Solvability of a Class of General Nonlinear Implicit Variational Inequalities Based on Perturbed Three-Step Iterative Processes with Errors

We introduce and study a new class of general nonlinear implicit variational inequalities, which includes several classes of variational inequalities and variational inclusions as special cases. By applying the resolvent operator technique and fixed point theorem, we suggest a new perturbed three-step iterative algorithm with errors for solving the class of variational inequalities. Several existence and uniqueness results of solutions for the general nonlinear implicit variational inequalities, and convergence and stability results of the sequence generated by the algorithm are obtained. The results presented in this paper extend, improve, and unify a host of results in recent literatures.

ON GENERALIZED VECTOR IMPLICIT VARIATIONAL INEQUALITIES AND COMPLEMENTARITY PROBLEMS

In this paper, two generalized vector implicit variational inequalities and three generalized vector implicit complementarity problems are introduced with a general order in ordered Banach spaces and the equivalence between them is studied under certain assumptions. Furthermore, some existence theorems for the generalized vector implicit variational inequalities are derived by using Lin’s result (1986) and Nadler’s result (1969). We affirmatively answer the open question proposed by Rapcsak (2000). Our results are the extension and improvement of the corresponding results in Huang and Li (2006).

An approach for solving fuzzy implicit variational inequalities with linear membership functions

In this paper, we consider a class of fuzzy implicit variational inequalities with linear membership functions. By using the “tolerance approach”, we show that solving such problems can be reduced to a semi-infinite programming problem. A version of the “method of centres” with “entropic regularization” techniques, only used a quasi-Newton line search using MATLAB software is required in our implementation. We also give a numerical example to illustrate the validity of our approach.

Existence of Solutions to General Quasiequilibrium Problems and Applications

A general quasiequilibrium problem is proposed including, among others, equilibrium problems, implicit variational inequalities, and quasivariational inequalities involving multifunctions. Sufficient conditions for the existence of solutions with and without relaxed pseudomonotonicity are established. Even semicontinuity may not be imposed. These conditions improve several recent results in the literature.

Vector [formula omitted]-implicit complementarity problems with corresponding variational inequality problems

In this work, we present some new existence theorems for solutions for a new class of generalized vector F -implicit complementarity problems and the corresponding generalized vector F -implicit variational inequality problems by using the Fan–KKM Theorem under some suitable assumptions without monotonicity, and establish the equivalence between those problems in Banach spaces.

Vector [formula omitted]-implicit complementarity problems in topological vector spaces

Recently, Huang and Li [J. Li, N.J. Huang, Vector F -implicit complementarity problems in Banach spaces, Appl. Math. Lett. 19 (2006) 464–471] introduced and studied a new class of vector F -implicit complementarity problems and vector F -implicit variational inequality problems in Banach spaces. In this work, we study this class in topological vector spaces and drive some existence theorems for the vector F -implicit variational inequality and vector F -implicit complementarity problem. Also, their equivalence is presented under certain conditions.

System of Generalized Implicit Vector Quasivariational Inequalities

We will introduce a system of generalized implicit vector quasivariational inequalities (in short, SGIVQVI) which generalizes and unifies the system of generalized implicit variational inequalities, the system of generalized vector quasivariational-like inequalities, the system of generalized vector variational inequalities, the system of variational inequalities, the generalized implicit vector quasivariational inequality, as well as various extensions of the classic variational inequalities in the literature, and we present some existence results of a solution for the SGIVQVI without any monotonicity conditions.

Generalized Vector Variational Inequalities in Topological Vector Spaces

The solvability of the generalized weak vector implicit variational inequality problem, generalized strong vector implicit variational inequality problem and generalized vector variational inequality problem are proved by using a generalized Fan's KKM theorem. Our results extend and unify corresponding results of other authors.

The solution existence of general variational inclusion problems

We propose general variational inclusion problems which are slightly different from corresponding problems considered in several recent papers in the literature and show that they are advantageous. Sufficient conditions for the solution existence are established. As applications we derive consequences for several special cases of variational inclusion problems, quasioptimization problems, equilibrium problems and implicit variational inequalities and show that they improve the results of some recent existing papers.

On Vector Implicit Variational Inequalities and Complementarity Problems

In this paper, two vector implicit variational inequalities and three vector implicit complementarity problems are introduced with a more general order in ordered Banach spaces and the equivalence between them is studied under certain assumptions. Furthermore, some existence theorems of the vector implicit variational inequalities are proved. We answered the open question proposed by Rapcsak [Nonconvex Optim., Appl., Vol. 38, Kluwer Acad. Publ., Dordrecht, 2000, pp. 371---380].

Vector [formula omitted]-implicit complementarity problems in Banach spaces

In this work, a new class of vector F -implicit complementarity problems and vector F -implicit variational inequality problems are introduced and studied, and the equivalence between of them is presented under certain assumptions in Banach spaces. We also derive some new existence theorems of solutions for the vector F -implicit complementarity problems and the vector F -implicit variational inequality problems by using the FKKM theorem under some suitable assumptions without monotonicity.