Multivalued Variational Inequalities Research Articles

Some Predictor-Corrector Algorithms for Multivalued Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest a new class of predictor-corrector algorithms for solving multivalued variational inequalities. The convergence of the proposed methods requires only the partially-relaxed strong monotonicity of the operator, which is weaker than cocoercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities.

System of multi-valued variational inequalities

System of multi-valued variational inequalities

Multivalued variational inequalities and resolvent equations

In this paper, we introduce and study some new classes of multivalued variational inequalities. These classes are more general and include the previously known classes of variational inequalities and related optimization problems as special cases. We also introduce some new resolvent equation problems. We prove that multivalued variational inequalities are equivalent to the fixed point problems and the resolvent equations. This equivalence is used to discuss the existence of the solution of the multivalued variational inequalities and suggest iterative algorithms for computing the approximate solutions. The convergence analysis is also studied.

On a class of multivalued variational inequalities

In this paper, we introduce and study a new class of variational inequalities, which are called multivalued variational inequalities. These variational inequalities include as special cases, the previously known classes of variational inequalities. Using projection techniques, we show that multivalued variational inequalities are equivalent to fixed point problems and Wiener‐Hopf equations. These alternate formulations are used to suggest a number of iterative algorithms for solving multivalued variational inequalities. We also consider the auxiliary principle technique to study the existence of a solution of multivalued variational inequalities and suggest a novel iterative algorithm. In addition, we have shown that the auxiliary principle technique can be used to find the equivalent differentiable optimization problems for multivalued variational inequalities. Convergence analysis is also discussed.

Generalized multivalued variational inequalities II

In this paper, we prove that the generalized multivalued variational inequalities are equivalent to the generalized multivalued Wiener-Hopf equations. This equivalence is used to suggest a number of iterative algorithms for solving generalized multivalued variational inequalities. We also discuss the convergence of these algorithms.

Multi-valued variational inequalities with K-pseudomonotone operators

In this paper, we first employ the 1961 celebrated Fan lemma to derive a very general existence result for multi-valued variational inequalities involving multi-valued K-pseudomonotone operators. It will be seen that this result improves and unifies existence results of variational inequalities for monotone operators. Next, we establish some uniqueness results for multi-valued variational inequalities by introducing the concepts of strict, α, and strong K-pseudomonotonicity of multi-valued operators, respectively. These uniqueness results appear to be new even if the underlying space is finite-dimensional.

Extension of generalized multi-valued variational inequalities

In this paper, we derive the existence result for a new class of generalized multi-valued variational inequality (1.1). Theorem 3.3 of this paper seems to be a natural extension of the celebrated result of Browder, Hartman and Stampacchia [1,2] variational inequalities for multi-valued operator in infinite-dimensional space.

Algorithm for generalized multivalued variational inequalities in hilbert spaces

We introduce a new class of generalized multivalued variational inequality problems (1.1) which contain several existing variational inequality problems as special cases. An iterative algorithm for finding approximate solutions of problem (1.1) is considered. Several convergence results for this algorithm are derived and in particular several existence results of problems (1.1) are obtained. We also introduce a new class of generalized multivalued complementarity problems (5.1) which also contain several known complementarity problems as special cases. It is shown that problem (5.1) is equivalent to problem (1.1) of special type from which several existence results of problem (5.1) are obtained.