Mixed quasi variational inequalities

Acta Mathematica Scientia, 2008

The general mixed quasi variational inequality containing a nonlinear term ϕ is a useful and an important generalization of variational inequalities. The projection method can not be applied to solve this problem due to the presence of nonlinear term. It is well known that the variational inequalities involving the nonlinear term ϕ are equivalent to the fixed point problems and resolvent equations. In this article, the authors use these alternative equivalent formulations to suggest and analyze a new self-adaptive iterative method for solving general mixed quasi variational inequalities. Global convergence of the new method is proved. An example is given to illustrate the efficiency of the proposed method.

In this paper, we consider a new class of quasi variational inequalities involving three operators, which is called the extended general quasi variational inequality. It is shown that the extended general quasi variational inequalities are equivalent to the fixed problems and the extended general implicit Wiener-Hopf equations. These alternative formulations are used to suggest and analyze some iterative methods. The convergence analysis of these new methods under some suitable conditions is investigated. Several special cases are discussed. Since the extended general quasi variational inequalities include general variational inequalities, quasi variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems. Results of this paper may stimulate further research in this fascinating area.

In this paper, we consider a new class of quasi variational inequalities involving three operators, which is called the extended general quasi variational inequality. It is shown that the extended general quasi variational inequalities are equivalent to the fixed problems. This equivalence is used to suggest and analyze some iterative methods for solving the extended general quasi variational inequalities. Convergence analysis is also considered. We have also shown that the extended general quasi variational inequalities are equivalent to the extended general implicit Wiener-Hopf equations. This alternative formulation is used to suggest and analyze some iterative methods. The convergence analysis of these new methods under some suitable conditions is investigated. Several special cases are discussed. Since the extended general quasi variational inequalities include general variational inequalities, quasi variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems. Results of this paper may stimulate further research in this fascinating area.

2016

In this paper, we introduce and study a new class of quasi variational inequalities, known as multivalued extended general quasi variational inequalities. It is shown that the multivalued extended general quasi variational inequalities are equivalent to the fixed point problems. We use this alternative equivalent formulation to suggest and analyze some iterative methods. We consider the convergence analysis of an iterative method under suitable conditions. We also introduce a new class of Wiener-Hopf equations, known as multivalued extended general implicit Wiener-Hopf equations. We establish the equivalence between the multivalued extended general quasi variational inequalities and multivalued extended general implicit Wiener-Hopf equations. Using this equivalence, we suggest and analyze some iterative methods. Several special cases are also discussed. The ideas and techniques of this paper may stimulate further research in this field.

Optimization Letters, 2014

In this paper we develop a new and efficient method for solving a quasivariational inequality problem (QVIP) by using an extragradient-type method. The strategy is to combine the well-known search directions in the correction step from literature with the direction defined by the current iterate and the trial point obtained in the prediction step. This new combined search direction allows us to improve the convergence of the sequence of iterates to the solution of the QVIP but under a slightly stronger assumption, namely the co-coercivity of the problem operator. The new algorithm is devised to solve problems where the projections onto the moving feasible set are not easy to obtain. This combined procedure is applied to three well-known search directions and numerical illustrations are given to show the improvements obtained thanks to this strategy.