On Algorithms For Quasi-variational Inequality Problem

In this dissertation, we mainly investigate the algorithms for quasi-variational inequality problem. We design three methods for solving quasi-variational in-equality problem. Four chapters are included in this thesis.Chapter 1 is the introduction. We describe the research situations of quasi-variational inequality problem. The main contributions of this paper are also stated briefly.In Chapter 2, based on the third section of [25], we do further research on the projection method for quasi-variational inequality problem. Under some appropriate conditions, we prove the Q-linear convergence rate of the related projection method.In Chapter 3, constructing a descent direction obtained from the direc-tional differentiability of the (generalized) regularized gap function, we present a derivative-free descent method for solving the quasi-variational inequality prob-lem. We do not need consider the gradient problem of the objective function. Compared with the ones in the related references, the method of this paper has the superiority that the application is more wider. Under some reasonable con-ditions, the convergence of the algorithm is proved.In Chapter 4, by constructing a strictly descent direction in the spirit of [26], we present another descent algorithm for solving the quasi-variational in-equality problem. We still do not need consider the gradient problem of the objective function. Compared with Chapter 3, the assumption that the▽F(·) is semi-positive definite is weaker than the one that▽F(·) is positive-definite. In addition, the effectiveness and convergence of this algorithm is guaranteed under some reasonable conditions.