Quantitative Stability Analysis Of Quasi-variational Inequality Problems

A special class of quasi-variational inequalities are considered in which the constraint set is a closed convex set defined by a conic constraint.The quasi-variational inequality problem can be expressed as a non-smooth equation described by the projection operator.In this way,the stability of the quasi-variational inequality problem can be explored by studying the stability of the corresponding non-smooth equation.Since the projection of a point over a convex set is the solution of a convex programming problem,the quantitative stability of the non-smooth equation is obtained by a stability theorem for a convex programming problem,and this gives the quantitative stability analysis for this class of quasi-variational inequality problem.The paper can be summarized as follows.Chapter 1 gives a generalization of the variational inequality,including the equivalence form of the variational inequality and the strong regularity and Aubin properties of the solution set.At the same time,we describe the strong stability of the KKT system by using the constraint specification and the second order condition of the important results.Chapter 2 presents preliminaries required in this thesis,including limits and continu-ity of set-valued mappings,metrical regularity,Lipschitz homeomorphism and the important Robinson-Ursescu stability theorem.Chapter 3,the quantitative stability of parametric conic convex optimization is studied.The Holder continuity of the constraint set mapping is established under Slater condition and the Holder continuity of the optimal solution mapping is proved under the second-order growth condition.As an application,this result is applied to an orthogonal projection operator,which provides a basis for the stability analysis of quasi-variational inequalities.Chapter 4,we use the projection operator to express the solution set of the quasi-variational inequality into the solution of the non-smooth equations.Using the stability results of the third chapter on the quantization of the projection operator,The results of quantization of variational inequality are obtained and the results are applied to the stability analysis of a set of semidefinite programming solutions.Chapter 5,a brief survey of this thesis is given and the quantitative stability problems for conic constrained quasi-variational inequalities studied in the future are suggested as well.