Some Research On Multiobjective Optimization And Stochastic Variational Inequalities Problems

In this thesis, the strong Karush-Kuhn-Tucker conditions of nonsmooth multiobjective optimization problems in terms of Clarke subdifferential and convexificators are investigated. And the method of weighted expected residuals for stochastic variational inequality problems is considered. The thesis is divided into seven chapters and organized as follows:In Chapter 1, the development on the topic of strong Karush-Kuhn-Tucker conditions of multiobjective optimization problems as well as the related researches about stochastic optimization problems are reviewed. Then we give the motivations and the main research work of this thesis.In Chapter 2, some definitions, notations and properties such as contingent cone, normal cone, constraint qualifications, expectation and density function, which will be frequently used, are presented.In Chapter 3, the strong Karush-Kuhn-Tucker conditions of multiobjective optimization problems with inequality, equality and set constraints, where the objective and constraint functions are locally Lipschitz, are considered. We firstly introduce a calmness condition associated with the objective function and the constraint system, and demonstrate that it is equivalent to two classes of exact penalty problems. Then based on above conclusions, the strong Karush-Kuhn-Tucker conditions for local weak efficient solutions are derived in terms of Clarke subdifferential. Finally, by virtue of the relationship between the calmness and error boundness of multiobjective optimization problem, we investigate the relationship between the generalized Mangasarian-Fromovitz constraint qualification and the calmness of multiobjective optimization problems in the case of smooth.In Chapter 4, for nonsmooth multiobjective optimization problems with inequality and set constraints, we introduce two constraint qualifications,(CQ1) and(CQ2). Firstly, we establish strong Karush-Kuhn-Tucker conditions for local efficient solutions in terms of convexificators for nonsmooth multiobjective programming problems under(CQ1) and(CQ2), respectively. Secondly, some examples are presented to illustrate that the convex hull and closure in strong Karush-Kuhn-Tucker conditions can not be removed, and the upper semi-regular convexificators of objective functions can not be replaced by upper convexificators. Finally, the relationships among(CQ1),(CQ2) and some other constraint qualifications proposed in recent papers are investigated.In Chapter 5, based on the convex combined expectations of the absolute deviation and least squares about the so-called regularized gap function, we transform the stochastic affine variational inequality problem with nonlinear perturbation into a weighted expected residual minimization problem. Some properties of the weighted expected residual minimization problem are derived under suitable conditions. Moreover, a discrete approximation problem of the weighted expected residual minimization problem is obtained by applying the quasi-Monte Carlo method, and the convergence of optimal solutions and stationary points of the approximation problem are also analyzed.In Chapter 6, for the nonlinear stochastic variational inequality problem, a method of convex combined expectations of the absolute deviation and least squares about the so-called regularized gap function is applied to transform it into a weighted expected residual minimization problem. Moreover, we present a discrete approximation of the weighted expected residual minimization problem by applying the quasi-Monte Carlo method when the sample space is compact, and give a compact approximation approach for the case that the sample space is noncompact. The convergence of optimal solutions of the discrete approximation problem and the compact approximation problem are analyzed, respectively.In Chapter 7, the main results of this thesis are summarized. Some problems which are deserved to consider and research in the future are put forward.