Weak Sharp Minima For Variational Inequalities And Sigma-convex Set-valued Mappings

Variational inequalities have wide applications in mechanics,cybernetics,optimization theory and financial mathematics.we mainly study the existence of solutions to generalized inverse mixed variational inequalities in Banach spaces,and give the concept of high-order weak sharp minima.Then we discuss the relationship between the original gap function and weak sharp minima in Banach spaces.By using differentiability,normal cones and approximate dual mappings,several sufficient and necessary conditions for the existence of weak sharp minima are obtained.Convexity plays a very important role in nonsmooth analysis and optimization theory.In this thesis,the concept of a sigma-convex set-valued mapping is given,and two equivalent conditions of sigma-convex set-valued mappings are obtained.The definition of sigma-coderivatives is given,and the calculus rules of sigma-coderivatives are established.The necessary condition for the existence of weak minimum solutions to sigma-convex set-valued optimization problems is given by using sigma-coderivatives.