Study On Characterizations Of Solution Set Of Nonsmooth Optimization Problems

Study on characterizations of solution set is very meaningful to characterize the struc-ture of solution set and design algorithm which solves nonlinear optimization problems for nonlinear optimization problems. This paper aims at some equivalent characterizations of solution sets of several kinds of nonlinear optimization problems including as differentiable pseudoconvex optimization problems, nonsmooth pseudoconvex optimization problems, nonsmooth pseudoinvex optimization problems and nonsmooth optimization problems with cone-constraints.In chapter2, we consider a class of differentiable nonlinear optimization problems with inequality constraints. We first prove the fact that Lagrange function is constant on the optimal solution set under the assumption of pseudoconvexity and then we give some equivalent characterizations of the optimal solution set of this kind of nonlinear optimization problems.In chapter3, we consider a class of nonsmooth optimization problems with inequality constraints and equality constraints. By means of some tools such as Clarke derivative and Clarke subdifferentials, we first prove the fact that the Lagrange function is constant on the optimal solution set under the assumption of pseudoinvexity and then give some equivalent characterizations of the optimal solution set of this kind of nonlinear optimiza-tion problems. Furthermore, we generalize corresponding results under pseudoconvexity to general pseudoinvexity case in terms of some tools such as Clarke derivative and Clarke subdifferentials, and obtain some equivalent characterizations of the optimal solution set.In chapter4, we consider a class of nonsmooth optimization problems with cone-constraints. By means of Clarke derivative and Clarke subdifferentials, we prove the fact that the Lagrange function is constant on the optimal solution set under the assumption of invexity and cone-invexity and then we also obtain some equivalent characterizations of the optimal solution set for this kind of nonsmooth optimization problems.