Generalized η-Convex Functions And Generalized Type Ⅰ Functions Multi-objective Optimization

For being of scientific and real sense and wide applied prospect, optimization problems have been paid more and more attention. We normally meet classical extremum problems, which can be solved using classical derivative and differential. This requires functions must be differentiable. For weakening sufficient optimality conditions in optimization problems and convexity assumptions of functions in dual problems, many kinds of generalized convexity have been presented. This paper extended the concept ofη- convexity,η- quasiconvexity andη- pseudoconvexity to the class of locally Lipschitz functions using the generalized gradient of Clarke, defined generalizedη- convex functions,η- quasiconvex functions andη- pseudoconvex functions, probed into their properties and interrelations among them, and gave sufficient optimality conditions for nonsmooth programming. Consequently, we defined generalized Type I function for nonsmooth multi-objective programming, and gave sufficient conditions for efficient solutions. For another hand, we know Wolf dual problem and Mond-weir dual problem are two kinds of dual problems which most widely used. If we can gain weak duality or strong duality in optimization problems, then we can obtain more efficient methods.This paper can be divided five chapters, the contents are as follow: in the first chapter, we introduced the history and researching present situation of generalized convex functions and multi-objective programming. In the second chapter, we introduced the definition and properties of convex set,convex function and three kinds of generalized convex functions. In the third chapter, we introducedη- convex function, defined generalizedη- convex functions using the generalized gradient of Clarke, and researched into their properties and interrelations among them. Also, we gave sufficient optimality conditions. In the fourth chapter, we first gave nonsmooth multi-objective programming model, introduced several kinds of solutions for multiobjective programming, and then we defined generalized Type I function and obtained sufficient conditions for efficient solutions. In the last chapter, we introduced duality theory and gave weak duality, strong duality and converse duality theories in generalized Type I function multi-objective programming.