Optimality Of Approximate Solutions For A Class Of Nonsmooth Multi-objective Optimization Problems

The multiobjective optimization problem has practical applications in economy,finance,investment portfolio,traffic network and algorithm design,and has been widely concerned by scholars.The optimality condition is one of the important contents of multiobjective optimization theory.Since most of the solutions of many problems obtained by algorithms are approximate solutions,it is very important to study the optimality conditions of approximate solutions of multi-objective optimization problems.In addition,in practical applications,the objective and constraint functions of multi-objective optimization problems are usually non-smooth,so it is of great significance to study non-smooth problems.In this thesis,the optimality conditions of approximate solutions for nonsmooth multiobjective optimization problems are studied,details of this article include:1.Optimality conditions for approximate quasi weakly efficient solutions of Mordukhovich subdifferential characterization.Firstly,the necessary conditions for optimality of approximate quasi weakly efficient solutions are established by using various properties of Mordukhovich subdifferentials.Secondly,a concept of generalized convexity is introduced,and under its hypothesis,sufficient conditions for optimality of approximate quasi weak efficient solutions are given.In addition,with the help of the concept of partial stabilization,it is obtained that the approximate quasi weak efficient solution of nonsmooth multiobjective optimization problem is a necessary condition for the approximate quasi-minimum solution of penalty problem.2.The optimality condition of approximate quasi weakly efficient solution given by image space analysis method.Firstly,a new separation function is defined and its properties are discussed.Secondly,based on the image space analysis method,the optimality condition and saddle point theorem of approximate weakly efficient solutions for nonsmooth multiobjective optimization problems are established.3.Standard quantization theorem of approximate quasi weakly efficient solution based on nonlinear function construction.On the one hand,using the properties of directed distance function and Tammer function respectively,the approximate quasi weakly efficient quantization theorem of scale resolution is given.On the other hand,by constructing an infeasible separate system and using image space analysis method,we give a quantization theorem for approximate quasi-weak efficient solutions.