Optimality Conditions And Scalarization Of Approximate Solutions For Nonsmooth Multiobjective Optimization

The multiobjective optimization problem is one of the important contents of optimization theory and method and its application research,and has been widely concerned by scholars.Among them,the optimality conditions of various approximate solutions,saddle point problem,dual theory,scalarization and so on are the core contents of multiobjective optimization theory.In practical application,the objective function and constraint function of multiobjective programming are mostly nonsmooth.Therefore,the research of nonsmooth theory has very important theoretical value and practical significance.This paper will focus on the optimality conditions of approximate solutions of nonsmooth multiobjective optimization problems and the nonlinear scalarization of approximate solutions,and discussed the relationship between the solutions of higher order strict minimum and the solutions of vector variational inequalities.The main contents are as follows:In part one,firstly,the concepts of two kinds of generalized convex functions are introduced.Secondly,based on Clarke subdifferential,under the condition of generalized convexity,the sufficient optimality conditions for robust approximate solutions of multiobjective optimization problems are obtained.Finally,the weak saddle point theorem for robust approximate solutions of multiobjective optimization problems is discussed.In part two,based on the extended weighted Chebyshev scalarization model and the modified weighted Chebyshev scalarization model,the scalarization property of cone approximation solution of multiobjective optimization problem is obtained,and for the given norm scalarization model,the norm scalarization result of the approximate solution of the multiobjective optimization problem is obtained.In part three,by introducing a class of generalized high order strongly pseudoconvex Lipschitz functions,which are called high order strongly pseudoconvex 0) functions.Under the assumption of generalized convexity,we give the characterization of the relationship among the solutions of high order strictly minimal solutions,vector key points and weak vector variational inequalities.