Optimality Conditions And Saddle-Point Theorems For Robust Solutions To Nonsmooth Multiobjective Programmings

The objectives or constraints to multiobjective programming mathematical model are usually nonsmooth,and are affected by various factors with uncertain information.Therefore,it is a very valuable work to study the nonsmooth uncertain optimization problems.Robust optimization is one of the effective methods to deal with uncertain optimization problems,this method is committed to ensuring the worst-case solution which is immunized against the data uncertainty to optimization problems.Convexity and its generalization play an important role in mathematical programming,especially in establishing optimality conditions.In this paper,the optimality conditions and saddle point theorems of several types to nonsmooth uncertain multiobjective problems are studied by using robust method.The specific contents are arranged as follows:1.The optimality conditions and saddle point theorems for approximate quasi-weak robust efficient solutions are studied in an uncertain multiobjective convex optimization problem(UMCOP).Firstly,the approximate quasi-weak robust efficient solution to problem(UMCOP)is defined and an example is given to illustrate its existence.Secondly,a scalarization theorem and the optimality conditions in sense of approximate quasi-weak robust efficiency are proposed by using an alternative theorem.Finally,the approximate quasiweak saddle points to problem(UMCOP)are defined,and the corresponding saddle point theorems are obtained.2.The optimality conditions of robust weak efficient solutions for nonsmooth uncertain multi-objective fractional programming problem(NUMFP)are investigated.First of all,the concepts of two types of generalized convex function pairs are introduced to problem(NUMFP),which are called type-I functions and pseudo quasi type-I functions.Secondly,under the Clarke subdifferential constrained qualification,optimality necessary conditions of robust weak efficient solution are given,and sufficient conditions are obtained under the hypothesis of pseudo quasi type-I generalized convexity.Finally,the concept of robust weak saddle points to problem(NUMFP)is introduced,saddle point theorems are obtained,and the main results in present paper are verified by concrete examples.3.The optimality conditions for robust approximate quasi weak efficient solutions are studied to a nonsmooth uncertain multiobjective programming problem(NUMP).Firstly,under the extended nonsmooth Mangasalian-Fromovitz constrained qualification assumption,the optimality necessary conditions of robust approximate quasi weak efficient solutions are given by using the alternative theorem.Secondly,the concept of pseudoquasi-type-I functions is introduced,its existence is illustrated by an example,and the optimality sufficient conditions for robust approximate quasi weak efficient solutions to problem(NUMP)are established.