Study On Robust Solutions For Multiobjective Optimization Problems

The multiobjective optimization problem is the vector extremum problem,which maximizes or minimizes the vector value function under certain conditions.The theoretical research of multiobjective optimization mainly includes the definition of the optimal solutions,the optimality conditions for these optimal solutions,the scalarization methods,the duality theory,the algorithm research and so on.And these theories are also applied to many practical problems,such as engineering,economy,management,military and system engineering.Therefore,multiobjective optimization is closely related to the real life.However,there are a lot of uncertain factors in many practical problems,which is called the uncertain optimization problems.Robust optimization is a new method to solve this kind of problem.Many scholars have also made some achievements in this field,so the research on robust solutions of multiobjective optimization problems is of great significance.This paper mainly studies the optimality conditions of robust solutions for multiobjective optimization,the duality theory and the relationship between two kinds of robust solutions and application.The main results are as follows:1.In the first part,we consider the optimality conditions of robust solutions of multiobjective optimization problems.Firstly,the concept of robust weakly efficient solutions for multiobjective optimization problems is given,the relationship between it and robust solutions and Borwein properly efficient solutions is studied,and several examples are given to illustrate our main results.Then,the relationship between the robust weakly efficient solutions and Benson properly efficient solutions is obtained under subconvexity,and the relationship between robust solutions and local Heing properly efficient solutions is studied under the general conic order.Finally,the necessary and sufficient optimality conditions for robust solutions are established under the subconvexity and strictly pseudoconvexity assumptions.2.In the second part,the duality theory of robust solutions for multiobjective optimization problems is studied.First of all,we give the optimality conditions of robust solutions under convexity assumption.Then,we establish Wolfe type dual for the original problem,and study the weak duality theorem and strong duality theorem.Finally,we derive Mond-Weir type dual for the original problem,and also study the duality theory.3.In the third part,the relationship between two kinds of robust solutions and application are studied.Firstly,we give the concept of W-robust strictly solutions based on W-robust solutions,study the relationship between it and the robust solutions,and give the corresponding examples to illustrate our main results.Secondly,we study the relationship between W-robust solutions and Benson properly efficient solutions under the condition of nearly conical subconvexlike.Then,the definition of W-robust solutions with respect to general point convex cone is given.Finally,the application of robust solutions to disaster impact planning and portfolio problems is presented.