Derivative-free Method For Non-convex And Non-smooth Multiobjective Programming Under Convex Constraints

Multiobjective optimization problem research is important in the field of optimizationwith rich research content and a wide range of practical application backgrounds.However,the traditional smooth multiobjective optimization algorithm is not applicable to reality.Actually,many social specific problems today are non-smooth multiobjective optimization problems.This paper focuses on the algorithm for solving non-smooth multiobjective problems.Considering that non-smooth multiobjective problems are more complicated to calculate and have slower convergence speed,this paper proposes the DFMOP algorithm.The specific research is as follows:In the first step,this paper first introduces the non-smooth multiobjective optimization problem,the domestic and foreign situation of it and the research of the exact penalty method.Then the concepts of non-smooth multiobjective optimization problems are combed,such as local Lipschitz continuity,Clarke generalized directional derivatives.The second part,for the non-smooth multiobjective optimization problem to be solved in this paper,we give a theoretical proof of the exact penalty function method for solving the problem with partially unbounded constraints,we first improve the mutiobjective problem we proposed in this paper,then prove the optimality conditions about the inequality constraint problems under the partially unbounded conditon and we finally give the exact penalty function of the partially unbounded non-smooth multiobjective programming problem in this paper,and prove the Pareto-Clarke stable point of the penalty function.In the third part,this paper proposes a exact-penalty-based line-search method,DFMOP algorithm,which gives the main algorithm and the sub-algorithms in detail,finally,gives the convergence of the DFMOP algorithm.At last part,two numerical examples verify the effectiveness of the DFMOP algorithm in solving non-smooth multiobjective problems.The conclusion shows that the DFMOP algorithm is suitable for solving non-smooth multiobjective problems under different scales of Halton matrices,at the same time,the comparison of DMS,CSDFNP and DFMOP algorithms proves that the DFMOP algorithm has a faster convergence speed.What's more,DFMOP still has the improvement,because the setting of the initial parameters of it will reduce the robustness.