Proximal Bundle Method For Nonsmooth Nonconvex Multi-Objective Optimization Problems

There are many urgent multi-objective optimization problems in economics,engineering,computer vision,and other fields.In the classical multi-objective optimization methods,the weighted summation method is often used to transform the multi-objective problem into a single objective problem.In the paper,we study the numerical algorithm for nonsmooth nonconvex multi-objective optimization problems without employing any scalarization.The basic idea is to transform constrained multi-objective optimization problem into unconstrained optimization problem by employing improvement function,and then employ proximal bundle method to optimize multiple objective functions simultaneously.Firstly,we propose unfeasible multi-objective proximal bundle method(UMPB),then improve the UMPB algorithm and we propose penalty unfeasible multi-objective proximal bundle method(PUMPB).The global convergence of the algorithm is proved,and there is a good balance between the decreasing of the objective function and the feasibility of the constraint function.We employ bundle compression algorithm into UMPB and PUMPB to solve large scale optimization problems.We highlight that UMPB and PUMPB are unfeasible algorithms,whose initial point is not required feasible.The numerical experiments show the effectiveness of UMPB and PUMPB.In the third chapter,we propose the UMPB algorithm.We give the optimality condition of multi-objective optimization problems based on classical improvement function,construct UMPB subproblem model via the classical improvement function,analyze the stability of parameters,and prove the global convergence of UMPB algorithm.The numerical experiments show the effectiveness of UMPB.In the fourth chapter,we propose the PUMPB algorithm.We give the study motivation of PUMPB,give the optimality condition of multi-objective optimization problems based on modified improvement function,construct PUMPB subproblem model via the modified improvement function,and prove the global convergence of PUMPB algorithm.The numerical experiments show the effectiveness of PUMPB.