Research On A Class Of Nonsmooth Multiobject DC Optimization Problems

In real life,we need to deal with many problems,usually not only the nonsmooth ones,but also in some cases,different objective functions exist at the same time.These objective functions are mutually constrained,and people can hardly guarantee all the objective functions are optimal at the same time.Therefore,people have to make a compromise for multicreteria optimization problems so that every objective can get a relatively good solution as much as possible.This thesis mainly studies a class of nonsmooth multiobjective DC optimization problems(?)where(?)and(?)Lare DC functions.For such problems,according to the Pareto optimality condition,we first use the objective and constraint functions to construct an improved function,transform the constrained optimization problem into an unconstrained optimization problem,then rewrite the improved function into a DC function with respect to a variable to generate new DC optimization problems.As to the new DC optimization problem,we first establish a cutting-planes model,and use the locally convexified and redistributed ideas to get the proximal sub-problem,thus transform the original sub-problem into a quadratic programming sub-problem for finding the search direction.Next,we design an algorithm to solve the original DC optimization based on this sub-problem.By theoretical analysis and derivation,it is proved that the algorithm has good convergence whether it generates infinitely many serious steps or finite serious steps followed by infinitely many null steps.The thesis is divided into four parts,the main contents are listed as follows:In the first chapter,we first give some basic concepts and existing conclusions related to nonsmooth DC optimization problems.Secondly,we introduce several basic methods for solving nonsmooth optimization problems,including the steepest-descent method,the black-box method,the subgradient method,the cutting-planes method and the bundle method and so on.Finally,we introduce the existing methods for solving non-smooth multiobjective optimization,which are the theoretical basis for further study in the next few chapters of this thesis.In the second chapter,for nonsmooth multiobjective DC optimization problem studied in this paper,firstly,we use the objective and constraint functions to construct the improved function,and the multiobjective optimization is transformed into the single objective optimization problem.Secondly,based on the cutting-planes model,the improved function is rewritten and approximated by the idea of local convexity and redistribution.Finally,the original problem is transformed into a series of sub-problems related to the convex piecewise linear model,laying the foundation for the further construction of the algorithm.In the third chapter,the parameter setting in the algorithm is introduced firstly.Next,the redistributed bundle algorithm is proposed to solve the nonsmooth multiobjective DC optimization.Finally,some remarks are given in detail.In the fourth chapter,the convergence analysis is carried out for the two situations that the algorithm may generate.In the end,it proves that the algorithm has better convergence whether the algorithm produces infinitely many serious steps or finite serious steps followed by infinitely many null steps.