Optimization Methods For Some Special DC Programming Problems

In the field of nonconvex programming,many nonconvex and nonsmooth problems can be transformed into DC(difference of convex)programming problems or approximated by DC programming problems.DC programming has good application prospects,such as economy,engineering,management,artificial intelligence and other fields.In practical application,it is often necessary to solve the global optimal solution of the problem,but at present,the biggest obstacle of solving the DC programming is that a feasible global optimality condition and optimal DC decomposition have not been found,and the existing algorithm is often unable to find the global optimal solution of the large-scale problem.Therefore,it is necessary to study the global optimization of DC programming.Codifferential is a kind of modification of quasidifferential,and widely used in nonconvex and nonsmooth functions.Because of the difficulty in calculating codifferential,many related optimality conditions cannot be verified,and the codifferential of polyhedral convex functions is easy to calculate.Therefore,this paper considers the DC programming problem in which one DC component is polyhedral convex function.Firstly,the problem of convex reduced polyhedral DC programming is considered,which is recorded as(PDC).In this paper,the special convex decomposition structure of DC pro-gramming problem is effectively used,and the original problem(PDC)is transformed into solving convex programming problem(Pi).On the basis of the concept of codifferential,the necessary and sufficient condition for the global optimality of the problem(PDC)is pro-posed,and based on this sufficient and necessary condition,the convex programming problem(Pi)global optimization method of[PGOMi]and prove[PGOMi].The global optimization method[DCGOM]for solving DC programming problem(PDC)is given.In particular,the first DC component of the objective function of the DC programming problem(PDC)is considered in the numerical experiment is the maximum value function of continuous differ-entiable convex and some continuous differentiable convex(?)hj(x).Finally,the numerical experiment results are compared with DCA algorithm to show the feasibility and effectiveness of the algorithmSecondly,the problem of polyhedral reduced continuous differentiable convex DC pro-gramming is considered,which is recorded as(MDC).Based on the concept of codifferential,a necessary condition of local optimality of the problem(MDC)is proposed,and the problem(MDC)is guaranteed to converge to a(?,?)stationary point.In this paper,the optimization method of the problem(MDC)is given,which is better than that of DCA.