Global Optimality Conditions And Optimization Methods For Some DC (Difference Of Convex Functions) Programming Problems

The objective function of the DC(Difference of convex)programming problem is the dif-ference of two convex functions.DC programming can be unconstrained or constrained opti-mization problems and it has a wide range of applications in the real world,such as economic,management and engineering etc.DC programming problem is known to be Nondeterministic Polynomial-time hard(NP-hard),in the past more than and thirty years,the research on the theory and algorithm of DC function and DC programming has attracted lots of attention of many scholars at home and abroad.In many literature,there are more and more algorithms to solve DC programming.In the various methods of solving DC programming,the cut-ting plane method,the outer approximation method,branch and bound algorithm,subgradient algorithm,DC algorithm are common methods.In this paper,based on the study of quadrat-ic programming,weakly convex programming and polynomial programming,combined with basic concepts and related properties,we study the global optimality conditions and global optimization methods for special kinds of DC programming problems.For these special kinds of DC programming problems,necessary global optimality con-ditions and sufficient global optimality conditions are investigated,.new local optimization methods and strongly local optimization methods are designed according to these conditions.These necessary global optimality conditions are generally stronger than Karush-Kuhn-Tucker(KKT)condition,therefore,these new local minimizers which is obtained by using the new local optimization methods may be better than the KKT points.Moreover,the filled function method is one of the practical auxiliary function methods which are used to achieve a global minimizer.In order to solve these special kinds of DC programming problems,we design the global optimization methods by combining the new local optimization methods,strongly local optimization methods and some auxiliary functions.We now proceed the contends of this paper as follows:In chapter 1,we briefly introduced the definition and properties of DC function and DC programming,and introduced the research actuality of the optimality conditions and optimiza-tion method of DC programming problems,to lay a good foundation for further study.In chapter 2,we consider a special DC programming problem with box constraints which are denoted by(DC1).A necessary global optimality conditions and sufficient global optimal-ity conditions for the problem(DC1)are established;And a new local optimization method(named strongly local optimization method)is proposed by exploiting the necessary global optimality conditions,this local optimization method is different from the traditional local optimization method,which is based on the Karush-Kuhn-Tucker(KKT)condition,and it's the improvement of the traditional local optimization method.Finally,combining the local optimization method,sufficient global optimality conditions and some auxiliary functions,we designed a global optimization method for the problem(DC1).Some numerical examples are given to show that this global optimization method is effective and stable.In chapter 3,based on the research of the chapter 2,we further investigated the opti-mality conditions and global optimization algorithms for a class of special DC programming problems(BDC)with box constraints.Firstly,we give the Alternating Direction Method of Multipliers for the problem(BDC)which are denoted by[NADMM],and strongly optimiza-tion method[SLOMA]is proposed by exploiting the necessary global optimality condition-s[GNC].Then,combining the algorithm[NADMM],strongly local optimization methods[SLOMA],sufficient global optimality conditions[GSC]and some auxiliary functions,we designed a global optimization method for the problem(BDC).Finally,some numerical examples are given to compare the global optimization algorithm[GOMA]with the global optimization algorithm[GOM],which is designed in the second chapter of this paper,and the numerical results demonstrate these two global optimization algorithms are effective and feasible.In chapter 4,we study the DC programming problem(QDC)with quadratic constraints.Firstly,by constructing small interval,the feasible region of the DC programming problem(QDC)is transformed into a box set.By constructing the box set in the global minimizer of the problem(QDC),and then considering the problem(QDC)in this box set,we derived the global necessary condition(named[QGNC])for the problem(QDC),And use the global necessary condition[QGNC],we design the strongly local algorithm for the problem(QDC).Then,combining the local algorithm and the filled function,we designed a global optimization algorithm of programming problem(QDC).Finally,some numerical examples are given to illustrate that the global optimization methods are efficient and stable.In chapter 5,we summarize the research in this article and make a prospect for the fourther research studying.