Based on theoretical and algorithmic tools of DC programming and DCA, the research in this thesis focus on the local and global approaches for non convex optimization and global mixed integer optimization. The thesis consists of 5 chapters. The first chapter presents fundamentals of DC programming and DCA, and techniques of Branch and Bound method (B&B) for global optimization (using the DC relaxation technique for calculating lower bounds of the optimal value). It shall include results concerning the exact penalty technique in mixed integer programming. The second chapter is devoted of a DCA method for solving a class of NP-hard nonconvex nonlinear mixed integer programs. These nonconvex problems are firstly reformulated as DC programs via penalty techniques in DC programming so that the resulting DC programs are effectively solved by DCA and B&B well adapted. As a first application in financial optimization, we modeled the problem pf portfolio selection under concave transaction costs and applied DCA and B&B to its solutions. In the next chapter we study the modeling of the problem of minimization of nonconvex discontinuous transaction costs in portfolio selection in two forms: the first is a DC program obtained by approximating the objective function of the original problem by a DC polyhedral function and the second is an equivalent mixed 0-1 DC program. And we present DCA, B&B algorithm, and a combined DCA-B&B algorithm for their solutions. Chapter 4 studied the exact solution for the multi-objective mixed zero-one linear programming problem and presents two practical applications of proposed method. We are interested int the last chapter two challenging problems: the linear integer least squares problem and the Nonnegative Mattrix Factorization problem (NMF). The NMF method is particularly important because of its many various applications of the first are in telecommunications. The numerical simulations show the robustness, speed (thus scalability), performance, and the globality of DCA in comparison to existent methods.
DC programming and DCA for nonconvex optimization/ global optimization in mixed integer programming : Codes and applications