Made available in DSpace on 2015-05-08T14:53:35Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 699035 bytes, checksum: 3300e24fe8473ff78b73d79a0e2d35cc (MD5) Previous issue date: 2014-02-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work an approach theory of geometrical problems and programming optimization theory Convex Difference Function (DC) is shown which is made of a class of geometric programming problems, known as Signomiais problems can be written as the difference convex functions and further, a DC problem can be written as CDC which is the canonical form of the problem DC. The advantage of writing in the form CDC is that one can find a global solution to this problem. The problem is solved by reference to Articles Nenad [20], Maranas [17] e Dembo [5]. / Neste trabalho é feita uma abordagem da teoria de Problemas de Programação Geométrica e da teoria de Otimização de Diferença de Funções Convexas(DC) onde mostrase que uma classe dos problemas de Programação Geométrica, conhecida por Problemas Signomiais, pode ser escrita como a diferença de funções convexas e mais adiante, que um problema DC pode ser escrito na forma CDC que é a forma canônica do problema DC. A vantagem de se escrever na forma CDC é que pode-se encontrar a solução global para este tipo de problema. Os problemas resolvidos tem como referência os artigos Nenad [20], Maranas [17] e Dembo [5].

Notre objectif dans cette thèse est de résoudre un problème quadratique binaire sous contraintes mixtes par les techniques d'optimisation DC. Puisque l'optimisation DC a prouvé son efficacité pour résoudre des problèmes de grandes tailles dans différents domaines, nous avons décidé d'appliquer cette approche d'optimisation pour résoudre ce problème. La partie la plus importante de l'optimisation DC est le choix d'une décomposition adéquate qui facilite la détermination et accélère la convergence de deux suites construites. La première suite converge vers la solution optimale du problème primal et la seconde converge vers la solution optimale du problème dual. Dans cette thèse, nous proposons deux décompositions DC efficaces et simples à manipuler. L'application de l'algorithme DC (DCA) nous conduit à résoudre à chaque itération un problème quadratique convexe avec des contraintes mixtes, linéaires et quadratiques. Pour cela, il faut trouver une méthode efficace et rapide pour résoudre ce dernier problème à chaque itération. Pour cela, nous appliquons trois méthodes différentes: la méthode de Newton, la programmation semi-définie positive et la méthode de points intérieurs. Nous présentons les résultats numériques comparatifs sur les mêmes repères de ces trois approches pour justifier notre choix de la méthode la plus rapide pour résoudre efficacement ce problème. / Our objective in this work is to solve a binary quadratic problem under mixed constraints by the techniques of DC optimization. As DC optimization has proved its efficiency to solve large-scale problems in different domains, we decided to apply this optimization approach to solve this problem. The most important part of D.C. optimization is the choice of an adequate decomposition that facilitates determination and speeds convergence of two constructed suites where the first converges to the optimal solution of the primal problem and the second converges to the optimal solution of the dual problem. In this work, we propose two efficient decompositions and simple to manipulate. The application of the DC Algorithm (DCA) leads us to solve at each iteration a convex quadratic problem with mixed, linear and quadratic constraints. For it, we must find an efficient and fast method to solve this last problem at each iteration. To do this, we apply three different methods: the Newton method, the semidefinite programing and interior point method. We present the comparative numerical results on the same benchmarks of these three approaches to justify our choice of the fastest method to effectively solve this problem.