Aspects théoriques et algorithmiques de l'optimisation semidéfinie. (Theoretical and Algorithmic Aspects in Semidefinite Programming)

This work deals with different subjects on nonlinear semidefinite programming (SDP). Thus, while in the first two chapters we show some algorithmic aspects, in chapters 3 and 4 we study theoretical aspects as the perturbation analysis of this problem. In the first chapter we develop a global algorithm that extends the local one S-SDP. This algorithm is based on a Han penalty function and a line search strategy. The second chapter is focused on penalty and barrier methods for solving convex semidefinite programming problems. We prove the convergence of primal and dual sequences obtained by this method. Moreover, we study the two parameters algorithm and extend to semidefinite case the results that are known in usual convex programming. In the second part, that involves chapters 3 and 4, we are interested on the characterization of the strong regularity property in function of second-order optimality conditions. So, in chapter 3, we mainly deal with second-order cone programming problems, whose are a particular instance of semidefinite programming problems. We thus obtain a characterization in this particular case. Finally in chapter 4, we give necessary and sufficient conditions to obtain the strong regularity property in the semidefinite case. However, its characterization is still an open problem.