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Aspects théoriques et algorithmiques de l'optimisation semidéfinie.

Abstract : This work deals with a number of subjects on nonlinear semidefinite programming (SDP). In the first two chapters, we consider the problem from an algorithmic standpoint while in chapters 3 and 4 we study theoretical aspects, in particular, giving a perturbation analysis of the problem. In the first chapter we develop a global algorithm that extends the local S-SDP algorithm. This algorithm is based on a Han penalty function and a line search strategy. The second chapter focuses on penalty and barrier methods for solving convex semidefinite programming problems. We prove the convergence of primal and dual sequences obtained by this method. In addition, we study the two-parameter algorithm and extend results from the usual convex programming setting to the semidefinite case. In the second part, which comprises chapters 3 and 4, we are interested in the characterization of the strong regularity property in terms of second-order optimality conditions. In chapter 3, we restrict our attention to second-order cone programming problems. These are a particular instance of semidefinite programming problems and we are able to obtain a characterization in this particular case. Finally in chapter 4, we give necessary and sufficient conditionsto obtain the strong regularity property in the semidefinite case. However, its characterization in terms of second-order optimality conditions is still an open problem.
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Submitted on : Tuesday, July 27, 2010 - 10:37:47 AM
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Hector Ramirez-Cabrera. Aspects théoriques et algorithmiques de l'optimisation semidéfinie.. Algorithme et structure de données [cs.DS]. Ecole Polytechnique X, 2005. Français. ⟨pastel-00001048⟩

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