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Problèmes d'approximation matricielle linéaires coniques: Approches par Projections et via Optimisation sous contraintes de semi-définie positivité

Abstract : In this thesis, we consider the study of the so-called linear conic nearness problems and the derivation of differents numerical approachs for solving them. We focused our attention on the projections based approach and the SemiDefinite Programming (SDP) one. In a normed vector space of matrices, a matrix nearness problem consists in finding a matrix having a known property $\mathcal(P)$, that is nearest, according to the space's norm, to a given matrix $A$. These problems, which appear in numerous differents fields, have been studied by \textsc(Higham) who proposed the following three-step solving procedure : existence and unicty of (optimal) solutions, caracterisation and explicit form of solutions, if possible, efficients algorithms for computing these solution. We have taken an euclidean framework, and considered the cases where the set of matrices having property $\mathcal(P)$ is described by affine and conic (convex) constraints. We call those problems (\it linear conic matrix nearness problems). We have taken as examples two nearness problems corresponding to convex sets well known in Convex Analysis for their "good" structure, but where an explicit solution of the nearness problem is hard. The first of these example is motivated by problems from Operations Researchs and Quantum Mechanics. It consists in finding the nearest doubly stochastic matrix to a given square matrix. The second one is the problem of calibrating a correlation matrice. This have a major importance in the analysis of the financial risk taken with a given choice of a stock asset. We study and derive for these linear conic matrix nearness problems two differents approachs. The first is primal : it consists in rewritting the problem as the one of finding a projection onto a convex set which is the intersection of "simple" convexs, meaning projections onto are easy. It allow us to propose an alternating projections algorithm, inspired by \textsc(Boyle and Dykstra)'s modified version of \textsc(Von Neumann)'s classical algorithm. The second is primal-dual, and is in the lineage of the recents advances in SemiDefinite Programming. It consists in deriving an interior point algorithm, within a new framework where Gauss-Newton search directions, computed by preconditionned conjugated gradients, are used insead of Newton ones and a crossover technique is introduced at the end of the algorithm leading asymptotically to a superlinear convergence. Numerical tests are performed as an illustration of the differents approachs and show the comparison between them from differents points of view. As an application, we present a generalization of \textsc(Blin)'s aggregation of preferences procedure using the nearest doubly stochastic matrix idea.
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Contributor : Pawoumodom Ledogada Takouda <>
Submitted on : Thursday, March 25, 2004 - 10:57:27 PM
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Pawoumodom Ledogada Takouda. Problèmes d'approximation matricielle linéaires coniques: Approches par Projections et via Optimisation sous contraintes de semi-définie positivité. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2003. Français. ⟨tel-00005469⟩



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