# Méthodes de programmation en nombres mixtes pour l’optimisation parcimonieuse en traitement du signal

Abstract : Sparse approximation aims to fit a linear model in a least-squares sense, with a small number of non-zero components (the L0 norm''). Due to its combinatorial nature, it is often addressed by suboptimal methods. It was recently shown, however, that exact resolution could be performed through a mixed-integer program (MIP) reformulation solved by a generic solver, implementing branch-and-bound techniques. This thesis addresses the L0-norm sparse approximation problem with tailored branch-and-bound resolution methods, exploiting the mathematical structures of the problem. First, we show that each node evaluation amounts to solving an L1-norm problem, for which we propose dedicated methods. Then, we build an efficient exploration strategy exploiting the sparsity of the solution, by activating first the non-zero variables in the tree search. The proposed method outperforms the CPLEX solver, reducing the computation time and making it possible to address larger problems. In a second part of the thesis, we propose and study the MIP reformulations of the spectral unmixing problem with L0-norm sparsity more advanced structured sparsity constraints, which are usually addressed through relaxations in the literature. We show that, for problems with limited complexity (highly sparse solutions, good signal-to-noise ratio), such constraints can be accounted for exactly and improve the estimation quality over standard approaches.
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Cited literature [89 references]

https://hal.archives-ouvertes.fr/tel-02733897
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Submitted on : Monday, July 13, 2020 - 4:11:13 PM
Last modification on : Wednesday, October 13, 2021 - 3:52:05 PM

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• HAL Id : tel-02733897, version 2

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Ramzi Ben Mhenni. Méthodes de programmation en nombres mixtes pour l’optimisation parcimonieuse en traitement du signal. Traitement du signal et de l'image [eess.SP]. École Centrale de Nantes (ECN), 2020. Français. ⟨tel-02733897v2⟩

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