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Descent dynamical systems and algorithms for tame optimization and multi-objective problems.

Abstract : In a first part, we focus on gradient dynamical systems governed by nonsmooth but also nonconvex functions, satisfying the so-called Kurdyka- Lojasiewicz inequality. After obtaining preliminary results for a continuous steepest descent dynamic, we study a general descent algorithm. We prove, under a compactness assumption, that any sequence generated by this general scheme converges to a critical point of the function to be minimized. We also obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward-backward method, with variable metric and relative errors. As an example, a nonsmooth and nonconvex version of the Levenberg-Marquardt algorithm is detailed. Applications to nonconvex feasibility problems, and to sparse inverse problems are discussed. In a second part, the thesis explores descent dynamics associated to constrained vector optimization problems. For this, we adapt the classic steepest descent dynamic to functions with values in a vector space ordered by a closed convex cone with nonempty interior. It can be seen as the continuous analogue of various descent algorithms developed in the last years. We have a particular interest for multi-objective decision problems, for which the dynamic make decrease all the objective functions along time. We prove the existence of trajectories for this continuous dynamic, and show their convergence to weak effcient points. Then, we explore an inertial dynamic for multi-objective problems, with the aim to provide fast methods converging to Pareto points.
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Contributor : Guillaume Garrigos Connect in order to contact the contributor
Submitted on : Friday, December 18, 2015 - 12:54:08 PM
Last modification on : Tuesday, July 5, 2022 - 10:22:18 AM
Long-term archiving on: : Saturday, March 19, 2016 - 11:00:18 AM


  • HAL Id : tel-01245406, version 1



Guillaume Garrigos. Descent dynamical systems and algorithms for tame optimization and multi-objective problems.. Optimization and Control [math.OC]. Université de Montpellier; Universidad Tecnica Federico Santa Maria, 2015. English. ⟨tel-01245406⟩



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