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Contribution à l'étude de la robustesse et à la dualité en optimisation

Abstract : Duality and robustness are two important tools in decision making process. This thesis deals with tree topics : duality for an uncertain convex conical optimization problem, duality and regularity in generalized convexity, and the maximization of the stability radius. In the first part of this work, we consider the notions of worst value and robust value of an uncertain convex conical optimization problem. We give a necessary and sufficient condition to obtain the equality between the robust value and the worst value with exactness for the worst value. We derive a sufficient condition to obtain a robust strong duality property for this problem. The second part of this work is devoted to duality and regularity of the extended real-valued functions. Two points of view are considered: the sub-level set approach and the epigraphical approach. We then extend some recent results concerning the passage from the quasi-convex duality to convex duality to the generalized convexity. We apply this theory to an optimization problem to derive a strong duality property for this problem. The third part of this work is devoted to the study of the problem of maximization of the stability radius. We define the stability radius for a decision problem under data uncertainty, and study some of its analytical properties (e.g concavity and upper semi-continuity). The robust counterpart of an uncertain optimization problem according to the stability radius is introduced. We study the relation between the solution set of this counterpart and the solution set of the robust counterpart according to the robust optimization approach. We propose a generic model of the maximization of stability radius which covers a large class of applications. We study this problem in a polyhedral case, in the case of regression and in quadratic case. In each case, we compute the stability radius and/ or transform the problem of maximization of the stability radius to a tractable problem. An application to a circular antenna design problem is given in the regression case, and an application to compute a robust estimator is provided in the quadratic case.
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Moussa Barro. Contribution à l'étude de la robustesse et à la dualité en optimisation. Analyse classique [math.CA]. Université d'Avignon, 2016. Français. ⟨NNT : 2016AVIG0416⟩. ⟨tel-01668600⟩

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