Stochastic optimization problems : decomposition and coordination under risk

Abstract : We consider stochastic optimization and game theory problems with risk measures. In a first part, we focus on time consistency. We begin by proving an equivalence between time consistent mappings and the existence of a nested formula. Motivated by well-known examples in risk measures, we investigate three classes of mappings: translation invariant, Fenchel-Moreau transform and supremum mappings. Then, we extend the concept of time consistency to player consistency, by replacing the sequential time by any unordered set and mappings by any relations. Finally, we show how player consistency relates to sequential and parallel forms of decomposition in optimization. In a second part, we study how risk measures impact the multiplicity of equilibria in dynamic game problems in complete and incomplete markets. We design an example where the introduction of risk measures leads to the existence of three equilibria instead of one in the risk neutral case. We analyze the ability of two different algorithms to recover the different equilibria. We discuss links between player consistency and equilibrium problems in games. In a third part, we study distribution ally robust optimization in machine learning. Using convex risk measures, we provide a unified framework and propose an adapted algorithm covering three ambiguity sets discussed in the literature
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Henri Gérard. Stochastic optimization problems : decomposition and coordination under risk. Optimization and Control [math.OC]. Université Paris-Est, 2018. English. ⟨NNT : 2018PESC1111⟩. ⟨tel-02067326⟩

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