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Journal of Convex Analysis 29 (2022), No. 2, 411--442
Copyright Heldermann Verlag 2022

On the Convexity of Level-Sets of Probability Functions

Yassine Laguel
Université Grenoble Alpes, CNRS, Grenoble INP, LJK, France

Wim Van Ackooij
EDF R&D OSIRIS, Palaiseau, France

Jérôme Malick
Université Grenoble Alpes, CNRS, Grenoble INP, LJK, France

Guilherme Matiussi Ramalho
Federal University of Santa Catarina, LABPLAN, Santa Catarina, Brazil

In decision-making problems under uncertainty, probabilistic constraints are a valuable tool to express safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision vector. Even if the original set of inequalities is convex, this favourable property is not immediately transferred to the probabilistically constrained feasible set and may in particular depend on the chosen safety level. In this paper, we provide results guaranteeing the convexity of feasible sets to probabilistic constraints when the safety level is greater than a computable threshold. Our results extend all the existing ones and also cover the case where decision vectors belong to Banach spaces. The key idea in our approach is to reveal the level of underlying convexity in the nominal problem data (e.g., concavity of the probability function) by auxiliary transforming functions. We provide several examples illustrating our theoretical developments.

Keywords: Probability constraints, convex analysis, elliptical distributions, stochastic optimization.

MSC: 90C15, 90C25.

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