Geodesic
On the Convexity of Level-Sets of Probability Functions
Journal of convex analysis, Tome 29 (2022) no. 2, pp. 411-442
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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.
Classification : 90C15, 90C25
Mots-clés : Probability constraints, convex analysis, elliptical distributions, stochastic optimization 
@article{JCA_2022_29_2_JCA_2022_29_2_a7,
     author = {Y. Laguel and W. Van Ackooij and J. Malick and G. Matiussi Ramalho},
     title = {On the {Convexity} of {Level-Sets} of {Probability} {Functions}},
     journal = {Journal of convex analysis},
     pages = {411--442},
     year = {2022},
     volume = {29},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2022_29_2_JCA_2022_29_2_a7/}
}
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Y. Laguel; W. Van Ackooij; J. Malick; G. Matiussi Ramalho. On the Convexity of Level-Sets of Probability Functions. Journal of convex analysis, Tome 29 (2022) no. 2, pp. 411-442. http://geodesic.mathdoc.fr/item/JCA_2022_29_2_JCA_2022_29_2_a7/