Generalized Differentiation of Probability Functions: Parameter Dependent Sets Given by Intersections of Convex Sets and Complements of Convex Sets

In this work we consider probability functions working on parameter dependent sets that are given as an intersection of convex sets and their complements. Such an underlying structure naturally arises when having to handle bilateral inequality systems in various applications, such as energy. We prov...

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Bibliographic Details
Published in:Applied mathematics & optimization 2022-02, Vol.85 (1), Article 2
Main Authors: van Ackooij, Wim, Pérez-Aros, Pedro
Format: Article
Language:English
Subjects:
Applied mathematics
Approximation
Calculus of Variations and Optimal Control
Optimization
Control
Convexity
Intersections
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Normal distribution
Numerical and Computational Physics
Optimization
Parameters
Simulation
Systems Theory
Theoretical
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Summary:In this work we consider probability functions working on parameter dependent sets that are given as an intersection of convex sets and their complements. Such an underlying structure naturally arises when having to handle bilateral inequality systems in various applications, such as energy. We provide conditions under which the probability functions are sub-differentiable as well as a constraint qualification condition under which the probability function turns out to be smooth. Our analysis allows for (nearly) arbitrary distributions of random vectors.
ISSN:0095-4616
1432-0606
DOI:10.1007/s00245-022-09844-5