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

Ackooij, Wim van; Pérez-Aros, Pedro

doi:10.1007/s00245-022-09844-5

Cited by 5 publications

(4 citation statements)

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“…The next result corresponds to a gradient formula for the probability function using the spherical radial decomposition. We refer to [48], for similar results and further extensions of the next theorem.…”

Section: Spherical Radial Decomposition and Gradient Formula For Prob...mentioning

confidence: 68%

“…The following technical condition is used to obtain formulae for the gradient of probability functions (see [48]). It is worth to emphasize that the condition below is general enough to cover most of the known distributions.…”

Section: Spherical Radial Decomposition and Gradient Formula For Prob...mentioning

confidence: 99%

“…Theorem 2.4 (Corollary 3.2 [48]) Let H be a finite-dimensional Hilbert space, let x ∈ H be such that g(x, 0) < 0, and assume that g satisfies the η θ -growth condition at (x, v) for all v ∈ I(x). Then the probability function ϕ defined in (6) is continuously differentiable on an appropriate neighbourhood U ′ of x with…”

Section: Spherical Radial Decomposition and Gradient Formula For Prob...mentioning

confidence: 99%

“…. , s and consider the vector valued function Φ given in (48). Then the following are equivalent a) There exists a continuously differentiable convex function h : H → R such that Φ satisfies (10).…”

Section: Nonsmooth Inequality Constraintmentioning

confidence: 99%

See 3 more Smart Citations

Inner Moreau envelope of nonsmooth conic chance constrained optimization problems

Ackooij¹,

Pérez-Aros²,

Soto³

et al. 2023

Preprint

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Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth. This motivates us to propose a regularization employing the Moreau envelope of a scalar representation of the vector inequality. More precisely, we consider a probability function which covers most of the general classes of probabilistic constraints:where K is a convex cone of a Banach space. The conic inclusion Φ(x, ξ) ∈ −K represents an abstract system of inequalities, and ξ is a random vector . We propose a regularization by applying the Moreau envelope to the scalarization of the function Φ. In this paper, we demonstrate, under mild assumptions, the smoothness of such a regularization and that it satisfies a type of variational convergence to the original probability function. Consequently, when considering an appropriately structured problem involving probabilistic constraints, we can thus entail the convergence of the minimizers of the regularized approximate problems to the minimizers of the original problem. Finally, we illustrate our results with examples and applications in the field of (nonsmooth) joint, semidefinite and probust chance constrained optimization problems.

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Section: Spherical Radial Decomposition and Gradient Formula For Prob...mentioning

confidence: 68%

Section: Spherical Radial Decomposition and Gradient Formula For Prob...mentioning

confidence: 99%

Section: Spherical Radial Decomposition and Gradient Formula For Prob...mentioning

confidence: 99%

Section: Nonsmooth Inequality Constraintmentioning

confidence: 99%

See 2 more Smart Citations

Inner Moreau envelope of nonsmooth conic chance constrained optimization problems

Ackooij¹,

Pérez-Aros²,

Soto³

et al. 2023

Preprint

View full text Add to dashboard Cite

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Second-Order Numerical Variational Analysis

Mordukhovich

2024

Springer Series in Operations Research and Financial Engineering

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Control in Probability for SDE Models of Growth Population

2022

Self Cite

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In this paper, we consider a (control) optimization problem, which involves a stochastic dynamic. The model proposes selecting the best control function that keeps bounded a stochastic process over an interval of time with a high probability level. Here, the stochastic process is governed by a stochastic differential equation affected by a stochastic process. This setting becomes a chance-constrained control optimization problem, where the constraint is given by the probability level of infinitely many random inequalities. Since such a model is challenging, we discretize the dynamic and restrict the space of control functions to piecewise mappings. On the one hand, it transforms the infinite-dimensional optimization problem into a finite-dimensional one. On the other hand, it allows us to provide the well-posedness of the problem and approximation. Finally, the results are illustrated with numerical results, where classical model for the growth of a population are considered.

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

Cited by 5 publications

References 60 publications

Inner Moreau envelope of nonsmooth conic chance constrained optimization problems

Inner Moreau envelope of nonsmooth conic chance constrained optimization problems

Second-Order Numerical Variational Analysis

Control in Probability for SDE Models of Growth Population

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