Chance-constrained programs with convex underlying functions: a bilevel convex optimization perspective

Chance constraints are a valuable tool for the design of safe decisions in uncertain environments; they are used to model satisfaction of a constraint with a target probability. However, because of possible non-convexity and non-smoothness, optimizing over a chance constrained set is challenging. In...

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Vydáno v:	Computational optimization and applications Ročník 88; číslo 3; s. 819 - 847
Hlavní autoři:	Laguel, Yassine, Malick, Jérôme, van Ackooij, Wim
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.07.2024 Springer Nature B.V
Témata:	Algorithms Approximation Constraints Convex analysis Convex and Discrete Geometry Convexity Management Science Mathematics Mathematics and Statistics Operations Research Operations Research/Decision Theory Optimization Random variables Smoothness Statistics DC programming Chance constraints Bilevel optimization Convex optimization Stochastic programming
ISSN:	0926-6003, 1573-2894
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Abstract	Chance constraints are a valuable tool for the design of safe decisions in uncertain environments; they are used to model satisfaction of a constraint with a target probability. However, because of possible non-convexity and non-smoothness, optimizing over a chance constrained set is challenging. In this paper, we consider chance constrained programs where the objective function and the constraints are convex with respect to the decision parameter. We establish an exact reformulation of such a problem as a bilevel problem with a convex lower-level. Then we leverage this bilevel formulation to propose a tractable penalty approach, in the setting of finitely supported random variables. The penalized objective is a difference-of-convex function that we minimize with a suitable bundle algorithm. We release an easy-to-use open-source python toolbox implementing the approach, with a special emphasis on fast computational subroutines.
AbstractList	Chance constraints are a valuable tool for the design of safe decisions in uncertain environments; they are used to model satisfaction of a constraint with a target probability. However, because of possible non-convexity and non-smoothness, optimizing over a chance constrained set is challenging. In this paper, we consider chance constrained programs where the objective function and the constraints are convex with respect to the decision parameter. We establish an exact reformulation of such a problem as a bilevel problem with a convex lower-level. Then we leverage this bilevel formulation to propose a tractable penalty approach, in the setting of finitely supported random variables. The penalized objective is a difference-of-convex function that we minimize with a suitable bundle algorithm. We release an easy-to-use open-source python toolbox implementing the approach, with a special emphasis on fast computational subroutines.
Author	Malick, Jérôme Laguel, Yassine van Ackooij, Wim
Author_xml	– sequence: 1 givenname: Yassine orcidid: 0000-0001-6289-9590 surname: Laguel fullname: Laguel, Yassine email: yassine.laguel@univ-cotedazur.fr organization: Université Côte d’Azur – sequence: 2 givenname: Jérôme surname: Malick fullname: Malick, Jérôme organization: CNRS, LJK – sequence: 3 givenname: Wim orcidid: 0000-0002-9943-3572 surname: van Ackooij fullname: van Ackooij, Wim organization: EDF R &D. OSIRIS
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Cites_doi	10.1137/070702928 10.1007/s10957-009-9523-6 10.1007/s11228-020-00552-2 10.1007/BFb0120831 10.1287/educ.2013.0111 10.1287/educ.1080.0048 10.1007/s11228-020-00566-w 10.1007/s10107-004-0552-5 10.1007/978-3-319-93031-2_27 10.1007/s10898-019-00755-4 10.1007/s007800200072 10.1137/1.9781611971309 10.1023/A:1018901022726 10.1007/s10898-010-9579-8 10.1287/opre.2021.2225 10.1007/1-84628-095-8_2 10.1287/mnsc.6.1.73 10.1007/PL00011393 10.1287/moor.1050.0186 10.1287/opre.2019.1942 10.1007/s10107-014-0801-1 10.1007/s12532-020-00199-y 10.1137/050622328 10.1109/MLSP49062.2020.9231909 10.1109/TAC.2006.875041 10.1287/opre.1100.0910 10.1137/0331063 10.1111/j.1467-9965.2007.00311.x 10.1145/2833157.2833162 10.1137/19M1261985 10.1137/S1052623493257344 10.21314/JOR.2000.038 10.1007/s10589-007-9105-1 10.1007/978-94-017-3087-7
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Risk20002214210.21314/JOR.2000.038 – reference: JiangNXieWAlso-x and also-x+: better convex approximations for chance constrained programsOper. Res.202270635813600453853410.1287/opre.2021.2225 – reference: Kröger, O., Coffrin, C., Hijazi, H., Nagarajan, H.: Juniper: An open-source nonlinear branch-and-bound solver in julia. In: Integration of Constraint Programming, Artificial Intelligence, and Operations Research. Springer International Publishing (2018) – reference: Laguel, Y., Malick, J., Harchaoui, Z.: First-order optimization for superquantile-based supervised learning. In: 2020 IEEE 30th International Workshop on Machine Learning for Signal Processing (MLSP), pp. 1–6. IEEE (2020) – reference: Prékopa, A.: Stochastic Programming. Kluwer, Dordrecht (1995). https://doi.org/10.1007/978-94-017-3087-7 – reference: LaguelYvan AckooijWMalickJMatiussi RamalhoGOn the convexity of level-sets of probability functionsJ. 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Snippet	Chance constraints are a valuable tool for the design of safe decisions in uncertain environments; they are used to model satisfaction of a constraint with a...
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SubjectTerms	Algorithms Approximation Constraints Convex analysis Convex and Discrete Geometry Convexity Management Science Mathematics Mathematics and Statistics Operations Research Operations Research/Decision Theory Optimization Random variables Smoothness Statistics
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Title	Chance-constrained programs with convex underlying functions: a bilevel convex optimization perspective
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