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Nonlinear Chance Constrained Problems: Optimality Conditions, Regularization and Solvers

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  • Lukáš Adam

    (Czech Academy of Sciences)

  • Martin Branda

    (Czech Academy of Sciences
    Charles University)

Abstract

We deal with chance constrained problems with differentiable nonlinear random functions and discrete distribution. We allow nonconvex functions both in the constraints and in the objective. We reformulate the problem as a mixed-integer nonlinear program and relax the integer variables into continuous ones. We approach the relaxed problem as a mathematical problem with complementarity constraints and regularize it by enlarging the set of feasible solutions. For all considered problems, we derive necessary optimality conditions based on Fréchet objects corresponding to strong stationarity. We discuss relations between stationary points and minima. We propose two iterative algorithms for finding a stationary point of the original problem. The first is based on the relaxed reformulation, while the second one employs its regularized version. Under validity of a constraint qualification, we show that the stationary points of the regularized problem converge to a stationary point of the relaxed reformulation and under additional condition it is even a stationary point of the original problem. We conclude the paper by a numerical example.

Suggested Citation

  • Lukáš Adam & Martin Branda, 2016. "Nonlinear Chance Constrained Problems: Optimality Conditions, Regularization and Solvers," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 419-436, August.
    • Handle: RePEc:spr:joptap:v:170:y:2016:i:2:d:10.1007_s10957-016-0943-9
    DOI: 10.1007/s10957-016-0943-9
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    References listed on IDEAS

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    Cited by:

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    4. Lukáš Adam & Martin Branda & Holger Heitsch & René Henrion, 2020. "Solving joint chance constrained problems using regularization and Benders’ decomposition," Annals of Operations Research, Springer, vol. 292(2), pages 683-709, September.
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    6. Roya Karimi & Jianqiang Cheng & Miguel A. Lejeune, 2021. "A Framework for Solving Chance-Constrained Linear Matrix Inequality Programs," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1015-1036, July.

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