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Journal of Optimization Theory and Applications

, Volume 170, Issue 2, pp 419–436 | Cite as

Nonlinear Chance Constrained Problems: Optimality Conditions, Regularization and Solvers

  • Lukáš Adam
  • Martin Branda
Article

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.

Keywords

Chance constrained programming Optimality conditions  Regularization Algorithms Free MATLAB codes 

Mathematics Subject Classification

90C15 90C26 49M05 

Notes

Acknowledgments

The authors gratefully acknowledge the support from the Grant Agency of the Czech Republic under Project 15-00735S.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Information Theory and Automation (ÚTIA)Czech Academy of SciencesPrague 8Czech Republic
  2. 2.Department of Probability and Mathematical Statistics, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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