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Nonlinear chance constrained problems: optimality conditions, regularization and solvers

Lukas Adam (adam***at***utia.cas.cz)
Martin Branda (branda***at***utia.cas.cz)

Abstract: We deal with chance constrained problems (CCP) 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 (MPCC) and regularize it by enlarging the set of feasible solutions. For all considered problems, we derive necessary optimality conditions based on Fr\'echet 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, whilst 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

Category 1: Stochastic Programming

Citation: L. Adam, M. Branda: Nonlinear chance constrained problems: optimality conditions, regularization and solvers. Journal of Optimization Theory and Applications, DOI 10.1007/s10957-016-0943-9

Download: [PDF]

Entry Submitted: 07/08/2015
Entry Accepted: 07/08/2015
Entry Last Modified: 05/16/2016

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