Nonconvex and Nonsmooth Approaches for Affine Chance-Constrained Stochastic Programs

TL;DR

This paper introduces a novel approach for solving complex affine chance-constrained stochastic programs by combining approximation, sampling, penalization, and convexification techniques to handle nonconvexity and nondifferentiability.

Contribution

It develops a unified framework for affine chance constraints involving disjunctive nonconvex events and mixed-signed probabilities, integrating multiple known methods.

Findings

Effective convexification of nonconvex constraints.

Flexible approximation techniques for probability functions.

Enhanced computational strategies for complex stochastic programs.

Abstract

Chance-constrained programs (CCPs) constitute a difficult class of stochastic programs due to its possible nondifferentiability and nonconvexity even with simple linear random functionals. Existing approaches for solving the CCPs mainly deal with convex random functionals within the probability function. In the present paper, we consider two generalizations of the class of chance constraints commonly studied in the literature; one generalization involves probabilities of disjunctive nonconvex functional events and the other generalization involves mixed-signed affine combinations of the resulting probabilities; together, we coin the term affine chance constraint (ACC) system for these generalized chance constraints. Our proposed treatment of such an ACC system involves the fusion of several individually known ideas: (a) parameterized upper and lower approximations of the indicator function in the expectation formulation of probability; (b) external (i.e., fixed) versus internal (i.e., sequential) sampling-based approximation of the expectation operator; (c) constraint penalization as relaxations of feasibility; and (d) convexification of nonconvexity and nondifferentiability via surrogation. The integration of these techniques for solving the affine chance-constrained stochastic program (ACC-SP) with various degrees of practicality and computational efforts is the main contribution of this paper.