On Solving Chance-Constrained Models with Gaussian Mixture Distribution

TL;DR

This paper develops mixed-integer quadratic programming methods to efficiently approximate and solve linear chance-constrained problems with Gaussian mixture coefficients, achieving high accuracy and near-optimal solutions in reasonable time.

Contribution

It introduces piecewise linear approximations for Gaussian mixture chance constraints and demonstrates their effectiveness through extensive computational experiments.

Findings

Problems with up to 1000 coefficients and 15 mixture components solved within 18 hours.

Fewer coefficients and mixture components lead to faster solution times.

Sample average approximation methods are less effective for these problems.

Abstract

We study linear chance-constrained problems where the coefficients follow a Gaussian mixture distribution. We provide mixed-binary quadratic programs that give inner and outer approximations of the chance constraint based on piecewise linear approximations of the standard normal cumulative density function. We show that $O\left(\sqrt{\ln(1/\tau)/\tau} \right)$ pieces are sufficient to attain $\tau$-accuracy in the chance constraint. We also show that any desired optimality gap can be achieved under a constraint qualification condition by controlling the approximation accuracy. Extensive computations using a commercial solver show that problems with up to one thousand random coefficients specified with up to fifteen Gaussian mixture components, generated under diverse settings, can be solved to near optimality within 18 hours, while satisfying chance constraint satisfaction probabilities of up to $0.999$. The solution times are significantly lower for problems with fewer random coefficients and mixture terms. For example, problems with one hundred random coefficients, ten mixture terms, and a constraint satisfaction probability of $0.999$ can be solved in a minute or less. Sample average approximations fail to provide meaningful solutions even for the smaller problems.