On Solving Chance-Constrained Models with Gaussian Mixture Distribution

TL;DR

This work tackles linear chance-constrained optimization under Gaussian mixture uncertainty by introducing mixed-integer quadratic programs that approximate the chance constraint through piecewise-linear approximations of the standard normal CDF $\Phi$. It proves that $O\left(\sqrt{\ln(1/\tau)/\tau}\right)$ breakpoints suffice to achieve $\tau$-accuracy and that, under a constraint qualification, any desired optimality gap can be obtained by controlling approximation fidelity. The authors develop two complementary formulations, PWL-O (outer) and PWL-I (inner), enabling provably valid outer bounds and feasible inner approximations solved with standard MIQP/MILP solvers, respectively. Extensive computational experiments with up to $n=1000$ variables and $K\le 15$ mixture components show that PWL methods rapidly approach near-optimal solutions with high probability of constraint satisfaction (up to $0.999$) within 18 hours, while sample-based SAA struggles to deliver reliable feasibility or optimality. The practical impact is a scalable, accurate framework for risk-constrained decision-making under multi-modal uncertainties that outperforms SAA in both efficiency and reliability, and lays groundwork for extensions to discrete decisions and distributionally robust settings.

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 $O\left(\sqrt{\ln(1/τ)/τ} \right)$ pieces are sufficient to attain $τ$-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.

Figures (6)

• Figure 1: Plots of $p_1(\bm{x})$, $p_2(\bm{x})$ (left) and $p(\bm{x})$ (right) over $\bm{x} \in \mathcal{X}$.
• Figure 2: Contour plots of $p_1(\bm{x})$, $p_2(\bm{x})$ (left) and $p(\bm{x})$ (right) over $\bm{x} \in \mathcal{X}$.
• Figure 3: Probability density of $\bm{\xi}_k^\top \bm{x}$ for $k=1, 2, \ldots, 5$ in blue, green, cyan, magenta, and orange colors, respectively, and of $\bm{\xi}^\top \bm{x}$ in black color. Each plot uses a distinct $\bm{x}$ sampled from the feasible region.
• Figure 4: Piecewise linear outer approximation of $\Phi$.
• Figure 5: CDF of standard Normal distribution and its second derivative

Theorems & Definitions (31)

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