Presolve techniques for quasi-convex chance constraints with finite-support low-dimensional uncertainty
Guillaume Van Dessel, François Glineur
TL;DR
The paper tackles chance-constrained optimization with finite-support, where the constraint $c(x,\xi)\le 0$ is quasi-convex in $\xi$ and the uncertain vector $\xi$ has discrete support, by reformulating the problem as a big-$M$ mixed-integer convex program (MICP) using binaries $z_s$ and big-$M$ bounds $M^{(s)}$ to enforce the probabilistic constraint. To tame the combinatorial blow-up from many scenarios, it introduces presolve heuristics based on computational geometry that partition scenarios into a safe set $\oplus$ and a pruned set $\ominus$ (plus a selectable remainder), derive valid inequalities, and tighten the big-$M$ bounds using information collected during presolve, under Assumptions (A) and (B) that ensure feasibility and permit exploitation of convex sublevel sets of $c(x,\cdot)$. A key result shows that a scenario not in the selected set but lying in the convex hull of selected scenarios can be safely handled, enabling a representation of the optimum as $F^*_{\tau}=\min_{S\in\mathbf{T}(\oplus,\ominus)} \nu(S)$ and, equivalently, a domain-separable formulation $\mathcal{F}_{\tau}(x)=\min_{S\in\mathbf{S}_{\tau}} f_S(x)$ over convex domains $\mathcal{X}(S)$. The approach is demonstrated on Probabilistic Facility Location problems in dimensions $p=2$ and $p=3$, where presolve yields substantial speedups over direct solves and can solve instances within time limits that the naive approach struggles to reach, with the implementation and data publicly available on GitHub.
Abstract
Chance-constrained programs (CCP) represent a trade-off between conservatism and robustness in optimization. In many CCPs, one optimizes an objective under a probabilistic constraint continuously parameterized by a random vector $ξ$. In this work, we study the specific case where the constraint is quasi-convex with $ξ$. Moreover, the support of vector $ξ$ is a collection of $N$ scenarios in dimension $p=2$ or $p=3$. In general, even when both the constraint and the objective are convex in the decision variable, the feasible region of a CCP is nonconvex, turning it into a difficult problem. However, under mild assumptions, many CCPs can be recast as big-$M$ mixed-integer convex programs (MICP). Unfortunately, the difficulty of these MICPs explodes with the number of scenarios, restricting the instances practically solvable in decent time. To cut down the effective number of scenarios considered in MICP reformulations and accelerate their solving, we propose and test presolve techniques based on computational geometry. Our techniques produce certificates to discard or select a priori some scenarios before solving a regular MICP. Moreover, the information aggregated during presolve leverages the possibility to strengthen big-$M$ constants. Our numerical experiments suggest that spending some time in presolve is more efficient than a direct solve for a class of probabilistic projection problems, including an interesting type of facility location problem.