ALSO-X and ALSO-X+: Better Convex Approximations for Chance Constrained Programs

TL;DR

This paper improves convex approximation methods for chance constrained programs by analyzing, generalizing, and enhancing the ALSO-X approach, including extensions to distributionally robust cases, leading to better solutions and convergence guarantees.

Contribution

It provides a bilevel optimization interpretation of ALSO-X, proves its superiority over CVaR when constraints are convex, and introduces an enhanced ALSO-X+ method with convergence guarantees and robust extensions.

Findings

ALSO-X outperforms CVaR approximation for convex constraints.

Sufficient conditions for recovering optimal CCP solutions with ALSO-X.

ALSO-X+ converges via alternating minimization and extends to DRCCPs.

Abstract

In a chance constrained program (CCP), the decision-makers aim to seek the best decision whose probability of violating the uncertainty constraints is within the prespecified risk level. As a CCP is often nonconvex and is difficult to solve to optimality, much effort has been devoted to developing convex inner approximations for a CCP, among which the conditional value-at-risk (CVaR) has been known to be the best for more than a decade. This paper studies and generalizes the ALSO-X, originally proposed by Ahmed, Luedtke, SOng, and Xie (2017), for solving a CCP. We first show that the ALSO-X resembles a bilevel optimization, where the upper-level problem is to find the best objective function value and enforce the feasibility of a CCP for a given decision from the lower-level problem, and the lower-level problem is to minimize the expectation of constraint violations subject to the upper bound of the objective function value provided by the upper-level problem. This interpretation motivates us to prove that when uncertain constraints are convex in the decision variables, ALSO-X always outperforms the CVaR approximation. We further show (i) sufficient conditions under which ALSO-X can recover an optimal solution to a CCP; (ii) an equivalent bilinear programming formulation of a CCP, inspiring us to enhance ALSO-X with a convergent alternating minimization method (ALSO-X+); (iii) extensions of ALSO-X and ALSO-X+ to solve distributionally robust chance constrained programs (DRCCPs) under $\infty$-Wasserstein ambiguity set. Our numerical study demonstrates the effectiveness of the proposed methods.