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

In a chance constrained program (CCP), decision makers 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 ($\text{CVaR}$) has been known to be the best for more than a decade. This paper studies and generalizes the $\text{ALSO}-\text{X}$, originally proposed by Ahmed, Luedtke, SOng, and Xie in 2017, for solving a CCP. We first show that the $\text{ALSO}-\text{X}$ 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, $\text{ALSO}-\text{X}$ always outperforms the $\text{CVaR}$ approximation. We further show (i) sufficient conditions under which $\text{ALSO}-\text{X}$ can recover an optimal solution to a CCP; (ii) an equivalent bilinear programming formulation of a CCP, inspiring us to enhance $\text{ALSO}-\text{X}$ with a convergent alternating minimization method ($\text{ALSO}-\text{X}+$); and (iii) an extension of $\text{ALSO}-\text{X}$ and $\text{ALSO}-\text{X}+$ to distributionally robust chance constrained programs (DRCCPs) under the $\infty -$Wasserstein ambiguity set. Our numerical study demonstrates the effectiveness of the proposed methods.

Funding: This work was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2046426].

Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2021.2225.