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

Abstract

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 ([Formula: see text]) has been known to be the best for more than a decade. This paper studies and generalizes the [Formula: see text], originally proposed by Ahmed, Luedtke, SOng, and Xie in 2017 , for solving a CCP. We first show that the [Formula: see text] 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, [Formula: see text] always outperforms the [Formula: see text] approximation. We further show (i) sufficient conditions under which [Formula: see text] can recover an optimal solution to a CCP; (ii) an equivalent bilinear programming formulation of a CCP, inspiring us to enhance [Formula: see text] with a convergent alternating minimization method ([Formula: see text]); and (iii) an extension of [Formula: see text] and [Formula: see text] to distributionally robust chance constrained programs (DRCCPs) under the [Formula: see text]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 .