ALSO-X is Better than CVaR: Convex Approximations for Chance Constrained Programs Revisited

Chance constrained programs (CCPs) are generic frameworks for decision-making under uncertain constraints. The objective of a CCP aims to find the best decision that violates the uncertainty constraints within the prespecified risk level. A CCP is often nonconvex and thus is difficult to solve to optimality. Alternatively, much literature has been attempted to develop convex inner approximation algorithms 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 algorithm, 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 search for the best objective function value and enforce the feasibility of the 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 distributionally robust chance constrained programs (DRCCP) under Wasserstein ambiguity set. Our numerical study demonstrates the effectiveness of the proposed algorithms.