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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 (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 in 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+); and (iii) an extension of ALSO−X and ALSO−X+ to distributionally robust chance constrained programs (DRCCPs) under the ∞−Wasserstein ambiguity set. Our numerical study demonstrates the effectiveness of the proposed methods.
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