  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.
