标题： ALSO-X#：分布鲁棒机会约束规划的更好凸近似 显示英文标题

标题： ALSO-X#: Better Convex Approximations for Distributionally Robust Chance Constrained Programs

摘要： 本文研究了分布鲁棒机会约束规划（DRCCPs），其中不确定约束必须对于Wasserstein模糊集中的所有概率分布至少满足一个预设的阈值概率。 由于DRCCPs通常是非凸的且难以最优求解，研究人员一直在开发各种凸的内近似方法。 最近，ALSO-X已被证明在确定集为凸的情况下，优于常规机会约束规划的条件风险价值（CVaR）近似。 在这项工作中，我们通过引入一种新的ALSO-X#方法来解决DRCCPs，放松了这一假设。 即，在ALSO-X和CVaR近似的双层重构中，我们观察到下层的ALSO-X是下层CVaR近似的一个特例，而上层的CVaR近似比ALSO-X中的更受限制。 这一观察促使我们提出了ALSO-X#，它仍然类似于双层结构——在下层问题中，我们采用更一般的CVaR近似，而在上层问题中，我们选择更不受限的ALSO-X。 我们证明了ALSO-X#始终优于CVaR近似，并且在常规机会约束规划和类型$\infty-$Wasserstein模糊集下可以优于ALSO-X。 我们还提供了新的充分条件，使得ALSO-X#能够输出DRCCP的最优解。 我们将提出的ALSO-X#应用于无线通信问题，并数值证明了其解的质量甚至可以优于精确方法。 显示英文摘要

摘要： This paper studies distributionally robust chance constrained programs (DRCCPs), where the uncertain constraints must be satisfied with at least a probability of a prespecified threshold for all probability distributions from the Wasserstein ambiguity set. As DRCCPs are often nonconvex and challenging to solve optimally, researchers have been developing various convex inner approximations. Recently, ALSO-X has been proven to outperform the conditional value-at-risk (CVaR) approximation of a regular chance constrained program when the deterministic set is convex. In this work, we relax this assumption by introducing a new ALSO-X\# method for solving DRCCPs. Namely, in the bilevel reformulations of ALSO-X and CVaR approximation, we observe that the lower-level ALSO-X is a special case of the lower-level CVaR approximation and the upper-level CVaR approximation is more restricted than the one in ALSO-X. This observation motivates us to propose the ALSO-X\#, which still resembles a bilevel formulation -- in the lower-level problem, we adopt the more general CVaR approximation, and for the upper-level one, we choose the less restricted ALSO-X. We show that ALSO-X\# can always be better than the CVaR approximation and can outperform ALSO-X under regular chance constrained programs and type $\infty-$Wasserstein ambiguity set. We also provide new sufficient conditions under which ALSO-X\# outputs an optimal solution to a DRCCP. We apply the proposed ALSO-X\# to a wireless communication problem and numerically demonstrate that the solution quality can be even better than the exact method.