Distributionally robust chance-constrained programs with right-hand side uncertainty under Wasserstein ambiguity

Abstract

We consider exact deterministic mixed-integer programming (MIP) reformulations of distributionally robust chance-constrained programs (DR-CCP) with random right-hand sides over Wasserstein ambiguity sets. The existing MIP formulations are known to have weak continuous relaxation bounds, and, consequently, for hard instances with small radius, or with large problem sizes, the branch-and-bound based solution processes suffer from large optimality gaps even after hours of computation time. This significantly hinders the practical application of the DR-CCP paradigm. Motivated by these challenges, we conduct a polyhedral study to strengthen these formulations. We reveal several hidden connections between DR-CCP and its nominal counterpart (the sample average approximation), mixing sets, and robust 0–1 programming. By exploiting these connections in combination, we provide an improved formulation and two classes of valid inequalities for DR-CCP. We test the impact of our results on a stochastic transportation problem numerically. Our experiments demonstrate the effectiveness of our approach; in particular our improved formulation and proposed valid inequalities reduce the overall solution times remarkably. Moreover, this allows us to significantly scale up the problem sizes that can be handled in such DR-CCP formulations by reducing the solution times from hours to seconds.

Supplementary Numerical Results

In Table 5 we report the performance of the basic formulation (5) when combined with mixing (16) and path inequalities (21) for $F=5$, $D=50$, $\epsilon =0.1$. These results highlight that mixing and path inequalities are indeed useful when applied to the basic formulation, without all the other enhancements we propose.

Table 5 Supplementary results for $F=5$, $D=50$, $\epsilon =0.1$

We see that for $N=100$, Basic+Mixing+Path solves all instances (and with slightly quicker times on average) whereas Basic in Table 1 does not manage to solve any instances for $\theta _1$. For $N=1000$, we again see an improvement in the number of instances solved when using the inequalities, but for larger $\theta _8,\theta _9,\theta _{10}$, times are slightly slower. We believe this is due to the extra time required to solve the larger LP relaxations as a result of adding cuts. For $N=3000$, we now see that Basic+Mixing+Path is unable to find a feasible integer solution within 1 h for a larger number of instances than Basic (we again believe this is due to larger LP relaxations), but whenever it does, it often solves to optimality, which Basic never does.

In contrast to Improved, there are more cuts generated for Basic for all $\theta$ values. Nevertheless, as expected, the performance achieved by Basic+Mixing+Path is still worse than the performance of our improved formulation (20), with or without inequalities.