Parametric error bounds for convex approximations of two-stage mixed-integer recourse models with a random second-stage cost vector

TL;DR

This paper derives explicit, distribution-dependent error bounds for convex approximations of two-stage mixed-integer recourse models with uncertain second-stage costs, applicable to any distribution with finite expected -norm.

Contribution

It introduces parametric error bounds that explicitly depend on the distribution of the second-stage cost vector, extending previous bounds limited to finite support.

Findings

Error bounds scale linearly with the expected -norm of the cost vector

Bounds are valid for any distribution with finite expected -norm

Provides a more general understanding of approximation errors in stochastic mixed-integer models.

Abstract

We consider two-stage recourse models in which the second-stage problem has integer decision variables and uncertainty in the second-stage cost vector, technology matrix, and the right-hand side vector. Such mixed-integer recourse models are typically non-convex and thus hard to solve. There exist convex approximations of these models with accompanying error bounds. However, it is unclear how these error bounds depend on the distributions of the second-stage cost vector $q$. In fact, the only error bound that is known hinges on the assumption that $q$ has a finite support. In this paper, we derive parametric error bounds whose dependence on the distribution of $q$ is explicit and that hold for any distribution of $q$, provided it has a finite expected $\ell_1$-norm. We find that the error bounds scale linearly in the expected value of the $\ell_1$-norm of $q$.