Pseudo-Valid Cutting Planes for Two-Stage Mixed-Integer Stochastic Programs with Right-Hand-Side Uncertainty

Abstract

Cutting planes need not be valid in stochastic integer optimization. Many practical problems under uncertainty, for example, in energy, logistics, and healthcare, can be modeled as mixed-integer stochastic programs (MISPs). However, such problems are notoriously difficult to solve. In “Pseudo-Valid Cutting Planes for Two-Stage Mixed-Integer Stochastic Programs with Right-Hand-Side Uncertainty,” Romeijnders and van der Laan introduce a novel approach to solve two-stage MISPs. Instead of using exact cuts that are always valid, they propose to use pseudo-valid cutting planes for the second-stage feasible regions that may cut away feasible integer second-stage solutions for some scenarios and may be overly conservative for others. The advantage of using such cutting planes is that the approximating problem remains convex in the first-stage decision variables and thus can be solved efficiently. Moreover, the performance of these cutting planes is good if the variability of the random parameters in the model is large enough.

A loose Benders decomposition algorithm for approximating two-stage mixed-integer recourse models

“…This error bound converges to zero if the variability of the random parameters in the model increases. Romeijnders and van der Laan [21] derive similar error bounds for convex approximations that fit into a specific framework. In this framework, convex approximations are defined using pseudo-valid cutting planes for the second-stage feasible regions.…”

“…for some C > 0; see [21,Example 2] for details. Observe that the error bound goes to zero if σ i → ∞ for all i = 1, .…”

“…We illustrate these properties in Example 1. In this example, we also illustrate the difference between the generalized α-approximations and the pseudo-valid cutting plane approximation of [21].…”

“…We use this example, since it also appears in [23] and [21], allowing us to compare the generalized α-approximations to the shifted LP-relaxation and a pseudo-valid cutting plane approximation.…”

“…Both the shifted LP-relaxation of [23] and the cutting plane framework of [21], however, cannot be applied directly to efficiently solve MIR models in general. This is because the shifted LP-relaxation is very difficult to compute in general, as discussed in [23], and furthermore, the asymptotic error bound in the cutting plane framework of [21] only applies for tight pseudo-valid cutting planes, which are only available in special cases, e.g. for SIR models.…”