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

Abstract

We propose a new class of convex approximations for two-stage mixed-integer recourse models, the so-called generalized alpha-approximations. The advantage of these convex approximations over existing ones is that they are more suitable for efficient computations. Indeed, we construct a loose Benders decomposition algorithm that solves large problem instances in reasonable time. To guarantee the performance of the resulting solution, we derive corresponding error bounds that depend on the total variations of the probability density functions of the random variables in the model. The error bounds converge to zero if these total variations converge to zero. We empirically assess our solution method on several test instances, including the SIZES and SSLP instances from SIPLIB. We show that our method finds near-optimal solutions if the variability of the random parameters in the model is large. Moreover, our method outperforms existing methods in terms of computation time, especially for large problem instances.

An Improved Benders Dual Decomposition Method to Solve the Optimal Sizing Problem of Power-to-Ammonia Plants

“…The effectiveness of classical Benders decomposition relies on the convexity of the second stage cost function [16], which is not the case for the problem in this paper due to the existence of discrete variables in the second stage. To achieve optimality, non-linear cuts are usually required to establish a tight approximation to the second stage cost [17], [18]. The dual decomposition method is a canonical scenario-decomposition method based on Lagrangian relaxation.…”

“…V, the duality gap is not inconsequential if the BDD is used for some cases in this paper. The duality gap arises from the difference between the expected value of convex envelopes of the second stage cost function (which is obtained by the BDD method) and the convex envelopes of the expected value of the second stage cost function (which is needed) [17]. This key observation motivates us to use scenario bundling to establish a tighter convex envelope for the expected value of the second stage cost function, which will be illustrated in the next section.…”

On Generating Lagrangian Cuts for Two-Stage Stochastic Integer Programs

“…However, outside of those settings, additional cuts or a specialized branching scheme would be required to obtain a convergent algorithm. We refer the reader to, e.g., [25,12,14,11] for examples of methods that can be used to obtain a finitely convergent algorithm in other settings. Lagrangian cuts could potentially be added to enhance any of these approaches.…”

“…The primary difference between the exact separation ( 14) and the proposed restricted separation problem (15) is the restriction constraint (π, π 0 ) ∈ Π s . Ideally, we would like to choose Π s so that ( 15) is easy to solve, in particular avoiding evaluating the function Q * s too many times, while also yielding a cut (13) that has a similar strength compared with the Lagrangian cut corresponding to the optimal solution of (14). Our proposal that aims to satisfy these properties is to define Π s to be the span of past Benders cuts with some normalization.…”

“…Qi and Sen [12] use cuts valid for multi-term disjunctions introduced in [13] to derive an algorithm for SIPs with mixed-integer recourse. van der Laan and Romeijnders [14] propose a new class of cuts, scaled cuts, which integrates previously generated Lagrangian cuts into a cut-generation model, again leading to a finitely convergent cutting-plane method for SIPs with mixed-integer recourse. Bodur et al [15] compare the strength of split cuts derived in the Benders master problem space to those added in the scenario LP relaxation space.…”