Efficient Separation of RLT Cuts for Implicit and Explicit Bilinear Terms

TL;DR

This paper enhances the RLT technique for bilinear terms by detecting implicit products in MILPs and introducing a new efficient cut separation algorithm, significantly improving computational performance.

Contribution

It introduces a method to detect implicit bilinear relations in MILPs and a novel RLT cut separation algorithm to reduce computational costs.

Findings

Improved detection of implicit bilinear relations in MILPs.

Enhanced RLT cut separation efficiency across various bilinear terms.

Significant computational performance improvements demonstrated.

Abstract

The reformulation-linearization technique (RLT) is a prominent approach to constructing tight linear relaxations of non-convex continuous and mixed-integer optimization problems. The goal of this paper is to extend the applicability and improve the performance of RLT for bilinear product relations. First, a method for detecting bilinear product relations implicitly contained in mixed-integer linear programs is developed based on analyzing linear constraints with binary variables, thus enabling the application of bilinear RLT to a new class of problems. Our second contribution addresses the high computational cost of RLT cut separation, which presents one of the major difficulties in applying RLT efficiently in practice. We propose a new RLT cutting plane separation algorithm which identifies combinations of linear constraints and bound factors that are expected to yield an inequality that is violated by the current relaxation solution. This algorithm is applicable to RLT cuts generated for all types of bilinear terms, including but not limited to the detected implicit products. A detailed computational study based on implementations in two solvers evaluates the performance impact of the proposed methods.