Automating Idealness Proofs for Binary Programs with Application to Rectangle Packing

TL;DR

This paper introduces an optimization framework for identifying ideal mixed binary linear programs, applied to rectangle packing problems, including variants with clearances, and analyzes their computational performance.

Contribution

It develops new methods for finding ideal MBLPs in rectangle packing and examines their effectiveness with existing and novel formulations.

Findings

Identifies limitations of Gurobi's default branch-and-cut for these problems.

Proposes formulations that better capture the disjunctive structure.

Addresses packing with clearance constraints.

Abstract

We develop an optimization framework for identifying ideal Mixed Binary Linear Programs (MBLP) which is linear when using known input data and nonconvex quadratic over parametric input data. These techniques are applied to various formulations for rectangle packing, conjectured to be pairwise-ideal. Additionally, we address a variation of the rectangle packing problem which incorporates clearances along selected edges of the packed objects. We present both existing and novel MBLP formulations for the underlying disjunctive program and investigate the poor performance of Gurobi's default branch-and-cut methodology. We operate under a strip-packing objective that aims to minimize the overall height of the packed objects.