Axis-Aligned Relaxations for Mixed-Integer Nonlinear Programming

TL;DR

This paper introduces a new geometric relaxation framework for mixed-integer nonlinear programming that improves solution bounds by convexifying function graphs using computational geometry techniques, leading to better optimization performance.

Contribution

The paper presents a novel relaxation method based on convexifying finite point sets, new inequalities for product functions, and geometric tools to enhance MINLP solving.

Findings

Achieves 20-25% better optimality gap closure on polynomial problems.

Provides superior dual bounds on 30% of MINLPLib instances.

Reduces gaps by over 50% in some cases.

Abstract

We present a novel relaxation framework for general mixed-integer nonlinear programming (MINLP) grounded in computational geometry. Our approach constructs polyhedral relaxations by convexifying finite sets of strategically chosen points, iteratively refining the approximation to converge toward the simultaneous convex hull of factorable function graphs. The framework is underpinned by three key contributions: (i) a new class of explicit inequalities for products of functions that strictly improve upon standard factorable and composite relaxation schemes; (ii) a proof establishing that the simultaneous convex hull of multilinear functions over axis-aligned regions is fully determined by their values at corner points, thereby generalizing existing results from hypercubes to arbitrary axis-aligned domains; and (iii) the integration of computational geometry tools, specifically voxelization and QuickHull, to efficiently approximate feasible regions and function graphs. We implement this framework and evaluate it on randomly generated polynomial optimization problems and a suite of 619 instances from \texttt{MINLPLib}. Numerical results demonstrate significant improvements over state-of-the-art benchmarks: on polynomial instances, our relaxation closes an additional 20--25\% of the optimality gap relative to standard methods on half the instances. Furthermore, compared against an enhanced factorable programming baseline and Gurobi's root-node bounds, our approach yields superior dual bounds on approximately 30\% of \texttt{MINLPLib} instances, with roughly 10\% of cases exhibiting a gap reduction exceeding 50\%.