Exact Hull Reformulation for Quadratically Constrained Generalized Disjunctive Programs

TL;DR

This paper introduces exact hull reformulations for quadratic disjunctive constraints in generalized disjunctive programming, improving solution robustness and efficiency by avoiding approximation issues.

Contribution

It develops the General Exact Hull Reformulation (GEHR) for non-convex quadratic constraints and the Conic Exact Hull Reformulation (CEHR) for convex quadratic constraints, both preserving solver-friendly structures.

Findings

GEHR and CEHR reduce numerical failures.

CEHR is most reliable and fastest for convex problems.

GEHR improves performance on non-convex instances.

Abstract

Generalized Disjunctive Programming (GDP) provides a natural framework for optimization models that combine logical decisions with nonlinear constraints. The Hull Reformulation (HR) is attractive because it yields tight continuous relaxations, but for nonlinear disjunctive constraints, it is commonly implemented using an epsilon-approximation of the closure of the perspective function. This approximation introduces fractional expressions, enlarges the relaxation for any non-zero value, can cause numerical instability, and may hinder solver convexity recognition. This paper develops a framework for constructing exact hull reformulations for GDPs with quadratic disjunctive constraints that avoids approximations while preserving solver-friendly structure. For general (possibly non-convex) quadratic constraints, we derive a General Exact Hull Reformulation (GEHR) that eliminates perspective division and preserves quadratic degree, and we prove that it is equivalent to the standard closed-perspective hull in lifted space. For convex quadratic constraints, we propose using a Conic Exact Hull Reformulation (CEHR), re-derived in this work, that represents the perspective term with rotated second-order cone constraints, enabling conic-capable solvers to certify and exploit convexity directly. We implement both reformulations in Pyomo.GDP and evaluate them on random convex and non-convex instances, a CSTR network benchmark, k-means clustering, and constrained layout problems using Gurobi, SCIP, and BARON. Across these benchmarks, the proposed exact formulations reduce numerical failures and often improve runtime relative to the epsilon-approximation. In particular, CEHR is consistently the most reliable and typically the fastest hull reformulation on convex benchmarks, while GEHR performs best on non-convex benchmarks, improving both robustness and overall performance.