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Exact Hull Reformulation for Quadratically Constrained Generalized Disjunctive Programs

Sergey Gusev, David E. Bernal Neira

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.

Exact Hull Reformulation for Quadratically Constrained Generalized Disjunctive Programs

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.

Paper Structure

This paper contains 42 sections, 3 theorems, 55 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. GDP to MINLP Reformulation Strategies
  3. Big-M Reformulation
  4. Hull Reformulation
  5. Reformulation of Disjunctions
  6. Convex and Non-Convex Constraints
  7. Numerical Challenges and $\varepsilon$-approximation
  8. Trade-offs and Practical Considerations
  9. Binary Multiplication Reformulation
  10. Hull Reformulations of Quadratically Constrained GDP
  11. General Exact Hull Reformulation of Quadratically Constrained GDP
  12. Conic Exact Hull Reformulation of Convex Quadratically Constrained GDP
  13. Conic Exact Hull (without factorization)
  14. Explicit Rotated Second Order Cone form via factorization
  15. Extension to general second-order cone representable constraints
  16. ...and 27 more sections

Key Result

Proposition 1

For a quadratic constraint $h(\mathbf v)\;=\;\mathbf v^{\mathsf T}Q\mathbf v + \mathbf{c}^{\mathsf T}\mathbf v + d \leq 0,$ define the sets where and Then, the two sets coincide: $S_1 = S_2$

Figures (10)

  • Figure 1: Illustration of the effect of the $\varepsilon$-approximation on the feasible region of the continuous relaxation resulting from HR applied to convex and non-convex GDP. The figure shows the exact feasible region with the enlarged region obtained using the $\varepsilon$-approximation, highlighting how the approximation leads to a relaxation that is larger than the exact one and weakens as $\varepsilon$ increases. The Big-M relaxation is not shown, as its feasible region would significantly exceed the plotted area, taking up most of the figure at this scale and axis limits.
  • Figure 2: Cumulative number of instances solved versus solution time for different reformulations applied to random convex quadratically constrained GDP problems. Results are shown separately for Gurobi, BARON, and SCIP. Each curve represents a distinct reformulation strategy.
  • Figure 3: Cumulative number of instances solved versus solution time for different reformulations applied to random non-convex quadratically constrained GDP problems. Results are shown separately for Gurobi, BARON, and SCIP. Each curve represents a distinct reformulation strategy.
  • Figure 4: Superstructure representation for the continuously stirred tank reactor (CSTR) network problem. The configuration includes $N_T$ reactor stages arranged in series, where each stage can either be a reactor or a bypass. The flow proceeds from left to right, and a recycle stream from the last reactor is allowed to return to any upstream stage. This flexible structure enables the optimizer to choose the optimal number of reactors and the location of recycle to minimize the total reactor volume ovalleLogicBasedDiscreteSteepestDescent2025.
  • Figure 5: Cumulative number of instances solved versus solution time for different reformulations applied to the CSTR network problem formulated as GDP. Results are shown separately for Gurobi, BARON, and SCIP. Each curve represents a distinct reformulation strategy.
  • ...and 5 more figures

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2: Conic epigraph reformulation of HR
  • proof
  • Proposition 3
  • proof