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Mixed-Integer Reaggregated Hull Reformulation of Special Structured Generalized Linear Disjunctive Programs

Albert Joon Lee, David E. Bernal Neira

TL;DR

Generalized Disjunctive Programming (GDP) blends algebraic constraints with disjunctions, but classical reformulations either yield weak relaxations (Big-M) or bloated models (Hull). The authors introduce a reaggregated hull reformulation (RHR) for GLDP with shared left-hand-side matrices $A_k$, grounded in convex-hull theory and the time-slot approach, to produce tight yet compact MILPs. They extend the technique to two benchmark problems—single-unit scheduling and two-dimensional strip packing—and demonstrate through extensive experiments that RHR consistently tightens relaxations and reduces solve times, often outperforming Hull and matching or beating Big-M. The work provides a practical reformulation framework for GDPs with shared-coefficient structure and suggests broad applicability to other piecewise-linear and disjunctive optimization problems. The approach offers a scalable path to leverage strong convex hull properties while maintaining tractable model sizes for commercial solvers.

Abstract

Generalized Disjunctive Programming (GDP) provides a powerful framework for combining algebraic constraints with logical disjunctions. To solve these problems, mixed-integer reformulations are required, but traditional reformulation schemes, such as Big-M and Hull, either yield a weak continuous relaxation or result in a bloated model size. Castro and Grossmann showed that scheduling problems can be formulated as GDP by modeling task orderings as disjunctions with algebraic timing constraints. Moreover, in their work, a particular representation of the single-unit scheduling problem, namely using a time-slot concept, can be reformulated as a tight yet compact mixed-integer linear program with notable computational performance. Based on that observation, and focusing on the case where the constraints in disjunctions are linear and share the same coefficients, we connect the characterization of the convex hull of these disjunctive sets by Jeroslow and Blair with Castro and Grossmann's time-slot reaggregation strategy to derive a unified reformulation methodology. We test this reformulation in two problems, single-unit scheduling and two-dimensional strip-packing. We derive new formulations of the general precedence concept of single-unit scheduling and symmetry-breaking formulations of the strip-packing problem, yielding mixed-integer programs with strong theoretical guarantees, particularly compact formulations in terms of continuous variables, and efficient computational performance when solving them with commercial mixed-integer solvers for these problems.

Mixed-Integer Reaggregated Hull Reformulation of Special Structured Generalized Linear Disjunctive Programs

TL;DR

Generalized Disjunctive Programming (GDP) blends algebraic constraints with disjunctions, but classical reformulations either yield weak relaxations (Big-M) or bloated models (Hull). The authors introduce a reaggregated hull reformulation (RHR) for GLDP with shared left-hand-side matrices , grounded in convex-hull theory and the time-slot approach, to produce tight yet compact MILPs. They extend the technique to two benchmark problems—single-unit scheduling and two-dimensional strip packing—and demonstrate through extensive experiments that RHR consistently tightens relaxations and reduces solve times, often outperforming Hull and matching or beating Big-M. The work provides a practical reformulation framework for GDPs with shared-coefficient structure and suggests broad applicability to other piecewise-linear and disjunctive optimization problems. The approach offers a scalable path to leverage strong convex hull properties while maintaining tractable model sizes for commercial solvers.

Abstract

Generalized Disjunctive Programming (GDP) provides a powerful framework for combining algebraic constraints with logical disjunctions. To solve these problems, mixed-integer reformulations are required, but traditional reformulation schemes, such as Big-M and Hull, either yield a weak continuous relaxation or result in a bloated model size. Castro and Grossmann showed that scheduling problems can be formulated as GDP by modeling task orderings as disjunctions with algebraic timing constraints. Moreover, in their work, a particular representation of the single-unit scheduling problem, namely using a time-slot concept, can be reformulated as a tight yet compact mixed-integer linear program with notable computational performance. Based on that observation, and focusing on the case where the constraints in disjunctions are linear and share the same coefficients, we connect the characterization of the convex hull of these disjunctive sets by Jeroslow and Blair with Castro and Grossmann's time-slot reaggregation strategy to derive a unified reformulation methodology. We test this reformulation in two problems, single-unit scheduling and two-dimensional strip-packing. We derive new formulations of the general precedence concept of single-unit scheduling and symmetry-breaking formulations of the strip-packing problem, yielding mixed-integer programs with strong theoretical guarantees, particularly compact formulations in terms of continuous variables, and efficient computational performance when solving them with commercial mixed-integer solvers for these problems.
Paper Structure (15 sections, 3 theorems, 26 equations, 6 figures, 1 table)

This paper contains 15 sections, 3 theorems, 26 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalized Disjunctive Programming
  3. Mixed-integer Programming Reformulations of GDP
  4. Single-Unit Scheduling as a GDP
  5. General Precedence Concept
  6. Immediate Precedence Concept
  7. Time-Slot Concept
  8. Mixed-integer programming formulations from GDP models
  9. Applications of the Time-Slot Concept
  10. Contributions of this work
  11. Methodology: Reaggregation Approach
  12. Results and Discussion
  13. Single-Unit Scheduling
  14. Strip-Packing Model
  15. Conclusion and Future Work

Key Result

Theorem 2.1

Let $F = \bigcup_{j \in D_k} P_{jk}, \text{ where } P_{jk} = \{ \mathbf{x} | \mathbf{A}_{jk}\mathbf{x}\leq \mathbf{b}_{jk} \} \text{ and } P_{jk} \neq \emptyset$. The convex hull of $F$ is given by the set of $(\hat{\mathbf{x}}_{jk},y_{jk}) \in \mathbb{R}^{n+1}, k \in K, \ j \in D_k$ and $\mathbf{x}

Figures (6)

  • Figure 1: General precedence for single unit problem. Each Boolean variable determines whether job $i$ is scheduled before job $j$, or job $j$ before job $i$.
  • Figure 2: Immediate precedence concept for single unit problem. Disjunctions assign each job a relative position by enforcing pairwise orderings and selecting exactly one first and one last job.
  • Figure 3: Time slot concept for single unit problem. Each job is assigned to exactly one time slot, establishing an implicit job sequence based on slot order.
  • Figure 4: Two-dimensional strip-packing problem. (a) Schematic of the strip-packing setup, where unrotatable rectangles are placed in a strip of fixed width $W$ and unbounded length; variables $(x_i, y_i)$ specify the bottom-left corner of each rectangle. (b) Example of a feasible packing configuration, where the resulting strip length $\ell_t$ is minimized subject to boundary and non-overlap constraints.
  • Figure 5: Benchmark run times for strip packing formulations (Gurobi). The $x$-axis label “Number of Rectangles” indicates the problem size. At each size, six bars are shown in the following order: \ref{['strip:s0']} BM, \ref{['strip:s0']} HR, \ref{['strip:s0']} RHR, \ref{['strip:s1']} BM, \ref{['strip:s1']} HR, and \ref{['strip:s1']} RHR, where \ref{['strip:s0']} is the standard strip-packing model and \ref{['strip:s1']} is the symmetry-breaking model. Bars report the average solution times (seconds) over ten instances per size, with error bars indicating $\pm$1 standard deviation.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 2.1: Theorem 3.3 in balas1985disjunctive, also in balas1974disjunctive
  • proof
  • Corollary 2.1
  • Theorem 2.2: Theorem 2.2 in jeroslow1988simplification, Theorem 3 and 4 in blair1990representation
  • proof