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arxiv: 2506.03561 · v3 · pith:LUR7PDLZnew · submitted 2025-06-04 · 🧮 math.OC

Disjunctive Benders Decomposition

Kaiwen Fang , Inho Sin , Geunyeong Byeon This is my paper

Pith reviewed 2026-05-22 00:50 UTC · model grok-4.3

classification 🧮 math.OC
keywords Benders decompositiondisjunctive programmingconvex hull inequalitiescut-generating oraclesnormalization frameworkmixed-binary linear programsoptimization algorithms
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The pith

Disjunctive Benders decomposition generates convex hull inequalities for the reformulation by using existing cut oracles without changes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes an improved version of Benders decomposition that produces valid inequalities describing the full convex hull of the Benders reformulation. It combines disjunctive programming with the decomposition and provides a routine to use standard cut-generating oracles directly under a common normalization framework. A reader would care because this change allows the master problem to be solved as a linear program even when variables are binary and subproblems are separable, leading to fewer branch-and-bound nodes in practice.

Core claim

Our method integrates disjunctive programming with BD and introduces a routine that leverages existing cut-generating oracles as-is to construct convex hull inequalities. For mixed-binary linear programs, the approach removes the need to solve the master problem as a mixed-integer program, even with separable subproblems. It builds on a unified normalization framework for cut-generating programs, encompassing norm-based, reverse polar, and right-hand-side normalization.

What carries the argument

The routine leveraging existing cut-generating oracles as-is under a unified normalization framework to construct convex hull inequalities for the Benders reformulation.

If this is right

  • Master problems for mixed-binary linear programs can be solved as linear programs rather than mixed-integer programs.
  • Branch-and-bound requires substantially fewer nodes, often by orders of magnitude.
  • Outperforms commercial solvers on selected large-scale problem classes.
  • New normalization schemes can be designed with streamlined analysis of supporting cuts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach may enable decomposition methods to handle larger instances with more binary variables efficiently.
  • Connections to other disjunctive cutting plane methods in integer programming could be explored further.
  • Testing on problems with non-separable subproblems might reveal additional benefits or limitations.

Load-bearing premise

Cut-generating oracles can be applied directly without modification to yield valid convex hull inequalities when placed in the unified normalization framework.

What would settle it

Running the algorithm on a mixed-binary linear program and checking if the master problem requires solving as a mixed-integer program or if the cuts are not tight for the convex hull would falsify the claim.

Figures

Figures reproduced from arXiv: 2506.03561 by Geunyeong Byeon, Inho Sin, Kaiwen Fang.

Figure 1
Figure 1. Figure 1: Comparison between the proposed approach and conventional methods. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of BD’s behavior; horizontal (or vertical) axes for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of Routine 1 over iterations [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of Algorithm 3 on the motivating example with colors associate the iterates [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Progress of gap (%) to best known upper bounds: [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The graph of γ ′ x,j as a function of mj . Note that since lj = 0 for j ∈ I, δ 1 j and δ 2 j serve as slack variables in (2.1c) and (2.1d), respectively, for j ∈ I. Therefore, we can construct a valid cut γ ′ 0 − (γ ′ x ) ⊤x − γ ′ t t ≤ 0 for P (ϕ ′ ,ϕ′ 0 ) with    γ ′ 0 = ˆγ0, γ ′ x,j = max{aˆ 1 j − σˆ 1mj , aˆ 2 j + ˆσ 2mj}, ∀j ∈ I, γ ′ x,j = ˆγx,j , ∀j ∈ [nx] \ I, γ ′ t = ˆγt . Note that … view at source ↗
read the original abstract

We propose an enhancement to Benders decomposition (BD) that generates valid inequalities for the convex hull of the Benders reformulation, addressing the limitation that classical BD cuts are typically tight only for the continuous relaxation. Our method integrates disjunctive programming with BD and introduces a routine that leverages existing cut-generating oracles as-is to construct convex hull inequalities. For mixed-binary linear programs, the approach removes the need to solve the master problem as a mixed-integer program, even with separable subproblems. It builds on a unified normalization framework for cut-generating programs, encompassing norm-based, reverse polar, and right-hand-side normalization, and enabling the design of new normalization schemes with streamlined analysis of supporting cuts. Computational results on large-scale instances show substantial reductions in branch-and-bound nodes often by orders of magnitude, while consistently outperforming commercial solvers on selected problem classes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes Disjunctive Benders Decomposition, an enhancement to Benders decomposition that integrates disjunctive programming to generate valid inequalities for the convex hull of the Benders reformulation. It introduces a routine that leverages existing cut-generating oracles as-is under a unified normalization framework encompassing norm-based, reverse polar, and right-hand-side normalizations. For mixed-binary linear programs the approach removes the need to solve the master as a mixed-integer program even with separable subproblems, and computational results on large-scale instances report substantial reductions in branch-and-bound nodes, often by orders of magnitude, while outperforming commercial solvers on selected classes.

Significance. If the oracle-compatibility claim and convex-hull validity hold, the work could meaningfully advance practical Benders methods for large mixed-binary programs by allowing pure-LP master solves and delivering large node-count reductions. The unified normalization framework also supplies a clean setting for designing and analyzing new cut-generating schemes.

major comments (2)
  1. [§3.2] §3.2 (Disjunctive cut routine): the central claim that standard cut-generating oracles can be invoked 'as-is' under any of the three normalizations to produce inequalities valid and tight for the convex hull of the Benders master rests on an unverified interface assumption. The normalization change alters the right-hand side or constraint set of the oracle LP/MIP, yet the manuscript provides neither an explicit proof that the oracle output remains unchanged in validity nor a small-instance verification that the generated cuts are indeed supporting for the true convex hull when the disjunction is defined on the master binary variables.
  2. [§5] §5 (Computational experiments): the reported node reductions are load-bearing for the practical contribution, yet the section does not isolate whether the gains arise from the disjunctive convex-hull cuts versus other implementation details; a direct comparison against a baseline that adds generic disjunctive cuts without the Benders-specific oracle reuse would strengthen the cross-period and cross-solver claims.
minor comments (2)
  1. [Abstract] The abstract states that the method 'removes the need to solve the master problem as a mixed-integer program'; a forward reference to the precise theorem or proposition establishing this property would improve readability.
  2. [Notation] Notation for the Benders subproblem and the disjunction should be introduced once and used consistently; occasional redefinition of symbols across sections slows reading.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive review of our manuscript on Disjunctive Benders Decomposition. We address each major comment in detail below, clarifying the theoretical basis where appropriate and indicating revisions that will be incorporated to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Disjunctive cut routine): the central claim that standard cut-generating oracles can be invoked 'as-is' under any of the three normalizations to produce inequalities valid and tight for the convex hull of the Benders master rests on an unverified interface assumption. The normalization change alters the right-hand side or constraint set of the oracle LP/MIP, yet the manuscript provides neither an explicit proof that the oracle output remains unchanged in validity nor a small-instance verification that the generated cuts are indeed supporting for the true convex hull when the disjunction is defined on the master binary variables.

    Authors: We acknowledge that an explicit invariance argument would improve rigor. The unified normalization framework in §3.2 is constructed precisely so that each normalization (norm-based, reverse polar, RHS) is absorbed into the objective or right-hand side of the cut-generating program without changing the recession cone or the set of valid supporting inequalities for the disjunctive hull. In the revision we will insert a short lemma proving that any optimal solution of the normalized oracle yields a cut that remains valid and facet-defining for the convex hull of the Benders master when the disjunction is taken over the master binary variables. We will also add a small, fully worked numerical example (a 3-binary, 2-continuous instance) that explicitly compares the cuts obtained under each normalization against the true convex hull. revision: yes

  2. Referee: [§5] §5 (Computational experiments): the reported node reductions are load-bearing for the practical contribution, yet the section does not isolate whether the gains arise from the disjunctive convex-hull cuts versus other implementation details; a direct comparison against a baseline that adds generic disjunctive cuts without the Benders-specific oracle reuse would strengthen the cross-period and cross-solver claims.

    Authors: We agree that an explicit ablation would help readers attribute the observed speed-ups. The reported node-count reductions stem from two intertwined features: (i) the convex-hull cuts themselves and (ii) the fact that oracle reuse permits a pure LP master even when subproblems are separable. A generic disjunctive-cut generator without the Benders-specific normalization and oracle interface would not preserve this LP-master property. Nevertheless, to isolate the contribution we will add, in the revised §5, a controlled comparison on the same instance set against a baseline that applies standard disjunctive cuts (via the same normalization options) but without the Benders oracle-reuse mechanism. This will clarify the incremental benefit of the integrated framework. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on external disjunctive programming and normalization frameworks without self-referential reduction

full rationale

The paper's core contribution is an integration of disjunctive programming into Benders decomposition that reuses existing cut-generating oracles under a unified normalization scheme (norm-based, reverse polar, RHS). No quoted step defines a quantity in terms of itself, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose validity is internal to the present work. The method is presented as a distinct algorithmic routine whose correctness is argued via the properties of the cited external oracles and the disjunctive hull construction; these are treated as independent inputs rather than derived outputs. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach relies on standard properties of convex hulls and disjunctive programming but introduces a new routine for leveraging cut oracles; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Convex hulls of Benders reformulations admit valid inequalities generated via disjunctive programming
    Invoked to justify the generation of stronger cuts beyond continuous relaxations.

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Reference graph

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    35 Figure 6: The graph ofγ′ x,j as a function ofmj

    = (ei −m,0)for some integral vectorm∈Z nx withm j = 0forj̸∈ I: xi −m ⊤x≤0∨x i −m ⊤x≥1. 35 Figure 6: The graph ofγ′ x,j as a function ofmj. Note that sincelj = 0forj∈ I,δ 1 j andδ 2 j serve as slack variables in (2.1c) and (2.1d), respectively, forj∈ I. Therefore, we can construct a valid cutγ ′ 0 −(γ ′ x)⊤x−γ ′ tt≤0forP (ϕ′,ϕ′

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    Note thatγ ′ x,j is a function ofmj, as illustrated in Figure 6

    with    γ′ 0 = ˆγ0, γ′ x,j = max{ˆa1 j −ˆσ1mj,ˆa2 j + ˆσ2mj},∀j∈ I, γ′ x,j = ˆγx,j,∀j∈[n x]\ I, γ′ t = ˆγt. Note thatγ ′ x,j is a function ofmj, as illustrated in Figure 6. Sincexj ≥0forj∈ I, the lower the value of the correspondingγ′ x,j is, the tighter the cutγ′ 0 −(γ ′ x)⊤x−γ ′ tt≤0is. Thus, we choose the value ofm j so thatγ ′ x,j = max...

    work page