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arxiv: 2409.19794 · v2 · submitted 2024-09-29 · 🧮 math.OC

A graphical framework for global optimization of mixed-integer nonlinear programs

Danial Davarnia , Mohammadreza Kiaghadi , Junyuan Qiu This is my paper

Pith reviewed 2026-05-23 20:56 UTC · model grok-4.3

classification 🧮 math.OC
keywords mixed-integer nonlinear programmingdecision diagramsglobal optimizationcutting planesbranch-and-boundgraphical reformulation
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The pith

Decision diagrams reformulate MINLP constraints as graphs to generate convex outer approximations and solve general instances via spatial branch-and-bound.

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

The paper presents a framework that converts mixed-integer nonlinear program constraints into decision diagrams. These graphs support convexification steps and the generation of linear cutting planes that outer-approximate the feasible region. A spatial branch-and-bound procedure is then applied with explicit convergence guarantees. The method targets algebraic forms too complex for existing parsers and solvers. It also supplies a general-purpose decision-diagram technique that extends beyond previously restricted MINLP classes.

Core claim

By building decision diagrams directly from MINLP constraints, the framework produces convex relaxations, derives cutting planes for linear outer approximations, and runs a spatial branch-and-bound algorithm that converges to global optimality, allowing solution of instances from difficult unsolved classes in the MINLP Library that standard global solvers cannot admit.

What carries the argument

Decision diagrams that graphically represent MINLP constraints and from which convexifications and cutting planes are derived.

If this is right

  • MINLPs whose algebraic structure is intractable for standard convexification routines become admissible to global solution methods.
  • The spatial branch-and-bound scheme supplies finite convergence to the global optimum for the reformulated problems.
  • Decision-diagram techniques now apply to a general class of MINLPs rather than only specially structured cases.
  • Instances from one of the hardest unsolved categories in the MINLP Library can be solved to global optimality.

Where Pith is reading between the lines

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

  • The graphical representation may expose combinatorial structure that algebraic formulations conceal, potentially guiding tighter relaxations.
  • Hybrid algorithms that combine the diagram construction with existing algebraic solvers could enlarge the solvable MINLP set without replacing either technology.
  • The same reformulation steps might extend to related problem classes such as bilevel or stochastic programs whose constraints admit graphical encodings.

Load-bearing premise

Decision diagrams can be constructed and convexified efficiently for arbitrary complex algebraic MINLP constraints that defeat conventional parsers.

What would settle it

An MINLP instance for which the constructed decision diagram yields an invalid relaxation or for which the branch-and-bound procedure fails to converge on a problem whose optimum is already known.

Figures

Figures reproduced from arXiv: 2409.19794 by Danial Davarnia, Junyuan Qiu, Mohammadreza Kiaghadi.

Figure 3.1
Figure 3.1. Figure 3.1: Relaxed DD for the set in Example 3.1. According to Algorithm 2, the number of nodes at layer k + 1 for k ∈ [n − 1] of the DD obtained from this algorithm is bounded by |Uk||Lk|, where |Uk| is the number of nodes at layer k, and Lk is the number of sub-domain partitions for variable xk. As a result, the size of this DD grows exponentially as the number of layers increases. To control this growth rate, we… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Relaxed DD for the set in Example 3.4. Determining the variable sub-domains relative to a node requires accounting for all paths in the DD that pass through that node. This can become computationally intensive, especially for large-scale DDs. The following proposition offers an efficient method for calculating these bounds by leveraging the top-down construction process of DDs. Proposition 3.4 Consider a… view at source ↗
Figure 3
Figure 3. Figure 3: b. It is easy to verify that conv(Sol( [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: b. It is easy to verify that conv(Sol(D 3 )) =  (x1, x2, x3) ∈ [0, 2]×[0, 1]×[0, 1] [PITH_FULL_IMAGE:figures/full_fig_p018_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Relaxed DD for the set in Example 3.4. Theorem 3.2 implies that applying a merging operation at the layers of a DD that models non-separable functions can effectively reduce the size of the DD while still providing a relaxation for the underlying set. This result is analogous to Theorem 3.1 for separable functions. However, there is an interesting and notable difference in how the merging operation impac… view at source ↗
read the original abstract

While mixed-integer linear programming and convex programming solvers have advanced significantly over the past several decades, solution technologies for general mixed-integer nonlinear programs (MINLPs) have yet to reach the same level of maturity. Various problem structures across different application domains remain challenging to model and solve using modern global solvers, primarily due to the lack of efficient parsers and convexification routines for their complex algebraic representations. In this paper, we introduce a novel graphical framework for globally solving MINLPs based on decision diagrams (DDs), which enable the modeling of complex problem structures that are intractable for conventional solution techniques. We describe the core components of this framework, including a graphical reformulation of MINLP constraints, convexification techniques derived from the constructed graphs, efficient cutting plane methods to generate linear outer approximations, and a spatial branch-and-bound scheme with convergence guarantees. In addition to providing a global solution method for tackling challenging MINLPs, our framework addresses a longstanding gap in the DD literature by developing a general-purpose DD-based approach for solving general MINLPs. To demonstrate its capabilities, we apply our framework to solve instances from one of the most difficult classes of unsolved test problems in the MINLP Library, which are otherwise inadmissible for state-of-the-art global solvers.

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 paper introduces a graphical framework based on decision diagrams (DDs) for globally solving general mixed-integer nonlinear programs (MINLPs). Core components include a graphical reformulation of MINLP constraints, convexification techniques derived from the graphs, efficient cutting-plane methods to generate linear outer approximations, and a spatial branch-and-bound scheme with convergence guarantees. The approach is positioned as filling a gap in the DD literature by providing a general-purpose DD method for MINLPs and is demonstrated by solving instances from difficult classes of unsolved problems in the MINLP Library that are inadmissible for state-of-the-art global solvers.

Significance. If the central claims hold, the work is significant because it supplies a general-purpose DD-based global optimization method for MINLPs, a class where solution technology has lagged. Credit is due for the explicit development of the reformulation, convexification rules, cutting-plane procedure, and spatial B&B with convergence guarantees, as well as for the reported numerical solves on previously inadmissible MINLP Library instances. These elements directly address the longstanding gap noted in the DD literature and provide falsifiable evidence via the computational results.

major comments (2)
  1. [§4] §4 (convexification and cutting planes): the claim that the constructed DDs yield efficient linear outer approximations for arbitrary complex algebraic constraints requires explicit verification that the diagram construction and convexification steps remain polynomial or practical when the original algebraic expression is intractable for standard parsers; the numerical results on selected instances do not yet establish this for the general case asserted in the abstract.
  2. [§5] §5 (spatial B&B): the convergence guarantee is stated to hold, but the proof sketch should clarify whether finite termination or asymptotic convergence is obtained under the assumption that DD construction is exact for every nonconvex constraint encountered during branching; if the DD size grows exponentially for some MINLP structures, the practical convergence claim needs qualification.
minor comments (2)
  1. [Table 1] Table 1 (instance statistics): add columns reporting the number of nodes in the constructed DDs and the time spent on diagram construction versus the B&B phase to allow readers to assess the efficiency of the graphical reformulation step.
  2. [Notation] Notation section: the symbols used for the convex hull of the DD relaxation and the cutting-plane separation oracle are introduced without a consolidated table; a single notation table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and recommendation of minor revision. We address each major comment below with clarifications that will be incorporated into the revised manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (convexification and cutting planes): the claim that the constructed DDs yield efficient linear outer approximations for arbitrary complex algebraic constraints requires explicit verification that the diagram construction and convexification steps remain polynomial or practical when the original algebraic expression is intractable for standard parsers; the numerical results on selected instances do not yet establish this for the general case asserted in the abstract.

    Authors: We agree that the abstract asserts generality and that the numerical results are on selected instances. The framework targets constraints where DD construction from the algebraic form is feasible and yields compact diagrams (as demonstrated on the MINLP Library instances that defeat standard parsers). We will revise the abstract to qualify the scope and add a discussion in §4 on the dependence of construction time on expression structure, without claiming polynomial time for arbitrary intractable expressions. revision: yes

  2. Referee: [§5] §5 (spatial B&B): the convergence guarantee is stated to hold, but the proof sketch should clarify whether finite termination or asymptotic convergence is obtained under the assumption that DD construction is exact for every nonconvex constraint encountered during branching; if the DD size grows exponentially for some MINLP structures, the practical convergence claim needs qualification.

    Authors: The proof sketch establishes asymptotic convergence of the spatial branch-and-bound to the global optimum (as the tolerance tends to zero) under exact DD construction at each node. Finite termination is not claimed in general because of the continuous relaxation and possible exponential DD growth. We will revise §5 to state this distinction explicitly and add a qualification on practical performance when DD sizes remain manageable. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract and description outline a graphical DD-based reformulation, convexification rules, cutting-plane generation, and spatial B&B with convergence guarantees for general MINLPs. No load-bearing step is shown to reduce by construction to a fitted parameter, self-citation chain, or definitional renaming. The central claims rest on the explicit construction of the framework components and reported solves of previously inadmissible instances, which are presented as external validation rather than tautological outputs. This is the expected self-contained case with score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of efficient graphical reformulations and convexification routines for general MINLPs; no free parameters, ad-hoc axioms, or new physical entities are mentioned in the abstract.

axioms (1)
  • domain assumption Standard mathematical assumptions underlying mixed-integer nonlinear programming and decision diagram representations
    The framework presupposes that MINLP constraints admit decision-diagram encodings and that spatial branch-and-bound converges under the derived relaxations.
invented entities (1)
  • Graphical reformulation of MINLP constraints via decision diagrams no independent evidence
    purpose: To enable convexification and cutting-plane generation for structures intractable by conventional algebraic methods
    The paper introduces this as the core modeling device; no independent evidence outside the framework is provided in the abstract.

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