arxiv: 2605.04850 · v1 · submitted 2026-05-06 · 🧮 math.OC

Recognition: unknown

Out-of-the-Box Global Optimization for Packing Problems: New Models and Improved Solutions

Timo Berthold , Dominik Kamp , Gioni Mexi , Sebastian Pokutta , Imre Polik

Authors on Pith no claims yet

Pith reviewed 2026-05-08 15:56 UTC · model grok-4.3

classification 🧮 math.OC

keywords packing problemsnonlinear optimizationglobal optimizationcircle packingpolygon packingS-lemmaFarkas lemmaPlatonic solids

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The pith

Off-the-shelf global solvers find new packings for circles, polygons, and solids using direct nonlinear models.

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

The paper develops straightforward nonlinear optimization models for geometric packing problems across several classes. These cover circles packed into squares and fixed-perimeter rectangles, circles into minimum-area ellipses with a new containment constraint, regular polygons into regular polygons, and Platonic solids into Platonic solids. Formulations draw on the S-lemma for ellipse containment and the Farkas lemma for non-overlap constraints in polygons and solids. A sympathetic reader would care because the models, solved without custom solver changes, produce new best packings and first solutions for unstudied variants, illustrating that global nonlinear optimization can handle these nonconvex geometry tasks effectively.

Core claim

The paper establishes that direct nonlinear formulations of packing problems, using the S-lemma for ellipse containment and Farkas-lemma-based non-overlap constraints for polygons and solids, enable off-the-shelf global optimization solvers to obtain numerous new incumbent solutions as well as first solutions for previously unstudied variants, without problem-specific modifications beyond writing down the models. Computational results also examine the impact of formulation choices, and the work supplies further evidence that global nonlinear optimization has matured into a practical technology for highly nonconvex problems.

What carries the argument

Nonlinear programming models of the packing constraints, with an S-lemma-based containment formulation for ellipses and compact Farkas-lemma-based non-overlap formulations for polygons and Platonic solids.

If this is right

New best-known solutions for several circle-packing instances in squares and rectangles.
First feasible solutions obtained for some previously unstudied packing variants of polygons and solids.
Direct evidence that formulation variants affect solver success on these nonconvex problems.
Support for treating global nonlinear optimization as a ready-to-use tool for geometric packing without custom algorithmic work.

Where Pith is reading between the lines

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

The same modeling pattern could be tested on related problems such as packing with rotation or irregular containers.
Success on these instances suggests global solvers may now scale to modestly larger packing tasks where only nonlinear models were previously considered intractable.
If the Farkas-lemma non-overlap constraints generalize cleanly, they could simplify modeling for other rigid-body packing problems in three dimensions.

Load-bearing premise

The nonlinear models must accurately capture the geometric rules for non-overlap and containment, and the global solvers must be able to locate improved or optimal solutions in the resulting nonconvex problems.

What would settle it

For a small, well-studied packing instance whose optimal value is known independently, if the solver applied to one of these models returns either a feasible solution that violates non-overlap or a worse objective value than the known optimum, the central claim would be falsified.

read the original abstract

Recent LLM-driven discoveries have renewed interest in geometric packing problems. In this paper, we study several classes of such packing problems through the lens of modern global nonlinear optimization. Starting from comparatively direct nonlinear formulations, we consider packing circles in squares and fixed-perimeter rectangles, packing circles into minimum-area ellipses, packing regular polygons into regular polygons, and packing Platonic solids into Platonic solids. For ellipse packing, we derive a novel containment formulation based on the S-lemma. For polygon and Platonic solid packing, we develop compact non-overlap formulations based on the Farkas lemma and study several natural modeling variants computationally. Using off-the-shelf global optimization solvers, namely FICO Xpress and SCIP, we obtain numerous new incumbent solutions as well as first solutions for previously unstudied variants without any problem-specific solver modifications beyond writing down the models. Our computational results analyze the impact of various formulation choices. Beyond the individual packing results, the paper illustrates a broader point. It provides further evidence that global nonlinear optimization has matured into an increasingly practical model-and-solve technology for highly nonconvex problems.

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.

Desk Editor's Note private letter to a colleague

The paper gives new S-lemma and Farkas reformulations that let standard global solvers produce better packings for ellipses, polygons, and solids.

read the letter

The main thing to know is that the authors get new incumbent solutions and first results on some unstudied variants by writing compact nonlinear models and handing them to FICO Xpress and SCIP, without any custom solver changes. For ellipse containment they use the S-lemma to produce quadratic constraints. For polygon and Platonic solid non-overlap they add Farkas variables to get a compact system. They also compare several natural modeling variants and report how those choices affect solver performance. This supports their point that global nonlinear optimization has become practical for these nonconvex geometric problems. The computational evidence is the real contribution here, since the models are derived from standard lemmas and the solvers are off-the-shelf. The results look like genuine progress on the packing instances they tried. The potential soft spot is whether the reformulations stay exactly equivalent to the geometry in every case. The S-lemma equivalence needs a strict feasibility point, and the Farkas certificates assume the inequality systems meet the theorem's conditions on recession directions. If those fail for some configurations, the model could accept invalid points or exclude valid ones. The paper finds improved solutions, which suggests the models are reliable enough for the tested instances, but explicit checks that the solutions are geometrically feasible or that the conditions hold would make the claims tighter. Without seeing the full tables and verification steps it is hard to judge how much of the improvement is due to the new models versus solver luck. This is aimed at people working on nonlinear optimization or geometric packing. A reader who wants formulation ideas or new benchmark results will get something concrete from it. It deserves peer review because the modeling approach is fresh and the computational outcomes are specific, even if the equivalence details need a closer look in review.

Referee Report

3 major / 2 minor

Summary. The paper develops direct nonlinear programming formulations for several geometric packing problems: circles in squares and fixed-perimeter rectangles, circles in minimum-area ellipses (via a novel S-lemma containment model), and regular polygons/Platonic solids into like solids (via compact Farkas-lemma non-overlap constraints). Using unmodified off-the-shelf global solvers (FICO Xpress and SCIP), the authors report numerous new incumbent solutions and first solutions for previously unstudied variants, together with computational comparisons of formulation variants.

Significance. If the reformulations are shown to be exactly equivalent to the geometric constraints and the reported solutions are independently verified, the work supplies concrete evidence that modern global NLP solvers can be applied out-of-the-box to nonconvex packing problems to produce improved or first solutions. The explicit analysis of modeling variants and the absence of problem-specific solver tuning are strengths that support the broader claim about the maturing practicality of model-and-solve technology for such problems.

major comments (3)

[ellipse-packing section] The S-lemma containment formulation for ellipse packing (described in the ellipse-packing section) is invoked to certify that one quadratic set lies inside another. The equivalence holds only when a strict-feasibility condition is satisfied (there exists a point at which the constraining quadratic is negative). The manuscript does not explicitly verify or prove that this condition holds for all solved instances; without it, the NLP may admit geometrically invalid points, undermining the claimed new solutions.
[polygon and Platonic solid packing sections] The Farkas-lemma non-overlap formulations for polygons and Platonic solids (polygon-packing and Platonic-solid sections) introduce auxiliary variables for separation certificates. Equivalence to non-intersection requires that the original inequality systems satisfy the hypotheses of Farkas' lemma (e.g., no recession directions that render the alternative non-strict). The paper does not discuss or check these conditions for the considered shapes; violation could allow invalid configurations or exclude feasible ones.
[computational results section] The computational results section reports new incumbents and first solutions but provides neither the complete model files, exact solver parameter settings, nor certificates (e.g., primal/dual bounds or geometric feasibility checks) that would allow independent verification that the NLP solutions satisfy the original non-overlap and containment constraints.

minor comments (2)

[abstract and results summary] A summary table listing the number of new solutions per problem class, the previous best-known values, and the improvement margins would make the impact of the results clearer.
[throughout] Notation for decision variables (centers, radii, orientations) is introduced separately for each problem class; a unified notation table would improve readability across sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on the equivalence conditions of our formulations and on reproducibility. We address each major comment below.

read point-by-point responses

Referee: [ellipse-packing section] The S-lemma containment formulation for ellipse packing (described in the ellipse-packing section) is invoked to certify that one quadratic set lies inside another. The equivalence holds only when a strict-feasibility condition is satisfied (there exists a point at which the constraining quadratic is negative). The manuscript does not explicitly verify or prove that this condition holds for all solved instances; without it, the NLP may admit geometrically invalid points, undermining the claimed new solutions.

Authors: The referee correctly notes the strict-feasibility hypothesis required for equivalence in the S-lemma. In the ellipse-packing models, the inner quadratic inequality for each disk is strictly feasible at the disk center, where it evaluates to -r² < 0. All instances use non-degenerate ellipses of positive area containing circles of positive radius, so the condition holds for every solved instance. We will add an explicit remark in the ellipse-packing section confirming this fact for the computational study. This strengthens the justification but does not alter any reported solutions or models. revision: partial
Referee: [polygon and Platonic solid packing sections] The Farkas-lemma non-overlap formulations for polygons and Platonic solids (polygon-packing and Platonic-solid sections) introduce auxiliary variables for separation certificates. Equivalence to non-intersection requires that the original inequality systems satisfy the hypotheses of Farkas' lemma (e.g., no recession directions that render the alternative non-strict). The paper does not discuss or check these conditions for the considered shapes; violation could allow invalid configurations or exclude feasible ones.

Authors: We agree that Farkas' lemma requires the inequality systems to have no nonzero recession directions for full equivalence. The regular polygons and Platonic solids are compact convex bodies, so their inequality descriptions are bounded and possess no recession directions. This guarantees the required equivalence. We will insert a short clarifying discussion in the polygon-packing and Platonic-solid sections to state these facts and confirm they hold for the shapes studied. No changes to the computational results or models are needed. revision: partial
Referee: [computational results section] The computational results section reports new incumbents and first solutions but provides neither the complete model files, exact solver parameter settings, nor certificates (e.g., primal/dual bounds or geometric feasibility checks) that would allow independent verification that the NLP solutions satisfy the original non-overlap and containment constraints.

Authors: We accept that additional details improve verifiability. The revised manuscript will list the precise solver parameter settings used with FICO Xpress and SCIP. Complete model files for all instances will be supplied as supplementary material on the arXiv posting. The global solvers return primal-feasible points to the NLP; once the equivalence conditions (now to be stated explicitly) are satisfied, these points meet the original geometric constraints. We will also tabulate the final primal and dual bounds to support verification of optimality gaps. revision: yes

Circularity Check

0 steps flagged

No circularity: models derived from standard lemmas and solved with external solvers

full rationale

The derivation chain consists of applying known theorems (S-lemma for ellipse containment, Farkas lemma for polygon/Platonic non-overlap) to produce explicit nonlinear constraints, then feeding the resulting NLPs to independent commercial solvers (Xpress, SCIP). No parameter is fitted to the target packing outcomes, no result is renamed as a prediction, and no load-bearing premise reduces to a self-citation or self-definition. The reported incumbents are direct solver outputs on the externally formulated models.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard optimization lemmas to build new models but does not introduce new free parameters, invented entities, or ad-hoc axioms beyond those lemmas; the main addition is the application of these lemmas to create compact formulations solved by existing solvers.

axioms (2)

standard math S-lemma
Invoked to derive a novel containment formulation for packing circles into minimum-area ellipses.
standard math Farkas lemma
Used to develop compact non-overlap formulations for packing regular polygons and Platonic solids.

pith-pipeline@v0.9.0 · 5504 in / 1319 out tokens · 57510 ms · 2026-05-08T15:56:56.638651+00:00 · methodology

discussion (0)

Reference graph

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