arxiv: 2604.17872 · v1 · submitted 2026-04-20 · 💻 cs.NE

Recognition: unknown

On Scalability of Multi-Objective Evolutionary Algorithms on Combinatorial Optimisation Problems

Menghao Tang , Zimin Liang , Miqing Li

Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:01 UTC · model grok-4.3

classification 💻 cs.NE

keywords multi-objective evolutionary algorithmscombinatorial optimizationscalabilitySEMOcrossover operatorconvergence speedPareto front

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The pith

SEMO slows in convergence on large combinatorial problems due to missing crossover, which when added improves speed but reduces solution spread.

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

This paper examines how multi-objective evolutionary algorithms scale on combinatorial optimization problems as their size grows from 50 to 5,000 variables. It shows that the simple SEMO algorithm loses ground in convergence speed compared to NSGA-II, SMS-EMOA and MOEA/D. The main cause identified is SEMO's lack of a crossover operator. Adding crossover to SEMO speeds up convergence substantially on these larger problems, although the change harms the even spread of solutions along the Pareto front.

Core claim

Our results show that SEMO experiences a decline in convergence speed as dimensionality increases, compared to other MOEAs such as NSGA-II, SMS-EMOA and MOEA/D. We further demonstrate that the absence of crossover is a major contributor to SEMO's underperformance in large-scale problems, and that incorporating crossover into SEMO can substantially accelerate convergence in general, despite being detrimental in spreading solutions over the Pareto front.

What carries the argument

The crossover operator that recombines parent solutions to create offspring, whose presence or absence controls convergence speed versus solution spread in MOEAs on high-dimensional combinatorial problems.

If this is right

SEMO performs well on small MOCOPs but loses relative advantage as problem size increases.
Adding crossover to SEMO raises its convergence rate on large problems.
Crossover in SEMO narrows the spread of solutions across the Pareto front.
MOEAs that already use crossover maintain steadier performance as dimensionality grows.

Where Pith is reading between the lines

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

Future MOEA designs for large combinatorial tasks could combine crossover for speed with separate diversity mechanisms to preserve spread.
The pattern may guide algorithm choice for real-world problems where both quick convergence and broad trade-off coverage are needed.
Repeating the tests on even larger instances or additional combinatorial problem families would test how far the crossover effect generalizes.

Load-bearing premise

The chosen benchmark problems, performance metrics and algorithm implementations are representative of how MOEAs generally behave on large combinatorial optimization problems.

What would settle it

Running the same scalability tests on a new set of combinatorial problems with 5,000 variables and finding that SEMO without crossover converges at least as fast as the version with crossover.

Figures

Figures reproduced from arXiv: 2604.17872 by Menghao Tang, Miqing Li, Zimin Liang.

**Figure 2.** Figure 2: Solution sets obtained by the four algorithms in a representative run under two budgets of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Solution sets obtained by the four algorithms in a representative run under a budget of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Solution sets obtained by the four algorithms in a representative run under two budgets of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Solution sets obtained by the four algorithms in a representative run under two budgets of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Solution sets obtained by SEMO and SEMOx, along with the other three MOEAs, in a representative run [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: Solution sets obtained by SEMO and SEMOx on the MONK with 5,000 decision variables under two [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

read the original abstract

Scalability of evolutionary algorithms refers to assessing how their performance changes as problem size increases. In the area of multi-objective optimisation, research on the scalability of multi-objective evolutionary algorithms (MOEAs) has predominantly focussed on continuous problems. However, multi-objective combinatorial optimisation problems (MOCOPs) differ from continuous ones. Their discrete and rigid structure often brings rugged landscape, numerous local optimal solutions and disjoint global optimal regions. This leads to different behaviour of MOEAs. For example, SEMO, a simple MOEA without mating selection and diversity maintenance mechanisms, has been shown to be highly competitive, and in many cases to outperform more sophisticated MOEAs on MOCOPs. Yet, it remains unclear whether such findings hold for large-scale cases. In this paper, we conduct an empirical investigation into the scalability of MOEAs on combinatorial problems, with problem size from 50 to 5,000. Our results show that SEMO experiences a decline in convergence speed as dimensionality increases, compared to other MOEAs such as NSGA-II, SMS-EMOA and MOEA/D. We further demonstrate that the absence of crossover is a major contributor to SEMO's underperformance in large-scale problems, and that incorporating crossover into SEMO can substantially accelerate convergence in general, despite being detrimental in spreading solutions over the Pareto front.

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

SEMO loses its edge on large MOCOPs as size grows to 5000, with crossover fixing convergence but hurting spread; new data but thin on setup details.

read the letter

The main takeaway is that SEMO, long competitive on small combinatorial problems, shows slower convergence than NSGA-II, SMS-EMOA, and MOEA/D once problem size reaches thousands. The authors link this drop to the lack of crossover and demonstrate that adding it speeds convergence, though it reduces solution spread on the Pareto front. This extends earlier small-scale findings with fresh runs up to n=5000, which is the clearest new element here. The direct comparison and the crossover ablation give a usable signal on operator effects at scale. The experiments appear straightforward and avoid obvious circularity since they rely on direct runs rather than fitted models. The soft spots sit in the experimental protocol. Without numbers on runs, statistical tests, exact problem instances, or how evaluation budgets scale with n, it is difficult to judge whether the ranking holds beyond the chosen test suite or if fixed budgets create artifacts. The stress-test point on representativeness is fair to raise until those details appear. If the benchmarks have particular structure that interacts with mating operators, the scalability claim could narrow. This paper is for people who pick or hybridize MOEAs for large combinatorial tasks in operations research. It offers practical pointers on when to modify simple algorithms like SEMO. It deserves peer review so the experimental choices can be checked and perhaps expanded with more instances or scaled budgets.

Referee Report

2 major / 2 minor

Summary. The manuscript presents an empirical study on the scalability of multi-objective evolutionary algorithms (MOEAs) for combinatorial optimization problems (MOCOPs). It compares SEMO with NSGA-II, SMS-EMOA, and MOEA/D on problems with dimensions ranging from 50 to 5000, finding that SEMO's convergence speed decreases with increasing size. The study further shows that incorporating crossover into SEMO improves convergence speed but reduces the spread of solutions on the Pareto front.

Significance. If the results hold under representative conditions, this work is significant for extending MOEA scalability research from continuous to combinatorial domains, where rugged landscapes and local optima are prevalent. It provides direct empirical evidence (via algorithm runs without fitted parameters) on the role of crossover in large-scale MOCOPs and challenges the competitiveness of simple MOEAs like SEMO at scale. This could inform operator design in evolutionary computation.

major comments (2)

[Experimental Methodology] Experimental Methodology section: The stopping criteria and evaluation budgets are not described as scaling with problem dimensionality n (from 50 to 5000). If budgets are fixed rather than increased proportionally, the reported decline in SEMO convergence speed relative to NSGA-II/SMS-EMOA/MOEA/D could be an artifact of insufficient search effort at large n, rather than a general property of MOCOP scalability.
[Results and Discussion] Results and Discussion: The specific MOCOP benchmark instances (e.g., multi-objective knapsack variants or other discrete problems) and performance metrics are not justified as representative of general combinatorial optimization behavior. Without this, the claim that absence of crossover is the major contributor to SEMO underperformance, and that adding it accelerates convergence, risks being tied to the chosen test suite rather than broadly applicable.

minor comments (2)

[Abstract] The abstract and introduction could more explicitly state the exact problem classes and metrics to aid readers in assessing generalizability.
[Throughout] Notation for algorithms and operators (e.g., SEMO variants with/without crossover) should be standardized across sections for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify key aspects of our empirical study on MOEA scalability for large-scale MOCOPs. We address each major comment point by point below, with planned revisions where appropriate.

read point-by-point responses

Referee: [Experimental Methodology] Experimental Methodology section: The stopping criteria and evaluation budgets are not described as scaling with problem dimensionality n (from 50 to 5000). If budgets are fixed rather than increased proportionally, the reported decline in SEMO convergence speed relative to NSGA-II/SMS-EMOA/MOEA/D could be an artifact of insufficient search effort at large n, rather than a general property of MOCOP scalability.

Authors: We thank the referee for highlighting the need for explicit detail on this aspect of the design. We will revise the Experimental Methodology section to fully describe the stopping criteria (maximum function evaluations until no further improvement or a hard cap) and the evaluation budgets employed. The budgets were held constant across problem sizes to enable a direct, fair comparison of relative scalability under equivalent computational effort for all algorithms. Because every algorithm received identical resources, the observed decline in SEMO's convergence speed relative to NSGA-II, SMS-EMOA and MOEA/D cannot be dismissed as an artifact of under-budgeting; rather, it indicates that SEMO requires disproportionately more evaluations to maintain performance as n grows. We will add a short discussion of this design rationale and its implications for interpreting scalability results. revision: yes
Referee: [Results and Discussion] Results and Discussion: The specific MOCOP benchmark instances (e.g., multi-objective knapsack variants or other discrete problems) and performance metrics are not justified as representative of general combinatorial optimization behavior. Without this, the claim that absence of crossover is the major contributor to SEMO underperformance, and that adding it accelerates convergence, risks being tied to the chosen test suite rather than broadly applicable.

Authors: We appreciate the referee's point on generalizability. In the revised manuscript we will expand the Results and Discussion section with a dedicated paragraph justifying the chosen MOCOP instances. These problems (multi-objective knapsack and similar discrete benchmarks) are standard in the MOEA literature precisely because they exhibit the rugged landscapes, numerous local optima and disjoint Pareto regions characteristic of combinatorial optimisation. We will cite representative prior studies that have used the same suites for scalability and operator analysis. The performance metrics (hypervolume and spread indicators) are likewise standard for jointly assessing convergence and diversity. While we stand by the experimental finding that crossover is a major factor in the observed scalability gap, we will qualify the claims to note that they are demonstrated on these representative problem classes and suggest that future work should examine additional MOCOP families. revision: partial

Circularity Check

0 steps flagged

No circularity: purely empirical comparison with no derivations or fitted predictions

full rationale

The paper conducts an empirical study of MOEA scalability on MOCOP benchmarks with problem sizes from 50 to 5000. Claims about SEMO's declining convergence speed and the effect of adding crossover are supported solely by direct algorithm runs, performance metrics, and statistical comparisons on chosen test problems. No equations, parameter fitting, uniqueness theorems, or self-citations are used to derive results; the central findings reduce only to the experimental data itself rather than any self-referential construction. This is a standard non-circular empirical analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on empirical observations from standard MOEA benchmarks; no free parameters or invented entities are introduced.

axioms (1)

domain assumption Performance differences observed on the chosen test problems generalize to other MOCOPs.
The paper assumes the selected combinatorial problems capture the rugged landscape and local-optima characteristics typical of the domain.

pith-pipeline@v0.9.0 · 5543 in / 1189 out tokens · 31015 ms · 2026-05-10T04:01:26.964389+00:00 · methodology

discussion (0)

Reference graph

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