Menghao Tang | Searcharxiv

arXiv2026

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

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.

cs.NE↗

arXiv2025

Maz'ya--Shaposhnikova Representation of Quasi-Norms of Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type with Weak Reverse Doubling Property

Let $Y(\mathcal{X})$ be a ball quasi-Banach function space on the space of homogeneous type $(\mathcal{X},\rho,\mu)$ satisfying some mild additional assumptions, $q\in(0,\infty)$, and $\dot{W}^{s,q}_Y(\mathcal{X})$ with $s\in(0,1)$ be the homogeneous fractional Sobolev space associated with $Y(\mathcal{X})$. In this article, we show that, for any $f\in Y(\mathcal{X})\cap\bigcup_{s\in(0,1)} \dot{W}^{s,q}_Y(\mathcal{X})$, \begin{align*} \|f\|_{Y(\mathcal{X})} &\lesssim\varliminf_{s \to 0^+} s^{\frac{1}{q}}\left\| \left\{\int_{\mathcal{X}} \frac{|f(\cdot)-f(y)|^q}{U(\cdot,y)[\rho(\cdot,y)]^{sq}} \, d\mu(y) \right\}^{\frac{1}{q}}\right\|_{{Y(\mathcal{X})}}\\ &\leq \varlimsup_{s \to 0^+} s^{\frac{1}{q}}\left\|\left\{\int_{\mathcal{X}} \frac{|f(\cdot)-f(y)|^q}{U(\cdot,y)[\rho(\cdot,y)]^{sq}} \, d\mu(y) \right\}^{\frac{1}{q}}\right\|_{{Y(\mathcal{X})}} \lesssim\|f\|_{Y(\mathcal{X})}, \end{align*} where $U(x,y):=\min\{\mu(B(x,\rho(x,y))),\,\mu(B(y,\rho(x,y)))\}$ for any $x,y\in\mathcal{X}$ and the implicit positive constants are independent of $f$, which is applied to ten specific ball quasi-Banach function spaces and hence is of wide generality. In particular, when $Y(\mathcal{X})=L^q(\mathbb{R}^n)$ with $q\in[1,\infty)$, the above formula is closely related to the celebrated result of Maz'ya and Shaposhnikova in 2002. We also establish the above representation formula on domains of $\mathcal{X}$. The main novelty lies in proposing two new concepts, namely the weak reverse doubling condition (for $\mathcal{X}$) and the weak measure density condition (for domains of $\mathcal{X}$), which are proved to be necessary in some sense. In addition, we find an interesting fact that, when the underlying space under consideration is bounded, the above Maz'ya--Shaposhnikova-type limit always tends to zero.

math.FA↗