While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems.
To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the LeadingOnes and Jump benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation $σ$ into another one $τ$, but also the precise cycle structure of $στ^{-1}$. For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of $Θ(n)$. Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order $m^{Θ(m)}$ on jump functions with jump size $m$. A short empirical analysis confirms these findings, but also reveals that small implementation details like the rate of void mutations can make an important difference.
Laboratoire d’Informatique (LIX)（计算机科学实验室（LIX））；CNRS（法国国家科学研究中心）；École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）
While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems. To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the \textsc{LeadingO…
Runtime Analysis of Evolutionary Algorithms for Multi-party Multi-objective Optimization
School of Artificial Intelligence, Nanjing University of Information Science and Technology（人工智能学院，南京信息工程大学）；Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies, Institute of Cyberspace Security, School of Computer Science and Technology, Harbin Institute of Technology（新型安全智能技术广东省重点实验室，网络空间安全学院，计算机科学与技术学院，哈尔滨工业大学）
First Mathematical Runtime Analyses of Evolutionary Multitask Multi-Objective Optimization
School of Computer Science and Technology, State Key Laboratory of Smart Farm Technologies and Systems, International Research Institute for Artificial Intelligence, Harbin Institute of Technology, Shenzhen, China（哈尔滨工业大学（深圳）计算机科学与技术学院、智慧农业技术与系统全国重点实验室、人工智能国际研究院）；Pengcheng Laboratory, Shenzhen, China（鹏城实验室）
dMFEA-II: An Adaptive Multifactorial Evolutionary Algorithm for Permutation-based Discrete Optimization Problems
TECNALIA, Basque Research and Technology Alliance (BRTA)（TECNALIA，巴斯克研究与技术联盟（BRTA））；Universidad de Cantabria（坎塔布里亚大学）；Toho University（东邦大学）；University of the Basque Country (UPV/EHU)（巴斯克大学（UPV/EHU））
We use an elementary argument building on group actions to prove that the selection-free steady state genetic algorithm analyzed by Sutton and Witt (GECCO 2019) takes an expected number of $Ω(2^n / \sqrt n)$ iterations to find any particular target search point. This bound is valid for all population sizes $μ$. Our result improves over the previous lower bound of $Ω(\exp(n^{δ/2}))$ valid for population sizes $μ= O(n^{1/2 - δ})$, $0 < δ< 1/2$.
Generating diverse populations of high quality solutions has gained interest as a promising extension to the traditional optimization tasks. This work contributes to this line of research with an investigation on evolutionary diversity optimization for three of the most well-studied permutation problems, namely the Traveling Salesperson Problem (TSP), both symmetric and asymmetric variants, and Quadratic Assignment Problem (QAP). It includes an analysis of the worst-case performance of a simple mutation-only evolutionary algorithm with different mutation operators, using an established diversi…
Evolutionary Algorithms and Multi-Objective Minimum Spanning Trees with Limited Distinct Weight Values
Univ. Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL（里尔大学、法国国家科学研究中心、中央理工学院、CRIStAL联合研究实验室）；University of Minnesota Duluth（明尼苏达大学德卢斯分校）；Adelaide University（阿德莱德大学）
Linear functions play a key role in the runtime analysis of evolutionary algorithms and studies have provided a wide range of new insights and techniques for analyzing evolutionary computation methods. Motivated by studies on separable functions and the optimization behaviour of evolutionary algorithms as well as objective functions from the area of chance constrained optimization, we study the class of objective functions that are weighted sums of two transformed linear functions. Our results show that the (1+1) EA, with a mutation rate depending on the number of overlapping bits of the funct…
Already Moderate Population Sizes Provably Yield Strong Robustness to Noise
The University of Adelaide（阿德莱德大学）；CNRS（法国国家科学研究中心）；École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；HSE University（高等经济大学）；Skoltech（斯科尔科沃科学技术研究院）
Runtime analysis aims at contributing to our understanding of evolutionary algorithms through mathematical analyses of their runtimes. In the context of discrete optimization problems, runtime analysis classically studies the time needed to find an optimal solution. However, both from a practical and from a theoretical viewpoint, more fine-grained performance measures are needed to gain a more detailed understanding of the main working principles and their resulting performance implications. Two complementary approaches have been suggested: fixed-budget analyses and fixed-target analyses. In t…
First Mathematical Runtime Analyses of Multi-Objective Evolutionary Algorithms for Multi-Valued Decision Variables
School of Computer Science and Technology, National Key Laboratory of Smart Farm Technologies and Systems, International Research Institute for Artificial Intelligence（计算机科学与技术学院，智慧农业技术与系统国家重点实验室，人工智能国际研究院）；Pengcheng Laboratory, Shenzhen, China（鹏城实验室）；Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳））；Institut Polytechnique de Paris, Palaiseau, France（巴黎理工学院）；Laboratoire d’Informatique (LIX), CNRS, École Polytechnique（计算机实验室（LIX），法国国家科学研究中心，巴黎综合理工学院）
Runtime Analysis of Single- and Multi-Objective Evolutionary Algorithms for Chance Constrained Optimization Problems with Normally Distributed Random Variables
CNRS（法国国家科学研究中心）；École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎综合理工学院）；The University of Adelaide（阿德莱德大学）
The theory of evolutionary computation for discrete search spaces has made significant progress in the last ten years. This survey summarizes some of the most important recent results in this research area. It discusses fine-grained models of runtime analysis of evolutionary algorithms, highlights recent theoretical insights on parameter tuning and parameter control, and summarizes the latest advances for stochastic and dynamic problems. We regard how evolutionary algorithms optimize submodular functions and we give an overview over the large body of recent results on estimation of distributio…
We argue that proven exponential upper bounds on runtimes, an established area in classic algorithms, are interesting also in heuristic search and we prove several such results. We show that any of the algorithms randomized local search, Metropolis algorithm, simulated annealing, and (1+1) evolutionary algorithm can optimize any pseudo-Boolean weakly monotonic function under a large set of noise assumptions in a runtime that is at most exponential in the problem dimension~$n$. This drastically extends a previous such result, limited to the (1+1) EA, the LeadingOnes function, and one-bit or bit…
The $(1 + (λ,λ))$ genetic algorithm is a younger evolutionary algorithm trying to profit also from inferior solutions. Rigorous runtime analyses on unimodal fitness functions showed that it can indeed be faster than classical evolutionary algorithms, though on these simple problems the gains were only moderate. In this work, we conduct the first runtime analysis of this algorithm on a multimodal problem class, the jump functions benchmark. We show that with the right parameters, the \ollga optimizes any jump function with jump size $2 \le k \le n/4$ in expected time $O(n^{(k+1)/2} e^{O(k)} k^{…
Runtime Analysis of the Univariate Marginal Distribution Algorithm under Low Selective Pressure and Prior Noise
University of Birmingham（伯明翰大学）
我们执行一个严格的运行时分析的单变量边际分布算法上的LeadingOnes功能，一个著名的基准功能，在理论界的演化计算与决策变量之间的高度相关性。对于大小为$n$的问题实例，当前已知的关于预期运行时间的上限是$\mathcal{O}(nλ\logλ+n^2)$（Dang and Lehre，GECCO 2015），而理解算法如何处理变量依赖关系所需的下限仍然缺失。出于这一动机，我们表明，该算法需要一个$e^{Ω(μ)}$运行时具有很高的概率，并在预期中，如果选择压力低，否则，我们得到一个下界的$Ω(\frac{nλ}{\log(λ-μ)})$的预期运行时间。此外，我们第一次考虑算法的功能下的先验噪声模型，并获得一个$\mathcal{O}(n^2)$预期运行时间的最佳参数设置。最后，我们的理论结果伴随着实证研究结果，不仅与严格的分析相匹配，而且还提供了新的见解算法的行为。
Evolutionary algorithms solve problems by simulating the evolution of a population of candidate solutions. We focus on evolving permutations for ordering problems like the traveling salesperson problem (TSP), as well as assignment problems like the quadratic assignment problem (QAP) and largest common subgraph (LCS). We propose cycle mutation, a new mutation operator whose inspiration is the well known cycle crossover operator, and the concept of a permutation cycle. We use fitness landscape analysis to explore the problem characteristics for which cycle mutation works best. As a prerequisite,…
Combinatorial designs provide an interesting source of optimization problems. Among them, permutation codes are particularly interesting given their applications in powerline communications, flash memories, and block ciphers. This paper addresses the design of permutation codes by evolutionary algorithms (EA) by developing an iterative approach. Starting from a single random permutation, new permutations satisfying the minimum distance constraint are incrementally added to the code by using a permutation-based EA. We investigate our approach against four different fitness functions targeting t…
Theoretical Analyses of Multiobjective Evolutionary Algorithms on Multimodal Objectives
School of Computer Science and Technology（计算机科学与技术学院）；International Research Institute for Artificial Intelligence（国际人工智能研究院）；Harbin Institute of Technology（哈尔滨工业大学）；Southern University of Science and Technology（南方科技大学）；Laboratoire d’Informatique (LIX)（信息学实验室）；École Polytechnique（巴黎综合理工学院）；CNRS（法国国家科学研究中心）；Institut Polytechnique de Paris（巴黎理工学院）
