Evolutionary algorithms for continuous-space optimisation

Huang

et al. 2019

Sci. China Inf. Sci.

The pigeon-inspired optimization (PIO) algorithm is a newly presented swarm intelligence optimization algorithm inspired by the homing behavior of pigeons. Although PIO has demonstrated effectiveness and superiority in numerous fields, particularly in practical engineering optimization, there have been few results concerning the theoretical foundations of PIO. This paper conducts convergence analysis of basic PIO in a continuous search space in two aspects. First, we analyze the convergence of each pigeon's expected position using a difference equation and prove that the average position of each pigeon in the swarm will converge to the same value. To further study the stochastic global convergence property of the pigeon swarm, we apply the martingale theory to investigate the basic PIO swarm sequence, and achieve a sufficient condition to guarantee global convergence of the basic PIO. Our theoretical analysis shows that this convergence depends upon the accumulation of the minimum probability with which the pigeon swarm jumps to the global-optimal region at each iteration. The mathematical methods proposed in this study, particularly the martingale technique, also provide a new effective approach for the theoretical analysis of bio-inspired algorithms in continuous optimization.

Section: Some Related Convergence Analyses On Continuous Easmentioning

confidence: 99%

Theoretical analysis of the convergence property of a basic pigeon-inspired optimizer in a continuous search space

Huang

et al. 2019

Sci. China Inf. Sci.

“…这两点修改分别是, 以步骤 3 的加法公式作为变异更新个体, 以及将步骤 5 的停机条件改为误差判定方式. 选择连续型 (1+1)EA 作为分析对象的原因有三: 首先, 选择 (1+1)EA 算法框架, 简化了种群规模、交叉算子和选择算子的影响, 主要研究点在于不同概率分布的变异步长对计算时间的影响; 其次, (1+1)EA 是离散型进化算法计算时间分析的经典对象 [3,4,8∼11] , 这里沿用其框架将使研究具有代表性; 再次, 流程中步骤 2 中 ∆ t 的随机性与步骤 3 的更新公式设计源于进化规划算法 [1,17] 的变异算子设计, 所以本分析结论将对进化规划算法的计算时间研究有积极意义. [18] 作为工具.…”

Section: 问题描述与算法简介unclassified

“…在此基础上, Jägersküpper [16] 还研究了 (1+1)ES 求解 Sphere 函数的计算时间分析框架. Agapie 等 [17] 完整描述了进化算法求解连续型优化问题的数学模型, 为计算时间分析作了铺垫.…”

unclassified

基于平均增益模型的连续型(1+1)进化算法计算时间复杂性分析

Huang

et al. 2014

Sci. Sin.-Inf.

Runtime analysis of continuous evolutionary algorithm (EA) is an open problem in theoretical foundation of evolutionary computation. There are fewer results about it than the runtime studies of discrete EA. For an example of (1+1)EA, an average gain model and its calculating method were proposed to produce a theory of runtime analysis as an index of computational time complexity. The average gain was computed to estimate the average runtime of two (1+1)EAs based on the mutation of standard normal distribution and uniform distribution, for Sphere function which is focused on by many researchers. The analysis result indicates that computational time complexity of the (1+1)EAs is exponential order. Furthermore, the solution speed of uniform-distribution mutation is faster than standard normal distribution with the same error accuracy and initial distance. Numerical results also verify the correctness of the proposed theory and the usefulness of the average gain model.

“…Jägersküpper conducted a rigorous runtime analysis on (1 + 1)ES, (1 + λ )ES minimizing the Sphere function [32, 33]. Agapie et al modeled the continuous EA as a renewal process under some strong assumption and analyzed the first hitting time of (1 + 1)EA on inclined plane and Sphere function [34, 35]. Chen et al [36] proposed general drift conditions to estimate the upper bound of the first hitting time for EAs to find ϵ -approximation solutions.…”

Section: Introductionmentioning

confidence: 99%

An Analytical Framework for Runtime of a Class of Continuous Evolutionary Algorithms

Computational Intelligence and Neuroscience

2015

Although there have been many studies on the runtime of evolutionary algorithms in discrete optimization, relatively few theoretical results have been proposed on continuous optimization, such as evolutionary programming (EP). This paper proposes an analysis of the runtime of two EP algorithms based on Gaussian and Cauchy mutations, using an absorbing Markov chain. Given a constant variation, we calculate the runtime upper bound of special Gaussian mutation EP and Cauchy mutation EP. Our analysis reveals that the upper bounds are impacted by individual number, problem dimension number n, searching range, and the Lebesgue measure of the optimal neighborhood. Furthermore, we provide conditions whereby the average runtime of the considered EP can be no more than a polynomial of n. The condition is that the Lebesgue measure of the optimal neighborhood is larger than a combinatorial calculation of an exponential and the given polynomial of n.