Random convergence analysis of particle swarm optimization algorithm with time‐varying attractor

doi:10.1016/j.swevo.2020.100819

Swarm and Evolutionary Computation

Volume 61, March 2021, 100819

https://doi.org/10.1016/j.swevo.2020.100819 Get rights and content

Highlights

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The mathematical model on PSO algorithm with time-varying attractor is provided.
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One spectral analysis on PSO algorithm with constant attractor is proposed.
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Spectral analysis on PSO algorithm with time-varying attractor is investigated.
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Mean and variance of PSO algorithm with time-varying attractor are also analyzed.
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Results show that spectral radius on PSO transfer matrix at some steps is not smaller than 1.

Abstract

The PSO convergence analysis is mainly based on the constant attractor, however, the attractor of PSO algorithm in the evolutionary process is time-varying point. So, the objective of this study is to mainly discuss the random convergence analysis of the standard and improved particle swarm optimization with the time-varying attractor. Its mathematical PSO model with the time-varying attractor is provided to calculate the convergence condition and the corresponding convergence speed. Specifically speaking, spectral radii of the random transfer matrix and the product of two adjacency random transfer matrices are calculated to determine the convergence or the divergence, together with the corresponding convergence speed. Additionally, it also calculates the mean and variance of the first and second order particle swarm optimization system with time-varying attractor. Numerical results highlight that the spectral analysis on some benchmark optimization functions is described to show the effectiveness of the obtained results, while the corresponding analysis is closely related to the objective fitness, the convergence speed, the time-varying attractor, the spectral radius of $M (t)$ and $M (t + 1) M (t),$ and swarm convergence behavior in the evolutionary process.

Introduction

Recently, particle swarm optimization (PSO) algorithm has been successfully and widely solved for different optimization problems, which are typically composed of the continuous optimization problem, the discrete optimization problem, the dynamic optimization problem, the multi-objectives optimization problem and the hybrid optimization problem, etc. Particle swarm optimization, which is originally developed by Kennedy and Ehbert in 1995 [1], [2], is mainly motivated by the swarm behavior of birds and fish. Due to its simple implement and high performance, the standard and improved PSO algorithms [3] have been widely attracted by a large number of experts and researchers in the area of evolutionary computation, computer science and big data, etc. Additionally, the PSO algorithm is widely applied in many realistic applications, such as electric power system [4], neural network [5], [6], electromagnetic [7], PID controller [8] and game theory [9], locating and tracking the targets [10], [11], [12], [13], [14], the optimization problem in the uncertain and dynamic scenario [15], [16], [17], [18], etc.

One of the challenging problems on the fundamental theory of the PSO algorithm is the convergence analysis and the convergence condition, since the convergence analysis of the PSO algorithm plays a role on parameter selection, the trajectory of each particle, the optimized result and the tradeoff between exploration ability and exploitation ability, etc. One aspect of the existing theoretical results on PSO algorithm is mainly conducted on the simplified PSO algorithm, which considers random variables including $r_{1}$ and $r_{2}$ as constant parameters. Ozcan and Mohan [19], [20] analyze the trajectory of one particle and some particles. Clerc and Kennedy [21] analyze the stability of the simplified PSO algorithm and the corresponding convergence condition of PSO algorithm, more importantly, the constriction factor method is introduced and has good optimization performance, etc. Trelea [22] calculates spectral radius of the transfer matrix, and provides convergence condition and convergence speed to balance the exploration ability and exploitation ability. Van den Bergh and Engelbrecht [23] also analyze the convergence trajectory and the corresponding convergence analysis. Samal and Konar [24] regard PSO algorithm as a closed loop control system and consider nonlinear element into the PSO model, while the stability analysis of PSO algorithm is utilized by Jury’s test and root locus technique.

Another aspect of the existing convergence results mainly concentrates on the random convergence analysis on the PSO system with random variables. Essentially, the typical and standard PSO algorithm is the random second-order discrete linear system, which is composed of two random variables including $r_{1}$ and $r_{2}$ at each step. Poli [25] and Milan [26] give the stability and convergence conditions of the standard PSO algorithm under random variables. Kadirkamanathan [27] analyzes the convergence analysis of PSO algorithm including random variables where PSO system is the nonlinear feedback system. Luis Fernandez and Esperanza Garcia [28] provide spectral radius and the trajectory where the PSO algorithm is considered as the stochastic damped mass-spring system. Pan and Zhang [29] discuss the consensus analysis of the random PSO algorithm from the viewpoint of multi-agents consensus, when the inertia weight $ω$ is set from -1 to 0 and is larger than $0.5 ϕ - 1,$ while the parameter $ϕ$ is larger than 0. Liu and Ma [30] introduce the joint spectral radius which denotes spectral radius of the product of all time-varying transfer matrices from the perspective of numerical analysis, furthermore, the joint spectral radius can denote the tradeoff between exploration ability and exploitation ability. Yuan and Yin [31] analyze the PSO asymptotic properties by stochastic approximation methods, while the convergence analysis and the convergence rate of PSO algorithm are used by the weak convergence method.

The existing convergence analysis on the PSO algorithm focuses on the constant convergence analysis on the constant transfer matrix and the random convergence analysis on basis of the random transfer matrix. More importantly, the corresponding assumption is that the attractor of PSO system is assumed to be the constant attractor. However, the attractor of PSO algorithm in fact is the time-varying point at each step, and few works concentrate on the random convergence analysis of the standard PSO algorithm with time-varying attractor. Therefore, the motivation of this paper is to discuss that the time-varying attractor plays a role on the convergence condition and the convergence speed, together with comparing the difference between the obtained results and the existing convergence analysis. Main contributions on the random convergence analysis of the random PSO algorithm can be highlighted as follows.

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The existing theoretical results on the PSO algorithm mainly discuss the random convergence analysis in the context of the constant attractor. However, the paper mainly considers the random convergence analysis in the presence of the time-varying attractor of PSO system.
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The convergence condition of the random PSO algorithm is calculated by the spectral radius of the random transfer matrix and the product of two adjacency transfer matrices in the random PSO system.
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It is also key to analyze the two-order convergence condition on the random PSO algorithm from the viewpoint of mean and variance.
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The spectral analysis on PSO algorithm is closely related to the time-varying attractor, the spectral radius, the distribution of spectral radius and the swarm convergence behavior, etc.
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Additionally, spectral radius of one random transfer matrix cannot determine the convergence behavior or the divergence behavior, but spectral radius of the product of two adjacency transfer matrices determines the convergence behavior or the divergence behavior in two steps.

The reminder of this paper is organized as follows. Section 2 gives the brief description of the general mathematical model of the random PSO system with time-varying attractor. Section 3 calculates the spectral radius of each time-varying matrix and the product of two adjacency transfer matrices of PSO algorithm with constant attractor. Section 4 calculates the spectral radius of one time-varying matrix and the product of two transfer matrices under the condition of time-varying attractor. Section 5 provides the spectral radius of the random PSO algorithm with time-varying attractor from the perspective of mean and variance. Section 6 highlights that simulation results, concerning some benchmark optimization functions, concentrate on the objective fitness and the convergence speed, together with time-varying attractor and spectral radius, etc. Section 7 concludes the interesting results and the future works on the convergence analysis on the random PSO algorithm.

Section snippets

Particle swarm optimization and its mathematical model

Particle swarm optimization can be widely applied in the fields of many realistic applications, and there are many improved PSO algorithms to reduce the computational time and obtain the good local optimum or the global optimum. In the standard PSO algorithm, there are two typical PSO algorithms, which are called the inertia weight method and the constriction factor method. Firstly, the particle’s velocity and the position in the inertia weight method are mathematically expressed by $v_{ij} (t + 1) = ω (t)$

Spectral analysis on the random PSO algorithm with constant attractor

In order to study and discuss the convergence analysis on the random PSO algorithm, the objective of this section is to conclude the convergence condition and the convergence speed under the constant attractor of PSO algorithm. Specifically speaking, it needs to calculate the spectral radii of one transfer matrix and the product of two transfer matrices, which mainly controls the convergence behavior and the divergence behavior.

The existing convergence analysis is mainly based on the constant

Spectral analysis on the random PSO algorithm with time‐varying attractor

The existing convergence analysis on the PSO algorithm does not consider the convergence analysis and the stability under the assumption of the time-varying attractor of PSO system. In fact, the attractor of the PSO algorithm, which essentially is the time-varying variable $Q (t),$ mainly depends on the equation of $ϕ_{1} P (t) + ϕ_{2} G (t),$ and the attractor at $t$ th step is not equal to the attractor at $t + 1$ th step, such as $Q (t + 1) \neq Q (t)$ . Additionally, the parameter $η$ is not equal to 0.

Mean and variance of the random PSO algorithm with time‐varying attractor

The PSO algorithm and the improved PSO algorithms are considered to be the random second-order discrete linear systems. Additionally, the velocity and the position of each particle in the PSO algorithm can be described by $v (t + 1) = ω v (t) + ϕ (t) (Q (t) - x (t))$ $x (t + 1) = x (t) + ω v (t) + ϕ (t) (Q (t) - x (t)) .$

The velocity and the position of each particle can be mathematically rewritten as $x (t + 1) = x (t) + ω (x (t) - x (t - 1)) + ϕ (t) (Q (t) - x (t)) .$

Then, the above-mentioned equation (74) can be rewritten by $\begin{matrix} \begin{matrix} Q (t + 1) - x (t + 1) = Q (t) - x (t) + ω (Q (t) - x (t \end{matrix} \end{matrix}$

Simulation results

In order to show the effectiveness on the convergence analysis of the random PSO algorithm with the time-varying attractor $Q (t + 1) \neq Q (t),$ the task of this section is to discuss the obtained results in the evolutionary process and the corresponding spectral radius of one transfer matrix and the product of two transfer matrices, together with the further discussion on the spectral radius on the benchmark optimization functions.

Conclusions and future works

This paper mainly concentrates on the random convergence analysis of the PSO algorithm with time-varying attractor $Q (t + 1) \neq Q (t)$ . Firstly, the convergence condition and the convergence speed are calculated by the spectral radius on the random transfer matrix and the product of two transfer matrices of the PSO algorithm with time-varying attractor. Secondly, the spectral radius on one transfer matrix cannot determine the convergence behavior and the divergence behavior, while the spectral radius

Declaration of Competing Interest

The authors declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowlgedgments

This work is mainly supported by National Key Research and Development Project (2020YFF0304104) and China Railway Corporation Key Project (P2019X001).

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Citation Excerpt :
The particle swarm optimization (PSO) algorithm, originally developed by Kennedy and Eberhart [1], has become a widely used optimization technique [2]. Given PSO’s popularity, it has undergone a considerable amount of theoretical investigation, to list just a few, [3–11]. While meta-heuristics have been used to solve numerous real world problems, they typically lack the theoretical underpinning and guarantees that classical optimization techniques have.
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¹: Dr Jun Liu is an associate researcher in the Institute of Computing Technologies, Chinese Academy of Railway Sciences. His current research interests include particle swarm optimization, cloud optimization, big data, intelligent railway and railway safety, etc.

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Survey PaperRandom convergence analysis of particle swarm optimization algorithm with time‐varying attractor

Highlights

Abstract

Introduction

Section snippets

Particle swarm optimization and its mathematical model

Spectral analysis on the random PSO algorithm with constant attractor

Spectral analysis on the random PSO algorithm with time‐varying attractor

Mean and variance of the random PSO algorithm with time‐varying attractor

Simulation results

Conclusions and future works

Declaration of Competing Interest

Acknowlgedgments

Appl. Soft Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Soft Comput.

Inf. Process. Lett.

Inf. Sci. (Ny)

Inf. Process. Lett.

Appl. Soft Comput.

Particle swarm optimization

Proceedings of IEEE International Conference on Neural Networks

A new optimizer using particle swarm theory

Proceedings of the Sixth International Symposium on Micro Machine and Human Science

A hybrid particle swarm optimization applied to loss power minimization

IEEE Trans. Power Syst.

A hybrid of genetic algorithm and particle swarm optimization for recurrent network design

IEEE Transactions on Systems Man and Cybernetics Part B-Cybernetics

Use of intelligent-particle swarm optimization in electromagnetics

IEEE Trans. Magn.

A particle swarm optimization approach for optimum design of PID controller in AVR system

IEEE Trans. Energy Convers.

Learning to play games using a PSO-based competitive learning approach

IEEE Trans. Evol. Comput.

Survey Paper
Random convergence analysis of particle swarm optimization algorithm with time‐varying attractor