Convergence proof of an enhanced Particle Swarm Optimisation method integrated with Evolutionary Game Theory

Abstract

This paper proposes an enhanced Particle Swarm Optimisation (PSO) algorithm and examines its performance. In the proposed PSO approach, PSO is combined with Evolutionary Game Theory to improve convergence. One of main challenges of such stochastic optimisation algorithms is the difficulty in the theoretical analysis of the convergence and performance. Therefore, this paper analytically investigates the convergence and performance of the proposed PSO algorithm. The analysis results show that convergence speed of the proposed PSO is superior to that of the Standard PSO approach. This paper also develops another algorithm combining the proposed PSO with the Standard PSO algorithm to mitigate the potential premature convergence issue in the proposed PSO algorithm. The combined approach consists of two types of particles, one follows Standard PSO and the other follows the proposed PSO. This enables exploitation of both diversification of the particles' exploration and adaptation of the search direction.

Pricing policies for a dual-channel retailer with cross-channel returns

“…Besides, an evolutionary game theoretic approach may be used to study different return policies in dualchannel retailing where a store has a set of different policies to choose from namely: no refund, partial refund, and full refund. Unlike the classical game theory used in this paper, players in the aforementioned game dynamically compete or interact until an evolutionary stable strategy (ESS) is successfully achieved (Leboucher et al 2016). …”

An Improved Convergence Particle Swarm Optimization Algorithm with Random Sampling of Control Parameters

“…However, in order to avoid the phenomenon of "oscillation" and "two steps forward, one step back," the inertia weight is selected around the center part of the convergence interval of . According to formula (8) and (9), Figure 1 shows the relationship between and for mean square convergence of PSO. For example, when the acceleration coefficients have been already sampled, c 1 =c 2 =2, and then parameters and 2 can be computed, =2, 2 =2/3.…”

“…PSO, proposed by Kennedy and Eberhart [1], is an evolutionary algorithm based on swarm intelligence which simulates birds or fish predation, and it has already attracted a lot of interest from scholars and researchers for the reason that PSO has simple structure, strong maneuverability, easy realization, and other characteristics. Up to now, PSO has been successfully applied in many areas [2][3][4][5][6], and meanwhile some improved versions of PSO have also been studied accordingly [7][8][9][10][11].…”

A framework for multi-objective optimisation based on a new self-adaptive particle swarm optimisation algorithm

“…Inspired by birds flocking and fish schooling, Kennedy and Eberhart first proposed PSO in 1995. The original aim of the basic PSO algorithm is to reproduce the social interactions among agents to solve some complex optimisation problems [29]. Each agent in PSO is called a particle and associated with a velocity, which is dynamically adjusted depending on its own flight experience and those of its companions.…”

“…Therefore, it is essential to address and guarantee the convergence of PSO when adjusting the three parameters for improving PSO [21,27,28]. However, like in most of the stochastic approaches, the stochastic nature of PSO imposes difficulties on the analytical investigation of its convergence [29].…”