Runtime Analysis of a Discrete Particle Swarm Optimization Algorithm on Sorting and OneMax

Abstract

Meta-heuristics are powerful tools for solving optimization problems whose structural properties are unknown or cannot be exploited algorithmically. We propose such a metaheuristic for a large class of optimization problems over discrete domains based on the particle swarm optimization (PSO) paradigm. We provide a comprehensive formal analysis of the performance of this algorithm on certain "easy" reference problems in a black-box setting, namely the sorting problem and the problem ONEMAX. In our analysis we use a Markov-model of the proposed algorithm to obtain upper and lower bounds on its expected optimization time. Our bounds are essentially tight with respect to the Markov-model. We show that for a suitable choice of algorithm parameters the expected optimization time is comparable to that of known algorithms and, furthermore, for other parameter regimes, the algorithm behaves less greedy and more explorative, which can be desirable in practice in order to escape local optima. Our analysis provides a precise insight on the tradeoff between optimization time and exploration. To obtain our results we introduce the notion of indistinguishability of states of a Markov chain and provide bounds on the solution of a recurrence equation with non-constant coefficients by integration.

Exact Markov chain-based runtime analysis of a discrete particle swarm optimization algorithm on sorting and OneMax

“…Extensions This article extends the conference paper (Mühlenthaler et al 2017) as follows. We present D-PSO, a discrete PSO algorithm with multiple particles in Sect.…”

“…The D-PSO algorithm essentially explores this graph, looking for an optimal vertex. In our analysis, we assume at first a swarm size of one as in Mühlenthaler et al (2017), similar to the analysis of EAs and ACO in Sudholt and Witt (2010). We refer to the corresponding specialization of D-PSO as ONEPSO.…”

“…5). Please note that a and b have been significantly improved compared to the conference version (Mühlenthaler et al 2017) and the upper bound on ONEMAX has also been reduced heavily to an exponential term with base bðcÞ. This means that the lower and upper bound on the base of the exponential expression for the expected optimization time is equal.…”

“…In addition, for determining the lower bound (Scharnow et al 2004) consider the Hamming distance to evaluate the distance between the current position and the optimum although the algorithm still uses the transposition distance to decide which position is better. In the conference version (Mühlenthaler et al 2017) we incorrectly claimed that the proof does not apply to this setting. Thanks to an anonymous reviewer we can correct this statement here.…”

Optimization of Day-Ahead Energy Storage System Scheduling in Microgrid Using Genetic Algorithm and Particle Swarm Optimization

“…It should also be noted that since metaheuristics like GA and PSO are highly probabilistic in nature, it is very hard to perform run-time analysis on them. There's only a small amount of literature available regarding the runtime analysis of simpler versions of GA [31]- [34] or PSO [35], [36], which are based on well known basic optimization problems like One Max Problem (maximizing the number of ones in an n bit string). There is no literature regarding the run-time analysis of GA or PSO for hard non-linear optimization problems so far.…”