# Exact Markov Chain-based Runtime Analysis of a Discrete Particle Swarm Optimization Algorithm on Sorting and OneMax

## TL;DR

This paper presents a rigorous Markov chain analysis of a discrete particle swarm optimization algorithm, providing bounds on its expected runtime on sorting and OneMax problems, and exploring the tradeoff between exploration and exploitation.

## Contribution

It offers the first formal Markov chain-based runtime analysis of a PSO algorithm on discrete problems, including bounds and parameter insights.

## Key findings

- Expected optimization time is comparable to known algorithms with suitable parameters.
- Algorithm behavior varies from greedy to explorative depending on parameter choices.
- Analysis introduces the notion of indistinguishability of Markov states and bounds on recurrence solutions.

## Abstract

Meta-heuristics are powerful tools for solving optimization problems whose structural properties are unknown or cannot be exploited algorithmically. We propose such a meta-heuristic 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.