An Analytic Expression of Relative Approximation Error for a Class of Evolutionary Algorithms

TL;DR

This paper derives an exact analytic expression for the relative approximation error of (1+1) elitist evolutionary algorithms, providing insights into their solution quality and convergence behavior over generations.

Contribution

It introduces a novel matrix analysis method to precisely quantify solution quality and convergence metrics for a specific class of evolutionary algorithms.

Findings

Exact expression for relative approximation error over generations

Analytic formulas for fitness value and convergence rate

Potential extension to non-elitist and population-based algorithms

Abstract

An important question in evolutionary computation is how good solutions evolutionary algorithms can produce. This paper aims to provide an analytic analysis of solution quality in terms of the relative approximation error, which is defined by the error between 1 and the approximation ratio of the solution found by an evolutionary algorithm. Since evolutionary algorithms are iterative methods, the relative approximation error is a function of generations. With the help of matrix analysis, it is possible to obtain an exact expression of such a function. In this paper, an analytic expression for calculating the relative approximation error is presented for a class of evolutionary algorithms, that is, (1+1) strictly elitist evolution algorithms. Furthermore, analytic expressions of the fitness value and the average convergence rate in each generation are also derived for this class of evolutionary algorithms. The approach is promising, and it can be extended to non-elitist or population-based algorithms too.