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

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.