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.
When for a difficult real-world optimisation problem no good problem-specific algorithm is available often randomised search heuristics are used. They are hoped to deliver good solutions in acceptable time. The theoretical analysis usually concentrates on the average time needed to find an optimal or approximately optimal solution. This matches neither the application in practice nor the empirical analysis since usually optimal solutions are not known and even if found cannot be recognised. More often the algorithms are stopped after some time. This motivates a theoretical analysis to concentrate on the quality of the best solution obtained after a pre-specified number of function evaluations called budget. Using this perspective two simple randomised search heuristics, random local search and the (1+1) evolutionary algorithm, are analysed on some well-known example problems. Upper and lower bounds on the expected quality of a solution for a fixed budget of function evaluations are proven. The analysis shows novel and challenging problems in the study of randomised search heuristics. It demonstrates the potential of this shift in perspective from expected run time to expected solution quality.