Theoretical analysis of stochastic search algorithms

Theoretical analyses of stochastic search algorithms, albeit few, have always existed since these algorithms became popular. Starting in the 1990s, a systematic approach to analyze the performance of stochastic search heuristics has been put in place. This quickly increasing basis of results allows, nowadays, the analysis of sophisticated algorithms such as population-based evolutionary algorithms, ant colony optimization, and artificial immune systems. Results are available concerning problems from various domains including classical combinatorial and continuous optimization, single and multi-objective optimization, and noisy and dynamic optimization. This chapter introduces the mathematical techniques that are most commonly used in the runtime analysis of stochastic search heuristics in finite, discrete spaces. Careful attention is given to the very popular artificial fitness levels and drift analyses techniques for which several variants are presented. To aid the reader’s comprehension of the presented mathematical methods, these are illustrated by analysis of simple evolutionary algorithms for artificial example functions. The chapter is concluded by providing references to more complex applications and further extensions of the techniques for the obtainment of advanced results.