  Theoretical analyses of stochastic search algorithms, albeit few, have always
existed since these algorithms became popular. Starting in the nineties a
systematic approach to analyse 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 optimisation and artificial immune systems.
Results are available concerning problems from various domains including
classical combinatorial and continuous optimisation, single and multi-objective
optimisation, and noisy and dynamic optimisation. This chapter introduces the
mathematical techniques that are most commonly used in the runtime analysis of
stochastic search heuristics. 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 applied to the 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.
