Hitting times of local and global optima in genetic algorithms with very high selection pressure

Abstract

The paper is devoted to upper bounds on the expected first hitting times of the sets of local or global optima for non-elitist genetic algorithms with very high selection pressure. The results of this paper extend the range of situations where the upper bounds on the expected runtime are known for genetic algorithms and apply, in particular, to the Canonical Genetic Algorithm. The obtained bounds do not require the probability of fitness-decreasing mutation to be bounded by a constant less than one.

Runtime Analysis of Fitness-Proportionate Selection on Linear Functions

“…Given appropriate scaling of fitness, the SGA without a crossover was shown to be capable of finding the optimum of OneMax within expected polynomial time [25]. A similar conclusion is made in [15] for the SGA with a constant crossover probability p c < 1 optimizing any pseudo-Boolean function without local optima which are not globally optimal. On OneMax, an expected polynomial runtime of the SGA without crossover was established in [6], assuming a reduction of the mutation probability to 1/(6n 2 ).…”

On Proportions of Fit Individuals in Population of Mutation-Based Evolutionary Algorithm with Tournament Selection

“…As corollaries of the general lower bounds on expected proportions of sufficiently fit individuals, we obtain polynomial upper bounds on the Randomized Local Search runtime on unimodal functions and upper bounds on runtime of EAs on 2-SAT problem and on a family of Set Cover problems proposed by Balas (1984). Unlike the upper bounds on runtime of evolutionary algorithms with tournament selection from (Corus et al, 2014;Eremeev, 2016;Lehre, 2011), which require sufficiently large tournament size, the upper bounds on runtime obtained here hold for any tournament size.…”

“…The tools for the non-elitist EA analysis from (Corus et al, 2014;Dang and Lehre, 2016;Eremeev, 2016) can be adjusted to upper-bound the runtime of the EA on B(n, n/2), but in such a case, a non-zero selection pressure would be required with a sufficiently large s and the results would hold only for λ = Ω(log n).…”

“…Here and below, by the runtime we mean the expected number of fitness evaluations made until an optimum is found for the first time. Upper bounds of the runtime of non-elitist GAs, involving the crossover operators, were obtained later in (Corus et al, 2014;Eremeev, 2016). The runtime bounds presented in (Corus et al, 2014;Lehre, 2011) are based on the drift analysis.…”