Convergence proofs and convergence rates for multi-modal multi-objective optimization
Yann Bonnemay, Olivier Teytaud and Michèle Sebag
In: EA 2005, 2005, Lille.

Abstract

Evolutionary algorithms are population-based. It's an advantage for optimization problems in which a solution set, and not only a solution, is expected. Therefore, they are in particular relevant for - multi-objective optimization (MOO), where the whole Pareto-front is interesting ; - multi-modal optimization (MMO), where all local maxima are interesting ; - multi-objective and multi-modal optimization (MOMMO). As far as we know, state of the art convergence results are of two types : - for stochastic algorithms, they concern convergence in distribution for multi-modal optimization (e.g. simulated annealing results), - for deterministic algorithms, they concern the inclusion of accumulation points in the set of substationary points (which includes the Pareto front). In this paper, we i) give convergence proofs for stochastic MMO, MOO and MOMMO algorithms (in the accumulation sense), ii) give convergence rates for stochastic MOO algorithms that can not be (without further hypothesis) extended to stochastic MMO or MOMMO algorithms, iii) provide convergence criterions for MOO problems. We then discuss the space complexity of population-based MOO, MMO and MOMMO algorithms. We then prove upper and lower bounds on the space complexity of population based MOO and MOMMO algorithms.

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