On the codimension of the set of optima: Large scale optimisation with few relevant variables

Abstract

The complexity of continuous optimisation by comparisonbased algorithms has been developed in several recent papers. Roughly speaking, these papers conclude that a precision ε can be reached with cost Θ(n log(1/ε)) in dimension n within polylogarithmic factors for the sphere function. Compared to other (non comparison-based) algorithms, this rate is not excellent; on the other hand, it is classically considered that comparison-based algorithms have some robustness advantages, as well as scalability on parallel machines and simplicity. In the present paper we show another advantage, namely resilience to useless variables, thanks to a complexity bound Θ(mlog(1/ε)) where m is the codimension of the set of optima, possibly m << n. In addition, experiments show that some evolutionary algorithms have a negligible computational complexity even in high dimension, making them practical for huge problems with many useless variables.