Inferential Performance Assessment of Stochastic Optimisers and the Attainment Function

The performance of stochastic optimisers can be assessed experimentally on given problems by performing multiple optimisation runs, and analysing the results. Since an optimiser may be viewed as an estimator for the (Pareto) minimum of a (vector) function, stochastic optimiser performance is discussed in the light of the criteria applicable to more usual statistical estimators. Multiobjective optimisers are shown to deviate considerably from standard point estimators, and to require special statistical methodology. The attainment function is formulated, and related results from random closed-set theory are presented, which cast the attainment function as a mean-like measure for the outcomes of multiobjective optimisers. Finally, a covariance-measure is defined, which should bring additional insight into the stochastic behaviour of multiobjective optimisers. Computational issues and directions for further work are discussed at the end of the paper.