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Direct Multisearch is a well-established class of algorithms, suited formultiobjective derivative-free optimization. In this work, we analyze theworst-case complexity of this class of methods in its most general formulationfor unconstrained optimization. Considering nonconvex smooth functions, we showthat to drive a given criticality measure below a specific positive threshold,Direct Multisearch takes at most a number of iterations proportional to thesquare of the inverse of the threshold, raised to the number of components ofthe objective function. This number is also proportional to the size of the setof linked sequences between the first unsuccessful iteration and the iterationimmediately before the one where the criticality condition is satisfied. Wethen focus on a particular instance of Direct Multisearch, which considers amore strict criterion for accepting new nondominated points. In this case, wecan establish a better worst-case complexity bound, simply proportional to thesquare of the inverse of the threshold, for driving the same criticalitymeasure below the considered threshold.

Worst-case Complexity Bounds of Directional Direct-search Methods for Multiobjective Optimization