Direct Multisearch is a well-established class of algorithms, suited for multiobjective derivative-free optimization. In this work, we analyze the worst-case complexity of this class of methods in its most general formulation for unconstrained optimization. Considering nonconvex smooth functions, we show that to drive a given criticality measure below a specific positive threshold, Direct Multisearch takes at most a number of iterations proportional to the square of the inverse of the threshold, raised to the number of components of the objective function. This number is also proportional to the size of the set of linked sequences between the first unsuccessful iteration and the iteration immediately before the one where the criticality condition is satisfied. We then focus on a particular instance of Direct Multisearch, which considers a more strict criterion for accepting new nondominated points. In this case, we can establish a better worst-case complexity bound, simply proportional to the square of the inverse of the threshold, for driving the same criticality measure below the considered threshold.

中文翻译：

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用于多目标优化的定向直接搜索方法的最坏情况复杂度界限

Direct Multisearch 是一类完善的算法，适用于多目标无导数优化。在这项工作中，我们分析了此类方法在其最通用的无约束优化公式中的最坏情况复杂性。考虑到非凸平滑函数，我们表明为了将给定的关键性度量驱动到特定的正阈值以下，直接多重搜索最多需要多次迭代，迭代次数与阈值的倒数平方成正比，增加到目标函数的分量数. 该数字还与第一次不成功迭代和紧接在满足临界条件的迭代之前的迭代之间的链接序列集的大小成正比。然后我们专注于 Direct Multisearch 的一个特定实例，它考虑了接受新的非支配点的更严格的标准。在这种情况下，我们可以建立一个更好的最坏情况复杂度界限，与阈值的倒数的平方成正比，以将相同的关键性度量驱动到所考虑的阈值以下。