Abstract
We propose a new black-box complexity model for search algorithms evaluating $\lambda $ search points in parallel. The parallel unary unbiased black-box complexity gives lower bounds on the number of function evaluations every parallel unary unbiased black-box algorithm needs to optimize a given problem. It captures the inertia caused by offspring populations in evolutionary algorithms and the total computational effort in parallel metaheuristics. 1 We present complexity results for LeadingOnes and OneMax. Our main result is a general performance limit: we prove that on every function every $\lambda $ -parallel unary unbiased algorithm needs at least a certain number of evaluations (a function of problem size and $\lambda $ ) to find any desired target set of up to exponential size, with an overwhelming probability. This yields lower bounds for the typical optimization time on unimodal and multimodal problems, for the time to find any local optimum, and for the time to even get close to any optimum. The power and versatility of this approach is shown for a wide range of illustrative problems from combinatorial optimization. Our performance limits can guide parameter choice and algorithm design; we demonstrate the latter by presenting an optimal $\lambda $ -parallel algorithm for OneMax that uses parallelism most effectively. 1 This article significantly extends preliminary results which appeared in [1] .
Highlights
B LACK-BOX optimisation describes a challenging realm of problems where no algebraic model or gradient information is available
General-purpose metaheuristics like evolutionary algorithms, simulated annealing, ant colony optimisers, tabu search, and particle swarm optimisers are well suited for black-box optimisation as they generally work well without any problem-dependent knowledge
A lot of research has focussed on designing powerful metaheuristics, yet it is often unclear which search paradigm works best for a particular problem class, and whether and how better performance can be obtained by tailoring a search paradigm to the problem class in hand
Summary
B LACK-BOX optimisation describes a challenging realm of problems where no algebraic model or gradient information is available. It describes the minimum number of function evaluations that every blackbox algorithm needs to make to optimise a problem from a given class It provides a rigorous theoretical foundation through capturing limits to the efficiency of all black-box search algorithms, providing a baseline for performance comparisons across all known and future metaheuristics as well as tailored black-box algorithms. These limits can be used to set stopping criteria appropriately, avoiding stopping an algorithm before it has had a chance to come close to local or global optima They are useful to set parameters such as the offspring population size λ: if we have a limited computational budget of T evaluations, (1) implies that we must choose λ satisfying λ/ ln+ λ ≤ T /(cn) as for larger values T is lower than (1), meaning that every λ-parallel unary unbiased black-box algorithm fails badly with overwhelming probability. The feasibility of this approach is demonstrated in this work as we present an optimal λ-parallel algorithm for ONEMAX that uses parallelism most effectively
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