Abstract
Rigorous runtime analysis is a major approach towards understanding evolutionary computing techniques, and in this area linear pseudo-Boolean objective functions play a central role. Having an additional linear constraint is then equivalent to the NP-hard Knapsack problem, certain classes thereof have been studied in recent works. In this article, we present a dynamic model of optimizing linear functions under uniform constraints. Starting from an optimal solution with respect to a given constraint bound, we investigate the runtimes that different evolutionary algorithms need to recompute an optimal solution when the constraint bound changes by a certain amount. The classical \((1{+}1)\) EA and several population-based algorithms are designed for that purpose, and are shown to recompute efficiently. Furthermore, a variant of the \((1{+}(\lambda ,\lambda ) )\) GA for the dynamic optimization problem is studied, whose performance is better when the change of the constraint bound is small.
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20 July 2020
In the article Reoptimization Time Analysis of Evolutionary Algorithms on Linear Functions Under Dynamic Uniform Constraints, we claimed a worst-case runtime of and for the Multi-Objective Evolutionary Algorithm and the Multi-Objective Genetic Algorithm, respectively, on linear profit functions under dynamic uniform constraint, where denotes the difference between the original constraint bound B and the new one. The technique used to prove these results contained an error. We correct this mistake and show a weaker bound of for both algorithms instead.
Notes
We use the term with high probability if there exists a constant \(c>0\) such that the probability is at least \(1 - 1/n^c\).
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Acknowledgements
The work leading up to this article has received funding from the German Research Foundation under Grant agreement FR2988 (TOSU) and the Australian Research Council under Grant agreements DP140103400 and DP160102401. We would like to thank the anonymous referees, whose comments and suggestions have greatly improved this paper.
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Shi, F., Schirneck, M., Friedrich, T. et al. Reoptimization Time Analysis of Evolutionary Algorithms on Linear Functions Under Dynamic Uniform Constraints. Algorithmica 81, 828–857 (2019). https://doi.org/10.1007/s00453-018-0451-4
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DOI: https://doi.org/10.1007/s00453-018-0451-4