Runtime Analysis of Evolutionary Diversity Optimization on the Multi-objective (LeadingOnes, TrailingZeros) Problem

• URL: http://arxiv.org/abs/2404.11496v2
• Date: Fri, 19 Apr 2024 00:31:21 GMT
• Title: Runtime Analysis of Evolutionary Diversity Optimization on the Multi-objective (LeadingOnes, TrailingZeros) Problem
• Authors: Denis Antipov, Aneta Neumann, Frank Neumann, Andrew M. Sutton,
• Abstract summary: We prove evolutionary diversity optimization (EDO) on a 3-objective function LOTZ$_k$. We show that the GSEMO computes a set of all Pareto-optimal solutions in $O(kn3)$ expected iterations. We complement our theoretical analysis with an empirical study, which shows a very similar behavior for both diversity measures.
• Score: 10.697103866816958
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The diversity optimization is the class of optimization problems, in which we aim at finding a diverse set of good solutions. One of the frequently used approaches to solve such problems is to use evolutionary algorithms which evolve a desired diverse population. This approach is called evolutionary diversity optimization (EDO). In this paper, we analyse EDO on a 3-objective function LOTZ$_k$, which is a modification of the 2-objective benchmark function (LeadingOnes, TrailingZeros). We prove that the GSEMO computes a set of all Pareto-optimal solutions in $O(kn^3)$ expected iterations. We also analyze the runtime of the GSEMO$_D$ (a modification of the GSEMO for diversity optimization) until it finds a population with the best possible diversity for two different diversity measures, the total imbalance and the sorted imbalances vector. For the first measure we show that the GSEMO$_D$ optimizes it asymptotically faster than it finds a Pareto-optimal population, in $O(kn^2\log(n))$ expected iterations, and for the second measure we show an upper bound of $O(k^2n^3\log(n))$ expected iterations. We complement our theoretical analysis with an empirical study, which shows a very similar behavior for both diversity measures that is close to the theory predictions.

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