Proven Runtime Guarantees for How the MOEA/D Computes the Pareto Front From the Subproblem Solutions
<|reference_start|>Proven Runtime Guarantees for How the MOEA/D Computes the Pareto Front From the Subproblem Solutions: The decomposition-based multi-objective evolutionary algorithm (MOEA/D) does not directly optimize a given multi-objective function $f$, but instead optimizes $N + 1$ single-objective subproblems of $f$ in a co-evolutionary manner. It maintains an archive of all non-dominated solutions found and outputs it as approximation to the Pareto front. Once the MOEA/D found all optima of the subproblems (the $g$-optima), it may still miss Pareto optima of $f$. The algorithm is then tasked to find the remaining Pareto optima directly by mutating the $g$-optima.   In this work, we analyze for the first time how the MOEA/D with only standard mutation operators computes the whole Pareto front of the OneMinMax benchmark when the $g$-optima are a strict subset of the Pareto front. For standard bit mutation, we prove an expected runtime of $O(n N \log n + n^{n/(2N)} N \log n)$ function evaluations. Especially for the second, more interesting phase when the algorithm start with all $g$-optima, we prove an $\Omega(n^{(1/2)(n/N + 1)} \sqrt{N} 2^{-n/N})$ expected runtime. This runtime is super-polynomial if $N = o(n)$, since this leaves large gaps between the $g$-optima, which require costly mutations to cover.   For power-law mutation with exponent $\beta \in (1, 2)$, we prove an expected runtime of $O\left(n N \log n + n^{\beta} \log n\right)$ function evaluations. The $O\left(n^{\beta} \log n\right)$ term stems from the second phase of starting with all $g$-optima, and it is independent of the number of subproblems $N$. This leads to a huge speedup compared to the lower bound for standard bit mutation. In general, our overall bound for power-law suggests that the MOEA/D performs best for $N = O(n^{\beta - 1})$, resulting in an $O(n^\beta \log n)$ bound. In contrast to standard bit mutation, smaller values of $N$ are better for power-law mutation, as it is capable of easily creating missing solutions.<|reference_end|>
@article{doerr2024proven,
    title={Proven Runtime Guarantees for How the MOEA/D Computes the Pareto Front
  From the Subproblem Solutions},
    author={Benjamin Doerr and Martin S. Krejca and No'e Weeks},
    journal={Parallel Problem Solving from Nature, PPSN 2024, 197-212. Springer},
    year={2024},
    doi={10.1007/978-3-031-70071-2\_13},
    archivePrefix={arXiv},
    eprint={2405.01014},
    primaryClass={cs.NE}
}