Proven Runtime Guarantees for How the MOEA/D Computes the Pareto Front From the Subproblem Solutions
arxiv(2024)
摘要
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 Ω(n^(1/2)(n/N + 1)√(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 β∈ (1, 2), we prove an expected
runtime of O(n N log n + n^βlog n) function evaluations.
The O(n^βlog n) 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^β - 1), resulting in an
O(n^βlog 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.
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