decomposition_ensemble

This is a Python implementation class based on the MOEA/D multi-objective optimization algorithm[7]. It decomposes multi-objective optimization into multiple single-objective problems and solves them with CMA-ES[8].
Unlike the original paper, it does not share information between neighboring problems; each decomposed single-objective problem is solved independently with CMA-ES.

Overview

MOEA/D decomposes a multi-objective optimization problem into multiple single-objective problems using an appropriate approach and solves each subproblem. The original paper[7] introduces the weighted-sum, Tchebycheff, and Penalty-based Boundary Intersection (PBI) approaches.

In this implementation, the decomposed single-objective problems are solved independently with CMA-ES. The solutions obtained during optimization are non-dominated sorted[6], and the rank-1 individuals are archived as Pareto-solution candidates.

info

When the problem is decomposed into single-objective problems, large differences in objective-function scales can bias the search.
When using decomposition_ensemble, normalizing the objective functions with coefficient is recommended. See the Dmodel multi-objective optimization example for a specific example.

Configurable keyword arguments

• num_decomposition: int ... Number of decompositions for multi-objective optimization. The default is 10.

Trade-off between computation time and Pareto-solution density with num_decomposition

Increasing num_decomposition creates a finer decomposition of the multi-objective problem and produces a denser set of Pareto solutions.
However, computation time increases in proportion to num_decomposition.

• decomposition_type: str ... Decomposition approach. Set to one of weighted_sum, tchebycheff, or pbi. The default is "tchebycheff".
• seed: int | null ... Random seed. The default is null (the random seed is not fixed).
• mean: np.ndarray ... Initial value of CMA-ES $\boldsymbol{m}$. The default is $\boldsymbol{m} = \boldsymbol{0}$.
• sigma: float ... Initial standard deviation of CMA-ES $\boldsymbol{C}$. The default is 1.
• bounds: tuple[float, float] | list[tuple[float, float]] ... Lower and upper bounds for each CMA-ES variable.

Behavior of bounds

• Default (when omitted) ... No bounds.
• With bounds: tuple[float, float] ... The specified pair is used as the bounds for every variable.
• With bounds: list[tuple[float, float]] ... The specified pairs are used as the bounds for each variable. The list may be truncated; bounds after the truncation point are automatically set to -1 to 1.

• population_size int ... Number of samples per CMA-ES iteration. The default is the CMA-ES recommendation, $4+\lfloor3\ln{n}\rfloor$.

info

The number of samples per optimization iteration is population_size × num_decomposition.