  We propose a novel constraint-handling technique for the covariance matrix
adaptation evolution strategy (CMA-ES). The proposed technique is aimed at
solving explicitly constrained black-box continuous optimization problems, in
which the explicit constraint is a constraint whereby the computational time
for the constraint violation and its (numerical) gradient are negligible
compared to that for the objective function. This method is designed to realize
two invariance properties: invariance to the affine transformation of the
search space, and invariance to the increasing transformation of the objective
and constraint functions. The CMA-ES is designed to possess these properties
for handling difficulties that appear in black-box optimization problems, such
as non-separability, ill-conditioning, ruggedness, and the different orders of
magnitude in the objective. The proposed constraint-handling technique (CHT),
known as ARCH, modifies the underlying CMA-ES only in terms of the ranking of
the candidate solutions. It employs a repair operator and an adaptive ranking
aggregation strategy to compute the ranking. We developed test problems to
evaluate the effects of the invariance properties, and performed experiments to
empirically verify the invariance of the algorithm. We compared the proposed
method with other CHTs on the CEC 2006 constrained optimization benchmark suite
to demonstrate its efficacy. Empirical studies reveal that ARCH is able to
exploit the explicitness of the constraint functions effectively, sometimes
even more efficiently than an existing box-constraint handling technique on
box-constrained problems, while exhibiting the invariance properties. Moreover,
ARCH overwhelmingly outperforms CHTs by not exploiting the explicit constraints
in terms of the number of objective function calls.
