  In several real-world applications in medical and control engineering, there
are unsafe solutions whose evaluations involve inherent risk. This optimization
setting is known as safe optimization and formulated as a specialized type of
constrained optimization problem with constraints for safety functions. Safe
optimization requires performing efficient optimization without evaluating
unsafe solutions. A few studies have proposed the optimization methods for safe
optimization based on Bayesian optimization and the evolutionary algorithm.
However, Bayesian optimization-based methods often struggle to achieve superior
solutions, and the evolutionary algorithm-based method fails to effectively
reduce unsafe evaluations. This study focuses on CMA-ES as an efficient
evolutionary algorithm and proposes an optimization method termed safe CMA-ES.
The safe CMA-ES is designed to achieve both safety and efficiency in safe
optimization. The safe CMA-ES estimates the Lipschitz constants of safety
functions transformed with the distribution parameters using the maximum norm
of the gradient in Gaussian process regression. Subsequently, the safe CMA-ES
projects the samples to the nearest point in the safe region constructed with
the estimated Lipschitz constants. The numerical simulation using the benchmark
functions shows that the safe CMA-ES successfully performs optimization,
suppressing the unsafe evaluations, while the existing methods struggle to
significantly reduce the unsafe evaluations.
