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Modified box constraint handling for the covariance matrix adaptation evolution strategy

We propose a modified box constraint handling technique for the covariance matrix adaptation evolution strategy (CMA-ES). The existing box constraint handling turns the box-constrained optimization problem into an unconstrained optimization by introducing an artificial fitness landscape, where a penalty function is added to the function values at the nearest feasible solutions. By adapting the penalty coefficients, that determine the sensitivity of constraints over the objective function value, it creates a reasonable virtual function landscape outside the feasible domain. In this paper, we address the issue of the original box constraint handling technique that it performs slow adaptation of the penalty coefficients when the objective function scales non-quadratically in particular when the objective function scales exponentially. The optimization is then stagnated until reasonable penalty coefficients are achieved. It is due to a relatively long history of the dispersion measure of the objective function values and the adaptation of the penalty coefficients using the median of the history. In the proposed algorithm, we look at a recent subsequence of the history when the dispersion measures in the history differ significantly. The current dispersion of the objective values is then estimated using the median of the computed subsequence of the history. Experimental results reveal that the proposed algorithm can converges without stagnation on a function with exponential factor, where the original algorithm exhibits stagnation.