Abstract 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.

中文翻译：

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用于显式约束黑盒优化的基于自适应排序的约束处理

摘要 我们为协方差矩阵自适应进化策略 (CMA-ES) 提出了一种新的约束处理技术。所提出的技术旨在解决显式约束的黑盒连续优化问题，其中显式约束是一种约束，与目标函数的计算时间相比，约束违反及其（数值）梯度的计算时间可以忽略不计。该方法旨在实现两个不变性：对搜索空间的仿射变换不变性，对目标函数和约束函数的递增变换不变性。CMA-ES 旨在拥有这些属性来处理黑盒优化问题中出现的困难，例如不可分离性、病态、鲁棒性、以及目标中的不同数量级。所提出的约束处理技术 (CHT)，称为 ARCH，仅根据候选解决方案的排名修改底层 CMA-ES。它采用修复运算符和自适应排名聚合策略来计算排名。我们开发了测试问题来评估不变性特性的影响，并进行了实验以凭经验验证算法的不变性。我们将所提出的方法与 CEC 2006 约束优化基准套件中的其他 CHT 进行了比较，以证明其有效性。实证研究表明，ARCH 能够有效地利用约束函数的明确性，有时甚至比现有的盒约束处理技术更有效地处理盒约束问题，同时表现出不变性。此外，ARCH 通过不利用目标函数调用次数方面的显式约束，压倒性地优于 CHT。