The domain of an optimization problem is seen as one of its most important characteristics. In particular, the distinction between continuous and discrete optimization is rather impactful. Based on this, the optimizing algorithm, analyzing method, and more are specified. However, in practice, no problem is ever truly continuous. Whether this is caused by computing limits or more tangible properties of the problem, most variables have a finite resolution. In this work, we use the notion of the resolution of continuous variables to discretize problems from the continuous domain. We explore how the resolution impacts the performance of continuous optimization algorithms. Through a mapping to integer space, we are able to compare these continuous optimizers to discrete algorithms on the exact same problems. We show that the standard $(\mu_W, \lambda)$-CMA-ES fails when discretization is added to the problem.

翻译：优化问题的定义域被视为其最重要的特征之一。特别是，连续优化与离散优化之间的区别具有显著影响。基于此，优化算法、分析方法等也随之确定。然而，在实践中，没有任何问题是真正连续的。无论是由于计算限制，还是问题本身更具体的属性，大多数变量都具备有限的分辨率。在本研究中，我们利用连续变量分辨率的理念，对连续域中的问题进行了离散化处理。我们探讨了分辨率如何影响连续优化算法的性能。通过映射到整数空间，我们能够在完全相同的问题上将这些连续优化器与离散算法进行比较。我们证明，标准的$(\mu_W, \lambda)$-CMA-ES在问题中加入离散化后会失效。