Finding a large set of optima in a multimodal optimization landscape is a challenging task. Classical population-based evolutionary algorithms typically converge only to a single solution. While this can be counteracted by applying niching strategies, the number of optima is nonetheless trivially bounded by the population size. Estimation-of-distribution algorithms (EDAs) are an alternative, maintaining a probabilistic model of the solution space instead of a population. Such a model is able to implicitly represent a solution set far larger than any realistic population size. To support the study of how optimization algorithms handle large sets of optima, we propose the test function EqualBlocksOneMax (EBOM). It has an easy fitness landscape with exponentially many optima. We show that the bivariate EDA mutual-information-maximizing input clustering, without any problem-specific modification, quickly generates a model that behaves very similarly to a theoretically ideal model for EBOM, which samples each of the exponentially many optima with the same maximal probability. We also prove via mathematical means that no univariate model can come close to having this property: If the probability to sample an optimum is at least inverse-polynomial, there is a Hamming ball of logarithmic radius such that, with high probability, each sample is in this ball.

翻译：在多模态优化景观中寻找大量最优解是一项具有挑战性的任务。经典的基于种群的进化算法通常只会收敛到单一解。尽管可以通过应用小生境策略来抵消这一局限，但最优解的数量仍受限于种群规模。估计分布算法（EDAs）提供了一种替代方案，它通过维护解空间的概率模型而非种群来工作。此类模型能够隐式地表示远超任何实际种群规模的解集。为支持研究优化算法如何处理大量最优解，我们提出了测试函数EqualBlocksOneMax（EBOM）。该函数具有易于优化的适应度景观，且包含指数级数量的最优解。我们证明，双变量EDA中的互信息最大化输入聚类方法，在无需任何问题特定修改的情况下，能快速生成一个与EBOM理论理想模型行为高度相似的模型——该模型能以相同最大概率采样每个指数级最优解。我们还通过数学手段证明，不存在任何单变量模型能够接近这一特性：若采样一个最优解的概率至少为反多项式量级，则存在一个对数半径的汉明球，使得每个样本高概率落在此球内。