In binary polynomial optimization, the goal is to find a binary point maximizing a given polynomial function. In this paper, we propose a novel way of formulating this general optimization problem, which we call factorized binary polynomial optimization. In this formulation, we assume that the variables are partitioned into a fixed number of sets, and that the objective function is written as a sum of r products of linear functions, each one involving only variables in one set of the partition. Our main result is an algorithm that solves factorized binary polynomial optimization in strongly polynomial time, when r is fixed. This result provides a vast new class of tractable instances of binary polynomial optimization, and it even improves on the state-of-the-art for quadratic objective functions, both in terms of generality and running time. We demonstrate the applicability of our result through the binary tensor factorization problem, which arises in mining discrete patterns in data, and that contains as a special case the rank-1 Boolean tensor factorization problem. Our main result implies that these problems can be solved in strongly polynomial time, if the input tensor has fixed rank, and a rank factorization is given. For the rank-1 Boolean matrix factorization problem, we only require that the input matrix has fixed rank.

翻译：在二元多项式优化中，目标在于寻找使给定多项式函数最大化的二元点。本文提出了一种表述该一般优化问题的新方法，称为因子化二元多项式优化。在此表述中，我们假设变量被划分为固定数量的集合，且目标函数被表示为r个线性函数乘积之和，其中每个线性函数仅涉及划分中一个集合内的变量。我们的主要结果是：当r固定时，可在强多项式时间内求解因子化二元多项式优化问题的算法。该结果为二元多项式优化提供了广阔的新型可处理实例类，甚至在二次目标函数方面，无论在一般性还是运行时间上都改进了现有技术水平。我们通过二元张量分解问题展示了所提结果的适用性，该问题产生于数据离散模式挖掘，并以秩-1布尔张量分解问题作为特例。我们的主要结果表明：若输入张量具有固定秩且给定秩分解，则这些问题可在强多项式时间内求解。对于秩-1布尔矩阵分解问题，我们仅要求输入矩阵具有固定秩。