We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to which an accelerated bisection and projection (BP) algorithm is applied. The COP involves a single equality constraint in a matrix variable which is restricted to the intersection of the positive semidefinite cone and a polyhedral cone representing the algebraic properties of binary and box constraints as well as the sparsity structure of the objective polynomial. Our DNN relaxation serves as a variant of the Lasserre moment-SOS hierarchy of relaxations for binary and box constrained POPs when the size of the cones is gradually increased to compute tighter lower bounds for their optimal values. A significant part of the paper is devoted to deriving and analyzing a class of polyhedral cones to strengthen the DNN relaxation, and the design of efficient computation of the metric projections onto these cones in the accelerated BP algorithm. Numerical results on various POPs are available in Ito et al. (ACM Trans Math Softw 45, Article 34, 2019).

中文翻译：

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具有二元和框约束的二次和多项式优化问题的双非负松弛

我们为具有二进制和框约束的多项式优化问题 (POP) 提出了双重非负 (DNN) 松弛。这项工作是 Kim、Kojima 和 Toh 在 2016 年从二次优化问题到 POPs 工作的延伸。密集和稀疏 DNN 松弛被简化为一个简单的圆锥优化问题 (COP)，加速二分和投影 (BP) 算法应用于该问题。COP 涉及矩阵变量中的单个等式约束，该约束限于正半定锥和多面锥的交集，表示二进制和框约束的代数性质以及目标多项式的稀疏结构。当锥体的大小逐渐增加以计算其最优值的更严格的下限时，我们的 DNN 松弛充当二元和框约束 POP 的 Lasserre 矩-SOS 松弛层次结构的变体。论文的一个重要部分致力于推导和分析一类多面体锥体以加强 DNN 松弛，以及在加速 BP 算法中设计对这些锥体的度量投影的有效计算。Ito 等人提供了各种 POPs 的数值结果。（ACM Trans Math Softw 45，第 34 条，2019 年）。以及在加速 BP 算法中有效计算这些锥体上的度量投影的设计。Ito 等人提供了各种 POPs 的数值结果。（ACM Trans Math Softw 45，第 34 条，2019 年）。以及在加速 BP 算法中有效计算这些锥体上的度量投影的设计。Ito 等人提供了各种 POPs 的数值结果。（ACM Trans Math Softw 45，第 34 条，2019 年）。