Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most \(n+d\). Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most \(n^{O(d)}\) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

来自理论计算机科学的各种关键问题可以表示为布尔超立方体上的多项式优化问题。证明这些类型问题的复杂性界限的一种特别成功的方法是基于平方和（SOS）作为非负性证明。在本文中，我们通过最近的另一种称为非负电路多项式之和（SONC）的替代证书，对布尔超立方体发起了优化问题。我们证明基于SOS的证书的关键结果仍然有效：首先，对于在度数为d的n变量布尔超立方体上为非负数的多项式，最多存在一个SONC度数\（n + d \）。其次，如果存在度d布尔超立方体上多项式的非负性的SONC证书，然后还存在一个短度d SONC证书，其中最多包含\（n ^ {O（d）} \）个非负电路多项式。此外，我们证明，与SOS相反，在对变量进行仿射变换后SONC锥不会闭合，并且对于SONC，不存在与SOS的普京纳尔Positivstellensatz等价的情况。我们从代数和优化角度讨论这些结果。