AI 中文总结
本文提出AHI多项式族，证明其兼具SOS与SONC性质，构造的Π_AHI锥属SOS锥却异于SONC、SDSOS锥，利用其稀疏性可大幅加速多项式优化计算。
AI 中文摘要
我们通过算术-调和不等式引入了一类新的非负多项式，称为AHI多项式。我们推导了该多项式族的显式代数条件，并证明对于AHI多项式，非负多项式锥与平方和（SOS）多项式锥重合。随后我们研究其凸性，表明尽管AHI多项式通常是非凸的，但某些单项式子结构是SOS凸的。我们进一步在标准非负性证书中精确定位该族：每个AHI多项式同时是SOS和非负回路多项式和（SONC），且属于这两个锥的交集是严格的。通过对该族进行乘法封闭，我们得到锥Π_AHI，其构造上仍属于SOS锥，但我们证明它同时位于SONC锥和更小的SDSOS锥之外。此外，该锥的成员资格具有闭式证书，无需求解任何半定规划。最后，我们展示这些结构在优化中的实用性：数值实验表明，利用AHI稀疏性可使计算时间比稠密SOS松弛快300倍以上，且能求解标准方法因计算限制无法处理的高次多项式优化问题（最高达40次），而Π_AHI的因子分解层次结构可将乘积分解为通用稀疏技术无法检测的独立小子问题。
英文摘要
We introduce a new family of non-negative polynomials, constructed via the arithmetic harmonic inequality, called AHI polynomials. We derive explicit algebraic conditions for this family and prove that, for AHI polynomials, the cone of non-negative polynomials coincides with the cone of sum of squares (SOS) polynomials. We then study their convexity, showing that although AHI polynomials are generally nonconvex, certain monomial substructures are SOS convex. We further locate the family precisely among the standard nonnegativities certificates; every AHI polynomial is simultaneously SOS and a sum of non negative circuit polynomials (SONC), and the containment in the intersection of these two cones is strict. By closing this family under multiplication, we obtain a cone Pi AHI that is, by construction, still SOS, yet we prove that it lies outside both the SONC cone and the smaller SDSOS cone. Moreover, membership in this cone admits a closed form certificate that does not require solving any semidefinite programs. Finally, we demonstrate the usefulness of these structures in optimization, numerical experiments indicate that exploiting AHI sparsity yields a computation time over 300 times faster than dense SOS relaxations and enables solving high degree polynomial optimization problems (up to degree 40) that standard methods cannot handle due to computational limits, and a factorized hierarchy for Pi AHI decomposes products into independent small subproblems that generic sparsity techniques do not detect.
Comments26 pages (including Appendices) and one figure
