The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits an SOC representation. Based on this, we give a new algorithm for unconstrained polynomial optimization via SOC programming. We also provide a hybrid numeric-symbolic scheme which combines the numerical procedure with a rounding-projection algorithm to obtain exact nonnegativity certificates. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with fairly large degree and number of variables.

中文翻译：

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通过二阶锥编程进行SONC优化和精确非负性证书

二阶锥（SOC）是一类简单的凸锥，与半定规划相比，可以更有效地对其进行优化。鉴于凸锥具有很强的表达能力，研究哪些凸锥使用SOC表示在理论上和实践上都很有趣。在本文中，我们以建设性的方式证明了非负电路和的锥（SONC）允许SOC表示。基于此，我们给出了一种通过SOC编程进行无约束多项式优化的新算法。我们还提供了一种混合的数字符号方案，该方案将数值过程与舍入投影算法相结合，以获得精确的非负证书。数值实验证明了我们的算法在具有较大程度和变量数量的情况下的效率。