发表机构
Faculty of Electrical Engineering, Czech Technical University in Prague（布拉格捷克理工大学电气工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明单变量多项式优化的矩-SOS 层级松弛误差以 $O(1/r^2)$ 收敛，并通过精确示例验证该二次速率最优。
AI 中文摘要
我们研究了在实直线有界子集上由任意多项式不等式描述的单变量多项式优化问题（POPs）的矩-SOS（平方和）层级收敛速率。我们证明，对于每个固定的单变量 POP，松弛误差以 $O(1/r^2)$ 为界，其中 $r$ 是松弛阶数。特别地，可行集多项式描述中的边界退化影响常数但不影响收敛指数。证明结合了有限生成单变量二次模的结构与切比雪夫多项式构造，该构造近似恢复可行集的自然生成元，同时控制证书的次数。我们还给出了一个初等的四次示例，其松弛误差恰好为 $1/(2r(r-1))$，表明二次速率是最优的。等价重述将该示例与一个三次单变量问题以及一个可行集在极小值点处具有尖点奇异的双变量 POP 联系起来。
英文摘要
We study the convergence rate of the moment-SOS (sum-of-squares) hierarchy for polynomial optimization problems (POPs) on a bounded subset of the real line described by arbitrary polynomial inequalities. We prove that, for every fixed univariate POP, the relaxation error is bounded by $O(1/r^2)$, where $r$ is the relaxation order. In particular, boundary degeneracies in the polynomial description of the feasible set affect the constant but not the convergence exponent. The proof combines the structure of finitely generated univariate quadratic modules with a Chebyshev polynomial construction that approximately recovers the natural generators of the feasible set while controlling the degree of the certificate. We also give an elementary degree-four example for which the relaxation error is exactly $1/(2r(r-1))$, showing that the quadratic rate is optimal. Equivalent reformulations connect this example to a cubic univariate problem and to a bivariate POP whose feasible set has a cusp singularity at the minimizer.
