AI 中文总结
本文将Ahmadi等人的d阶牛顿法扩展到带SOS-凸多项式约束的优化，证明其局部收敛阶为d，还分析了三阶牛顿法的最坏情况性能并扩展了全局收敛修正方法。
AI 中文摘要
我们将Ahmadi、Chaudhry和Zhang提出的用于无约束优化的d阶牛顿法[《数学进展》，452:109808]扩展到带有SOS-凸多项式约束的优化问题。考虑在SOS-凸多项式约束下，最小化光滑函数f:ℝⁿ→ℝ的问题。给定迭代点x∈ℝⁿ，Ahmadi等人将下一个迭代点x⁺定义为f在x处的d阶泰勒展开式的最小化者，该展开式带有d'次正则项，其中d'是大于d的最小偶数，且该多项式为SOS-凸，同时满足约束条件。构造该多项式及其在约束下的最小化均可在n的多项式时间内归约为半定规划（SDP）。我们证明，若f是强凸的，且f的d阶偏导数张量是Lipschitz连续的，则所提方法局部收敛到最优解x*，收敛阶为d。我们还证明，在特定约束规格下，x*处的有效约束集会在一次迭代中被局部识别。接下来，我们利用性能估计研究无约束情形下三阶牛顿法对两类单变量f的最坏情况性能。最后，我们将d阶牛顿法的全局收敛修正方法扩展到SOS-凸多项式约束的情形。
英文摘要
We extend a $d$th-order Newton method for unconstrained optimization by Ahmadi, Chaudhry, and Zhang [Advances in Mathematics, 452:109808] to optimization with SOS-convex polynomial constraints. Consider the problem of minimizing a smooth function $f:\mathbb{R}^{n}\to\mathbb{R}$ subject to SOS-convex polynomial constraints. Given an iterate $x\in\mathbb{R}^n$, Ahmadi et al. define the next iterate $x^{+}$ as the minimizer of the $d$th-order Taylor expansion of $f$ at $x$ with a regularization term of degree $d^{\prime}$, where $d^{\prime}$ is the smallest even number greater than $d$, chosen such that this polynomial is SOS-convex, subject to the constraints. Constructing this polynomial and minimizing it subject to the constraints can both be reduced in time polynomial in $n$ to a semidefinite program (SDP). We prove that, if $f$ is strongly convex and the tensor of the $d$th-order partial derivatives of $f$ is Lipschitz continuous, then our method converges locally to the optimal solution $x^{\ast}$ with order $d$. We further prove that, under certain constraint qualifications, the set of active constraints at $x^{\ast}$ is identified locally in a single iteration. Next, we study the worst-case performance of the third-order Newton method in the unconstrained setting for two classes of univariate $f$ using performance estimation. Finally, we extend a globally convergent modification of the $d$th-order Newton method to the setting of SOS-convex polynomial constraints.
