SIAM Journal on Optimization, Volume 30, Issue 4, Page 2750-2779, January 2020.
In this paper, we study $p$-order methods for unconstrained minimization of convex functions that are $p$-times differentiable ($p\geq 2$) with $\nu$-Hölder continuous $p$th derivatives. We propose tensor schemes with and without acceleration. For the schemes without acceleration, we establish iteration complexity bounds of $\mathcal{O}\left(\epsilon^{-1/(p+\nu-1)}\right)$ for reducing the functional residual below a given $\epsilon\in (0,1)$. Assuming that $\nu$ is known, we obtain an improved complexity bound of $\mathcal{O}\left(\epsilon^{-1/(p+\nu)}\right)$ for the corresponding accelerated scheme. For the case in which $\nu$ is unknown, we present a universal accelerated tensor scheme with iteration complexity of $\mathcal{O}\left(\epsilon^{-p/[(p+1)(p+\nu-1)]}\right)$. A lower complexity bound of $\mathcal{O}\left(\epsilon^{-2/[3(p+\nu)-2]}\right)$ is also obtained for this problem class.

中文翻译：

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使用Hölder连续高阶导数最小化凸函数的张量方法

SIAM优化杂志，第30卷，第4期，第2750-2779页，2020年1月。
在本文中，我们研究了用$ \ nu $-Hölder连续$ p $ th个导数对凸函数进行无约束最小化的$ p $阶方法。我们提出了带有和不带有加速度的张量方案。对于没有加速的方案，我们建立$ \ mathcal {O} \ left（\ epsilon ^ {-1 /（p + \ nu-1）} \ right）$的迭代复杂度边界，以将功能残差降低到给定的$ \以下epsilon \ in（0,1）$。假设$ \ nu $是已知的，则对于相应的加速方案，我们可以获得$ \ mathcal {O} \ left（\ epsilon ^ {-1 /（p + \ nu）} \ right）$的改进复杂度范围。对于$ \ nu $未知的情况，我们提出了一种通用加速张量方案，其迭代复杂度为$ \ mathcal {O} \ left（\ epsilon ^ {-p / [（p + 1）（p + \ nu- 1）]} \ right）$。