Title: Nonmonotone higher-order Taylor approximation methods for composite problems Show Chinese title

Title: 非单调高阶泰勒逼近方法求解复合问题

Abstract: We study composite optimization problems in which the smooth part of the objective function is \( p \)-times continuously differentiable, where \( p \geq 1 \) is an integer. Higher-order methods are known to be effective for solving such problems, as they speed up convergence rates. These methods often require, or implicitly ensure, a monotonic decrease in the objective function across iterations. Maintaining this monotonicity typically requires that the \( p \)-th derivative of the smooth part of the objective function is globally Lipschitz or that the generated iterates remain bounded. In this paper, we propose nonmonotone higher-order Taylor approximation (NHOTA) method for composite problems. Our method achieves the same nice global and rate of convergence properties as traditional higher-order methods while eliminating the need for global Lipschitz continuity assumptions, strict descent condition, or explicit boundedness of the iterates. Specifically, for nonconvex composite problems, we derive global convergence rate to a stationary point of order \( \mathcal{O}(k^{-\frac{p}{p+1}}) \), where \( k \) is the iteration counter. Moreover, when the objective function satisfies the Kurdyka-{\L}ojasiewicz (KL) property, we obtain improved rates that depend on the KL parameter. Furthermore, for convex composite problems, our method achieves sublinear convergence rate of order \( \mathcal{O}(k^{-p}) \) in function values. Finally, preliminary numerical experiments on nonconvex phase retrieval problems highlight the promising performance of the proposed approach. Show Chinese abstract

Abstract: 我们研究了目标函数光滑部分为 \( p \)-次连续可微的复合优化问题，其中 \( p \geq 1 \) 是一个整数。 已知高阶方法对于解决此类问题非常有效，因为它们可以加快收敛速度。这些方法通常要求或隐式保证目标函数在迭代过程中单调下降。 为了保持这种单调性，通常需要目标函数光滑部分的 \( p \)-阶导数全局满足 Lipschitz 条件，或者生成的迭代点保持有界。 本文提出了一种针对复合问题的非单调高阶泰勒近似（NHOTA）方法。该方法在保持传统高阶方法的良好全局和收敛速率特性的同时，消除了对全局 Lipschitz 连续性假设、严格下降条件或显式迭代点有界的依赖。 具体而言，对于非凸复合问题，我们得到了到平稳点的全局收敛率为 \( \mathcal{O}(k^{-\frac{p}{p+1}}) \)，其中 \( k \) 是迭代计数器。 此外，当目标函数满足 Kurdyka-{\L }ojasiewicz (KL) 性质时，我们获得了依赖于 KL 参数的改进收敛率。 另外，对于凸复合问题，我们的方法在函数值上实现了次线性收敛率 \( \mathcal{O}(k^{-p}) \)。 最后，在非凸相位检索问题上的初步数值实验展示了所提出方法的有前景性能。