发表机构
University of Antwerp（安特卫普大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出非凸复合优化的张量包络（TEE），基于泰勒模型与幂正则化构建，保留全局极小与水平有界性，并作为精确张量迭代的 Lyapunov 函数，在无全局 Lipschitz 假设下给出复杂度界。
AI 中文摘要
我们针对复合目标函数 $\varphi=f+g$ 引入了一种张量包络（TEE），该包络由光滑项 $f$ 的 $q$ 阶泰勒模型经 $p$ 次幂（$p>q$）正则化而构建。为构造该包络，我们首先建立了张量下降引理，并刻画了其在 $q<p<q+1$、$p=q+1$ 和 $p>q+1$ 三种区间内的行为。在适当的 prox-有界性和参数条件下，相应的张量算子（TOP）具有非空紧值，TEE 为有限值且连续，保持全局极小值点，并继承 $\varphi$ 的水平有界性。我们进一步证明 TEE 是局部 Lipschitz 连续且方向可微的，刻画了其 Fréchet 可微性，并证明当 TOP 为单值时 TEE 是 $C^1$ 的。因此，TEE 保留了 $\varphi$ 的基本优化景观性质，同时具有更丰富的解析结构，这有助于基于张量的优化方法的设计与分析。最后，TEE 作为精确张量迭代的 Lyapunov 函数：每个聚点都是临界的，并且对于满足 $q<p\le q+1$ 的有界轨迹，在 $\mathcal{O}(\epsilon^{-p/(p-1)})$ 次迭代内达到 $\epsilon$-稳定点。据我们所知，张量下降引理和该复杂度界均是在不假设 $f$ 的 $q$ 阶导数具有全局 Lipschitz 或 Hölder 连续性的情况下首次获得的。
英文摘要
We introduce a tensor envelope (TEE) for composite objectives $φ=f+g$, built from a $q$th-order Taylor model of the smooth term $f$ regularized by a $p$th power with $p>q$. To construct it, we first establish a tensor descent lemma and characterize its behavior in the three regimes $q<p<q+1$, $p=q+1$, and $p>q+1$. Under suitable prox-boundedness and parameter conditions, the associated tensor operator (TOP) has nonempty compact values, and TEE is finite-valued and continuous, preserves global minimizers, and inherits the level-boundedness of $φ$. We further show that TEE is locally Lipschitz continuous and directionally differentiable, characterize its Fréchet differentiability, and prove that it is $C^1$ whenever TOP is single-valued. As such, TEE preserves the essential optimization landscape properties of $φ$ while enjoying a richer analytical structure, which facilitates the design and analysis of tensor-based optimization methods. Finally, TEE serves as a Lyapunov function for the exact tensor iteration: every cluster point is critical, and, for bounded trajectories with $q<p\le q+1$, an $ε$-stationary point is reached within $\mathcal{O}(ε^{-p/(p-1)})$ iterations. To our knowledge, both the tensor descent lemma and this complexity bound are the first of their kind obtained without assuming global Lipschitz or Hölder continuity of the $q$th-order derivative of $f$.
