AI 中文总结
研究针对偶数p时计算ℓ_p非扩张映射的ε近似不动点问题，提出时间复杂度为poly(d, p, log(1/ε))的算法并给出首个固定p ≠ 2时运行时间为poly(d, log(1/ε))的算法，技术基于Sion定理并扩展到更一般问题。
AI 中文摘要
我们给出了一种时间复杂度为poly(d, p, log(1/ε))的算法，用于计算任意ℓ_p非扩张映射f: ℳ → ℳ的ε近似不动点，其中ℳ ⊂ ℝ^d是凸紧集且p是偶数整数。这是对于任意固定p ≠ 2的首个运行时间为poly(d, log(1/ε))的算法。我们的技术基于非紧极小极大问题的Sion定理的高效计算版本，并扩展到允许低次多项式势的更一般的全搜索问题。
英文摘要
We give a $\text{poly}(d, p, \log(1/ε))$-time algorithm that computes an $ε$-approximate fixed point of any $\ell_p$-nonexpansive map $f : \mathcal{X} \to \mathcal{X}$, where $\mathcal{X} \subset \mathbb R^d$ is a convex compact set and $p$ is an even integer. This is the first algorithm with $\text{poly}(d, \log(1/ε))$ runtime for any fixed $p \ne 2$. Our techniques are based on a computationally efficient version of Sion's theorem for non-compact minmax problems, and extend to more general total search problems that admit low-degree polynomial potentials.
