动态可行域优化中，新算法实现低遗憾与低移动成本的双赢。
Convex Optimization with Nested Evolving Feasible Sets
- 设计懒惰算法和节制算法，应对随时间缩小的可行域。
- 强凸情况下实现零遗憾，移动成本仅约O(log T)。
- 适用于在线优化场景，尤其适合资源受限的实时系统。
考虑凸优化中的嵌套演化可行集（CONES）问题：目标函数固定，但可行域随时间以嵌套序列 $S_1 igsupseteq S_2 igsupseteq \\
原文摘要 · Abstract (English)
\emph{Convex Optimization with Nested Evolving Feasible Sets (CONES)} is considered where the objective function \(f\) remains fixed but the feasible region evolves over time as a nested sequence \(S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T\). The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost $M_\cA(T)$ while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known \emph{nested convex body chasing} (NCBC). When the loss function is convex, we propose a lazy-algorithm and show that it achieves $O(T^{1-β}), O(T^β)$ simultaneous regret and movement cost for any $β\in (0,1]$, over a time horizon of $T$. When the loss function is strongly convex, we propose a \textsc{Frugal} algorithm that simultaneously achieves zero regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with $o(T)$ regret has a movement cost of $Ω\left(\sqrt{\frac{\log{T}}{\log \log T}}\right)$.
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