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 while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known nested convex body chasing problem. 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 or $α$-sharp, we propose an algorithm Frugal 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 $Ω(\log{T})$ for both cases, proving optimality of Frugal.

翻译：具有嵌套演化可行集的凸优化(CONES)问题中，目标函数$f$固定不变，但可行域随时间演化为嵌套序列$S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T$。在线算法的目标是在保证各时刻可行性的前提下，同时最小化相对于事后静态最优基准的遗憾值和总移动成本。CONES是著名的嵌套凸体追踪问题的优化导向泛化。当损失函数为凸函数时，我们提出一种惰性算法，并证明其在时间范围$T$内对任意$β\in (0,1]$可实现$O(T^{1-β})$的遗憾值和$O(T^β)$的移动成本。当损失函数为强凸或$α$-锐利时，我们提出Frugal算法，该算法同时实现零遗憾值和$O(\log T)$的移动成本。作为补充，我们证明在两种情形下，任何具有$o(T)$遗憾值的在线算法其移动成本至少为$Ω(\log{T})$，从而证实了Frugal的最优性。