Convex Optimization with Nested Evolving Feasible Sets

Abstract

\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 $O(T^{1-\beta}), O(T^\beta)$ simultaneous regret and movement cost for any $\beta \in (0,1]$, 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 $\Omega\left(\sqrt{\frac{\log{T}}{\log \log T}}\right)$.