Convex Optimization with Nested Evolving Feasible Sets

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 (S1 supseteq S2 supseteq cdots supseteq ST). 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 nested convex body chasing (NCBC). When the loss function is convex, we propose a lazy-algorithm and show that it achieves O(T1-β), O(Tβ) simultaneous regret and movement cost for any β ∈ (0,1], over a time horizon of T. When the loss function is strongly convex, we propose a textscFrugal 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 Omegaleft(√fraclogTlog log Tright).