Convex Optimization with Nested Evolving Feasible Sets

2026-05-08

AI Summary

This study addresses dynamic convex optimization problems where the objective function remains fixed while the feasible region undergoes nested, time-varying contractions. The goal is to simultaneously minimize static regret and total movement cost while maintaining feasibility of the iterates at all times. The work introduces Lazy and Frugal algorithms that extend nested convex body chasing to a general convex optimization framework. For strongly convex or α-sharp loss functions, the proposed methods achieve, for the first time, zero regret with $O(\log T)$ movement cost, and establish $\Omega(\log T)$ as a lower bound on movement cost. In the case of general convex functions, the algorithms attain a Pareto-optimal trade-off between $O(T^{1-\beta})$ regret and $O(T^\beta)$ movement cost.

Abstract

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.

Problem

• Convex Optimization
• Nested Feasible Sets
• Online Algorithm
• Regret Minimization
• Movement Cost

Innovation

• nested convex optimization
• online learning
• movement cost
• regret minimization
• strongly convex