🤖 AI Summary
This paper addresses feasibility and optimization problems over smooth and/or strongly convex constraint sets. To overcome the computational bottleneck of conventional methods—relying on expensive projection or linear optimization oracles—we propose a scalable, accelerated first-order algorithm that requires neither projections nor linear oracles, but only adaptive one-dimensional line searches and normal vector computations. Our key contribution is the first establishment of structural inheritance: the squared Minkowski gauge inherits both smoothness and strong convexity from the underlying constraint set, thereby circumventing analytical difficulties arising from its inherent nonsmoothness and lack of strong convexity. Theoretically, the algorithm achieves an $O(1/T)$ convergence rate under strong convexity, $O(1/T^2)$ under smoothness, and accelerated linear convergence when both properties hold—matching known lower bounds. Moreover, per-iteration complexity is substantially reduced.
📝 Abstract
We consider feasibility and constrained optimization problems defined over smooth and/or strongly convex sets. These notions mirror their popular function counterparts but are much less explored in the first-order optimization literature. We propose new scalable, projection-free, accelerated first-order methods in these settings. Our methods avoid linear optimization or projection oracles, only using cheap one-dimensional linesearches and normal vector computations. Despite this, we derive optimal accelerated convergence guarantees of $O(1/T)$ for strongly convex problems, $O(1/T^2)$ for smooth problems, and accelerated linear convergence given both. Our algorithms and analysis are based on novel characterizations of the Minkowski gauge of smooth and/or strongly convex sets, which may be of independent interest: although the gauge is neither smooth nor strongly convex, we show the gauge squared inherits any structure present in the set.
