Gauges and Accelerated Optimization over Smooth and/or Strongly Convex Sets

TL;DR

This work addresses feasibility and constrained optimization over sets that exhibit $eta$-smoothness or $\\alpha$-strong convexity, introducing projection-free, accelerated first-order methods that rely on cheap one-dimensional line searches and boundary normal computations. The core idea is to leverage the Minkowski gauge $\\gamma_{S,e}$ and its squared form $\tfrac{1}{2}\\gamma^2_{S,e}$ to reformulate constraints and objectives via radial/gauge representations, enabling unconstrained optimization on gauges. The authors establish structural results linking set curvature to gauge-squared smoothness/strong convexity, prove tightness and converse properties, and develop accelerated algorithms (level-projection, accelerated generalized gradient) with rates $O(1/T)$, $O(1/T^2)$, and accelerated linear convergence under appropriate conditions. They validate the theory numerically on $Lg$-norm ellipsoid feasibility and trust-region problems, showing that projection-free radial methods can be competitive with traditional solvers in high dimensions. This work broadens projection-free optimization by exploiting geometric gauge properties, offering scalable tools for constrained problems where projections are expensive or unavailable.

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.

Figures (24)

• Figure : (a) $p=1.5$-norm ball
• Figure : (a) $(p_1,p_2)$=(1.5,1.8)
• Figure : (a) $(n,m)=(400,200)$
• Figure : (a) $(n,m)=(400,200)$
• Figure : (a) $(n,m)=(400,200)$

Theorems & Definitions (38)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• Lemma 8
• proof