The condition of a function relative to a polytope

TL;DR

This paper introduces a new concept of a relative condition number of a smooth convex function with respect to a polytope, extending classical properties and linking it to convergence rates of optimization algorithms.

Contribution

It defines a relative condition number for convex functions relative to a polytope, generalizing traditional condition number properties and relating it to geometric ratios of scaled polytopes.

Findings

The relative condition number equals the square of the diameter-to-facial-distance ratio of a scaled polytope.

The relative condition number bounds the linear convergence rate of first-order methods.

The concept extends classical condition number properties to constrained optimization settings.

Abstract

The condition number of a smooth convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex function is precisely the square of the diameter-to-width ratio of a canonical ellipsoid associated to the function. Furthermore, the condition number of a function bounds the linear rate of convergence of the gradient descent algorithm for unconstrained minimization. We propose a condition number of a smooth convex function relative to a reference polytope. This relative condition number is defined as the ratio of a relative smooth constant to a relative strong convexity constant of the function, where both constants are relative to the reference polytope. The relative condition number extends the main properties of the traditional condition number. In particular, we show that the condition number of a quadratic convex function relative to a polytope is precisely the square of the diameter-to-facial-distance ratio of a scaled polytope for a canonical scaling induced by the function. Furthermore, we illustrate how the relative condition number of a function bounds the linear rate of convergence of first-order methods for minimization of the function over the polytope.