# The condition number of a function relative to a set

## TL;DR

This paper introduces a new concept of a relative condition number for convex functions with respect to a set, extending classical notions to constrained optimization and providing bounds and characterizations for specific function-set pairs.

## Contribution

The paper defines a relative condition number for convex functions relative to a set, generalizing classical condition numbers and analyzing its properties and bounds in specific cases.

## Key findings

- The relative condition number extends classical properties and characterizations.
- Bounds are provided for functions of the form f = g ∘ A relative to convex sets.
- The relative condition number influences the convergence analysis of first-order methods.

## Abstract

The condition number of a differentiable 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 the square of the aspect 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 convex minimization. We propose a condition number of a differentiable convex function relative to a reference convex set and distance function pair. This relative condition number is defined as the ratio of a relative smoothness to a relative strong convexity constants. We show that the relative condition number extends the main properties of the traditional condition number both in terms of its geometric insight and in terms of its role in characterizing the linear convergence of first-order methods for constrained convex minimization. When the reference set $X$ is a convex cone or a polyhedron and the function $f$ is of the form $f = g\circ A$, we provide characterizations of and bounds on the condition number of $f$ relative to $X$ in terms of the usual condition number of $g$ and a suitable condition number of the pair $(A,X)$.