TY - JOUR

T1 - The condition number of a function relative to a set

AU - Gutman, David H.

AU - Peña, Javier F.

N1 - Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

PY - 2020

Y1 - 2020

N2 - 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 relative smoothness to 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∘ 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).

AB - 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 relative smoothness to 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∘ 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).

UR - http://www.scopus.com/inward/record.url?scp=85085072609&partnerID=8YFLogxK

U2 - 10.1007/s10107-020-01510-4

DO - 10.1007/s10107-020-01510-4

M3 - Article

AN - SCOPUS:85085072609

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

ER -