How does the contraction property fail for convex functions on normed spaces?

2023/11/26 by Ohta, Shin-ichi · 1 citation
#FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2311.15152

Abstract

On Euclidean and Hilbert spaces, Riemannian manifolds, and CAT(0)-spaces, gradient flows of convex functions are known to satisfy the contraction property, which plays a fundamental role in optimization theory and possesses fruitful analytic and geometric applications. On (non-inner product) normed spaces, however, gradient flows of convex functions do not satisfy the contraction property. We give a detailed proof of this characterization of inner products, and discuss a possible form of a weaker contraction property on normed spaces.

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• Discrete-time gradient flows for unbounded convex functions on Gromov hyperbolic spaces

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