On the diameter of subgradient sequences in o-minimal structures

Abstract

We study subgradient sequences of locally Lipschitz functions definable in a polynomially bounded o-minimal structure. We show that the diameter of any subgradient sequence is related to the variation in function values, with error terms dominated by a double summation of step sizes. Consequently, we prove that bounded subgradient sequences converge if the step sizes are of order $1/k$. The proof uses Lipschitz $L$-regular stratifications in o-minimal structures to analyze subgradient sequences via their projections onto different strata.