Goldstein growth and INGD convergence

Determine whether the linear growth of the Goldstein modulus, equivalently the absence of Goldstein flat directions, suffices to guarantee fast or linear convergence specifically for the Interpolated Normalized Gradient Descent (INGD) method.

Background

The paper studies Goldstein-subgradient-based optimization methods, focusing on the INGD algorithm with restarts and on conditions that may explain its empirically observed approximately linear convergence. The authors introduce Goldstein flatness as an obstruction to linear growth of the Goldstein modulus and prove that several structural assumptions—most notably piecewise twice continuously differentiable structure combined with strong convexity, or with quadratic growth and gradient regularity—rule out Goldstein flatness.

Although the absence of Goldstein flat directions has been used to support convergence results for related Goldstein-subgradient methods, the paper does not establish that this condition alone yields fast convergence for INGD. The question is therefore left unresolved and is directly identified as an open issue concerning the algorithmic significance of the new growth condition.