The convergence of Broyden algorithms for LC gradient function (Q1582577)

Broyden algorithms are very efficient methods for solving the unrestricted nonlinear programming problem \(\min \{f(x)\mid x\in\mathbb{R}^n\}\). In this paper the convergence of such algorithms is discussed for functions \(f\) which are non-twice-differentiable, but have LC-gradients. Assume \(f\) is uniformly convex and an LC-gradient function then the Broyden algorithms have linear convergence. It is proved that under some additional conditions the convergence is superlinear in the sense \(\|x_k\|\leq C\delta^k\), \(0< \delta< 1\). Unfortunately, the assumptions are rather strong, and there is no example!