On Nesterov’s Smooth Chebyshev-Rosenbrock Function

We discuss a modification of the chained Rosenbrock function introduced by Nesterov, a polynomial of degree four of $n$ variables. Its only stationary point is the global minimizer with optimal value zero. An initial point is given such that any continuous piecewise linear descent path consists of at least an exponential number of $0.72 \cdot 1.618^{n}$ linear segments before reducing the function value by 75\%. Moreover, there exists a uniform bound, independent of $n$, on the Lipschitz constant of the first and second derivatives of this modified Rosenbrock function along a descent path.

Citation

Technical Report, Dept. of Mathematics, University of Duesseldorf, May 2011