On Nesterov's smooth Chebyshev-Rosenbrock function

We discuss a modification of the chained Rosenbrock function introduced by Nesterov. This function rN is a polynomial of degree four defined for x ∈ R. Its only stationary point is the global minimizer x∗ = (1, 1, . . . , 1) with optimal value zero. A point x in the box B := {x | −1 ≤ xi ≤ 1 for 1 ≤ i ≤ n} with rN (x) = 1 is given such that there is a continuous descent path within B that starts at x and leads to x∗. It is shown that any continuous piecewise linear descent path starting at x consists of at least an exponential number of 0.72 · 1.618 linear segments before reducing the value of rN to 0.25. Moreover, there exists a uniform bound, independent of n, on the Lipschitz constant for the second derivative of rN within B.