On Nesterov's smooth Chebyshev-Rosenbrock function
We discuss a modification of the chained Rosenbrock function introduced by Nesterov. This function r(N) is a polynomial of degree 4 defined for x(n). Its only stationary point is the global minimizer x*=(1, 1, ..., 1)(T) with optimal value zero. A point x((0)) in the box B:={x |-1x(i)1 for 1in} with r(N)(x((0)))=1 is given such that there is a continuous piecewise linear descent path within B that starts at x((0)) and leads to x*. It is shown that any continuous piecewise linear descent path starting at x((0)) consists of at least an exponential number of 0.72 center dot 1.618(n) linear segments before reducing the value of r(N) to 0.25. Moreover, there exists a uniform bound, independent of n, on the Lipschitz constant for the second derivative of r(N) within B.
rosenbrock function descent path