On Nesterov’s Nonsmooth Chebyschev-Rosenbrock Functions

We discuss two nonsmooth functions on R^n introduced by Nesterov. We show that the first variant is partly smooth in the sense of [A.S. Lewis. Active sets, nonsmoothness and sensitivity. SIAM Journal on Optimization, 13:702–725, 2003.] and that its only stationary point is the global minimizer. In contrast, we show that the second variant has $2^{n-1}$ Clarke stationary points, none of them local minimizers except the global minimizer. Furthermore, its only stationary point in the sense of Mordukhovich [R.T. Rockafellar and R.J.B. Wets. Variational Analysis. Springer, New York, 1998.] is the global minimizer. Nonsmooth optimization algorithms from multiple starting points generate iterates that approximate all $2^{n-1}$ Clarke stationary points, not only the global minimizer, but it remains an open question as to whether the other Clarke stationary points are actually points of attraction for optimization algorithms.


Submitted. Jan 2011.



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