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.
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Submitted. Jan 2011.
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