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Identifying structure of nonsmooth convex functions by the bundle technique. (English) Zbl 1191.90066
Summary: We consider the problem of minimizing nonsmooth convex functions, defined piecewise by a finite number of functions each of which is either convex quadratic or twice continuously differentiable with positive definite Hessian on the set of interest. This is a particular case of functions with primal-dual gradient structure, a notion closely related to the so-called ${\cal {VU}}$ space decomposition: At a given point, nonsmoothness is locally restricted to the directions of the subspace ${\cal V}$, while along the subspace ${\cal U}$ the behavior of the function is twice differentiable. Constructive identification of the two subspaces is important, because it opens the way to devising fast algorithms for nonsmooth optimization (by following iteratively the manifold of smoothness on which superlinear ${\cal U}$-Newton steps can be computed). In this work we show that, for the class of functions in consideration, the information needed for this identification can be obtained from the output of a standard bundle method for computing proximal points, provided a minimizer satisfies the nondegeneracy and strong transversality conditions.

90C30Nonlinear programming
65K05Mathematical programming (numerical methods)
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