# zbMATH — the first resource for mathematics

$${\mathcal V}{\mathcal U}$$-decomposition derivatives for convex max-functions. (English) Zbl 0944.65069
Théra, Michel (ed.) et al., Ill-posed variational problems and regularization techniques. Proceedings of a workshop, Univ. of Trier, Germany, September 3-5, 1998. Berlin: Springer. Lect. Notes Econ. Math. Syst. 477, 167-186 (1999).
Summary: For minimizing a convex max-function $$f$$ we consider, at a minimizer, a space decomposition. That is, we distinguish a subspace $${\mathcal V}$$, where $$f'$$s nonsmoothness is concentrated, from its orthogonal complement, $${\mathcal U}$$. We characterize smooth trajectories, tangent to $${\mathcal U}$$, along which $$f$$ has a second-order expansion. We give conditions (weaker than typical strong second-order sufficient conditions for optimality) guaranteeing the existence of a Hessian of a related $${\mathcal U}$$-Lagrangian. We also prove, under weak assumptions and for a general convex function, superlinear convergence of a conceptual algorithm for minimizing $$f$$ using $${\mathcal V}{\mathcal U}$$-decomposition derivatives.
For the entire collection see [Zbl 0930.00059].

##### MSC:
 65K05 Numerical mathematical programming methods 90C25 Convex programming