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$𝒱𝒰$-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 $𝒱$, where ${f}^{\text{'}}$s nonsmoothness is concentrated, from its orthogonal complement, $𝒰$. We characterize smooth trajectories, tangent to $𝒰$, 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 $𝒰$-Lagrangian. We also prove, under weak assumptions and for a general convex function, superlinear convergence of a conceptual algorithm for minimizing $f$ using $𝒱𝒰$-decomposition derivatives.

##### MSC:
 65K05 Mathematical programming (numerical methods) 90C25 Convex programming