$\mathcal{V}\mathcal{U}$-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 ${f}^{\text{'}}$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.

##### MSC:

65K05 | Mathematical programming (numerical methods) |

90C25 | Convex programming |