Typesetting: Editing and reformatting of the authors' input files using a Springer TEX macro package SPIN: 11326120 4113111-5432 Printed on acid-free paper 4 First-Order Differentiation . . . . . . . . . . . . . 4.1 One-Sided Differentiability of Convex Functions 4.2 Basic Properties of Subderivatives 4.3 Calculus Rules .

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This paper relates some numerical experiments with variable storage quasi-Newton methods for the optimization of large-scale models. The basic idea of the recommended algorithm is to start bfgs updates on a diagonal matrix, itself generated by an update formula and adjusted to Rayleigh's ellipsoid of the local Hessian of the objective function in the direction of the change in the gradient. A variational derivation of some rank one and rank two updates in Hilbert spaces is also given.

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When computing the infimal convolution of a convex function f with the squared norm, the so-called Moreau-Yosida regularization of f is obtained. Among other things, this function has a Lipschitzian gradient. We investigate some more of its properties, relevant for optimization. The most important part of our study concerns second-order differentiability: existence of a secondorder development of f implies that its regularization has a Hessian. For the converse, we disclose the importance of the decomposition of R N along U (the subspace where f is "smooth") and V (the subspace parallel to the subdifferential of f).