: When computing the infimal convolution of a convex function f with the squared norm, one obtains the so-called Moreau-Yosida regularization of f . Among other things, this function has a Lipschitzian gradient. We investigate some more of its properties, relevant for optimization. Our main result concerns second-order differentiability and is as follows. Under assumptions that are quite reasonable in optimization, the Moreau-Yosida is twice diffferentiable if and only if f is twice differentiable as well. In the course of our development, we give some results of general interest in convex analysis. In particular, we establish primaldual relationship between the remainder terms in the first-order development of a convex function and its conjugate. Key-words: Convex optimization, mathematical programming, proximal point, secondorder differentiability. (R'esum'e : tsvp) Unite de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Telep...

This paper proposes an implementable proximal quasi-Newton method for minimizing a nondifferentiable convex function f in ! n . The method is based on Rockafellar's proximal point algorithm and a cutting-plane technique. At each step, we use an approximate proximal point p a (x k ) of x k to define a v k 2 @ ffl k f(p a (x k )) with ffl k ffkv k k; where ff is a constant. The method monitors the reduction in the value of kv k k to identify when a line search on f should be used. The quasi-Newton step is used to reduce the value of kv k k. Without the differentiability of f , the method converges globally and the rate of convergence is Q-linear. Superlinear convergence is also discussed to extend the characterization result of Dennis and Mor'e. Numerical results show the good performance of the method. Key words. nondifferentiable convex optimization, proximal point, quasi-Newton method, cutting-plane method, bundle methods. AMS subject classifications. 65K05, 90C30 Abbrevia...

Abstract. The question of second-order expansions is taken up for a class of functions of importance in optimization, namely Moreau envelope regularizations of nonsmooth functions f. It is shown that when f is prox-regular, which includes convex functions and the extended-real-valued functions representing problems of nonlinear programming, the many second-order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself. This clarifies the circumstances conducive to developing computational methods based on envelope functions, such as second-order approximations in nonsmooth optimization and variants of the proximal point algorithm. The results establish that generalized second-order expansions of Moreau envelopes, at least, can be counted on in most situations of interest in finite-dimensional optimization. Keywords. Prox-regularity, amenable functions, primal-lower-nice functions, Hessians, first- and second-order expansions, strict proto-derivatives, proximal mappings, Moreau envelopes, regularization, subgradient mappings, nonsmooth analysis, variational analysis, proto-derivatives, second-order epi-derivatives, Attouch’s theorem.

Problems of nonlinear programming are placed in a broader framework of composite optimization. This allows second-order smoothness in the data structure to be utilized despite apparent nonsmoothness in the objective. Second-order epi-derivatives are shown to exist as expressions of such underlying smoothness, and their connection with several kinds of second-order approximation is examined. Expansions of the Moreau envelope functions and proximal mappings associated with the essential objective functions for certain optimization problems in composite format are studied in particular.