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Unformatted text preview: GENERALIZED HESSIAN PROPERTIES OF REGULARIZED NONSMOOTH FUNCTIONS R. A. Poliquin 1 and R. T. Rockafellar 2 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 func- tions 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 prop- erties 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 gen- eralized 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, Attouchs theorem. 1980 Mathematics Subject Classification (1985 Revision ). Primary 49A52, 58C06, 58C20; Secondary 90C30 May, 1995 1 Work supported by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983. Dept. of Math. Sciences, Univ. of Alberta, Edmonton, Alberta, Canada T6G 2G1; firstname.lastname@example.org. 2 Work supported by the National Science Foundation under grant DMS9200303. Dept. of Math., Univ. of Washington, Seattle, WA 98195 USA; email@example.com. 1. Introduction Any problem of optimization in IR n can be posed abstractly as one of minimizing a function f : IR n IR := IR over all of IR n ; constraints are represented by infinite penalization. It is natural then that f be lower semicontinuous (l.s.c.). In this setting, useful for many purposes, consider a point x where f achieves its minimum, f ( x ) being finite. For any > 0, x also minimizes the Moreau envelope function e ( x ) := min x f ( x ) + 1 2 x- x 2 , (1 . 1) which provides a regularization of f . While f may have values and exhibit discontinu- ities, e is finite and locally Lipschitz continuous, and it approximates f in the sense that e increases pointwise to f as & 0. Among other regularizing effects, e has one-sided directional derivatives at all points, even Taylor expansions of degree 2 almost everywhere, and at x it is differentiable with e ( x ) = f ( x ) , e ( x ) = 0 . (1 . 2) When f is convex, e is convex too and actually of class...
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This note was uploaded on 02/04/2012 for the course ECE 445 taught by Professor Hert during the Spring '11 term at Maryland.
- Spring '11