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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 secondorder 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 proxregular, which includes convex functions and the extendedrealvalued functions representing problems of nonlinear programming, the many secondorder 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 secondorder approximations in nonsmooth optimization and variants of the proximal point algorithm. The results establish that gen eralized secondorder expansions of Moreau envelopes, at least, can be counted on in most situations of interest in finitedimensional optimization. Keywords. Proxregularity, amenable functions, primallowernice functions, Hessians, first and secondorder expansions, strict protoderivatives, proximal mappings, Moreau envelopes, regularization, subgradient mappings, nonsmooth analysis, variational analysis, protoderivatives, secondorder epiderivatives, 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; rene@fenchel.math.ualberta.ca. 2 Work supported by the National Science Foundation under grant DMS9200303. Dept. of Math., Univ. of Washington, Seattle, WA 98195 USA; rtr@math.washington.edu. 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 onesided 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.
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