Equicalmness and Epiderivatives That Are Pointwise Limits

D. Azé and R. A. Poliquin
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R. A. Poliquin: University of Alberta

Journal of Optimization Theory and Applications, 1998, vol. 96, issue 3, No 4, 555-573

Abstract: Abstract Recently, Moussaoui and Seeger (Ref. 1) studied the monotonicity of first-order and second-order difference quotients with primary goal the simplification of epilimits. It is well known that epilimits (lim inf and lim sup) can be written as pointwise limits in the case of a sequence of functions that is equi-lsc. In this paper, we introduce equicalmness as a condition that guarantees equi-lsc, and our primary goal is to give conditions that guarantee that first-order and second-order difference quotients are equicalm. We show that a piecewise-C 1 function f with convex domain is epidifferentiable at any point % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa!36EA! of its domain. We also show that a convex piecewise C 2-function (polyhedral pieces) is twice epidifferentiable. We thus obtain a modest extension of the Rockafellar result concerning the epidifferentiability of piecewise linear-quadratic convex functions.

Keywords: Piecewise-C k functions; equi-lsc functions; equicalmness; calmness; proxregularity; amenable functions; primal-lower-nice functions; Moreau envelopes; nonsmooth analysis; variational analysis; protoderivatives; epiderivatives (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1023/A:1022660427548

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