Second-order epi-derivatives of integral functionals
- A. B. Levy
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Epi-derivatives have many applications in optimization as approached through nonsmooth analysis. In particular, second-order epi-derivatives can be used to obtain optimality conditions and carry out sensitivity analysis. Therefore the existence of second-order epi-derivatives for various classes of functions is a topic of considerable interest. A broad class of composite functions on ℝ n called ‘fully amenable’ functions (which include general penalty functions composed withC 2 mappings, possibly under a constraint qualification) are now known to be twice epi-differentiable. Integral functionals appear widely in problems in infinite-dimensional optimization, yet to date, only integral functionals defined by convex integrands have been shown to be twice epi-differentiable, provided that the integrands are twice epi-differentiable. Here it is shown that integral functionals are twice epi-differentiable even without convexity, provided only that their defining integrands are twice epi-differentiable and satisfy a uniform lower boundedness condition. In particular, integral functionals defined by fully amenable integrands are twice epi-differentiable under mild conditions on the behavior of the integrands.
- Poliquin, R.A. and Rockafellar, R.T.: Amenable functions in optimization, preprint, 1991.
- Ioffe, A.D.: Variational analysis of a composite function: A formula for the lower second order epi-derivative,J. Math. Anal. Appl. 160 (1991), 379–405.
- Rockafellar, R.T.: First- and second-order epi-differentiability in nonlinear programming,Trans. A.M.S. 307 (1988), 75–107.
- Poliquin, R.A.: An extension of Attouch's theorem and its application to second-order epi-differentiation of convexly composite functions,Trans. Amer. Math. Soc. 332 (1992), 861–874.
- Cominetti, R.: On pseudo-differentiability,Trans. A.M.S. 324 (1991), 843–865.
- Do, C.N.: Generalized second-order derivatives of convex functions in reflexive Banach spaces,Trans. A.M.S. 334 (1992), 281–301.
- Attouch, H.: inVariational Convergence of Functions and Operators, Pitman, London, 1984.
- Rockafellar, R.T.: Integral functionals, normal integrands and measurable selections, in L. Waelbroeck (ed.),Nonlinear Operators and the Calculus of Variations, Springer-Verlag, New York, 1976, 157–207.
- Zheng, H. and Loewen, P.D.: Epi-derivatives of integral functionals with applications, preprint, 1992.
- Salinetti, G. and Wets, R.J.B.: On the relations between two types of convergence for convex functions,J. Math. Anal. Appl. 60 (1977), 211–226.
- Rockafellar, R.T.: inConjugate Duality and Optimization, SIAM, Philadelphia, 1974.
- Rockafellar, R.T.: Linear-quadratic programming and optimal control,SIAM J. Control Optim. 25 (1987), 781–814.
- Rockafellar, R.T.: Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives,Math. Oper. Res. 14 (1989), 462–484.
- Second-order epi-derivatives of integral functionals
Volume 1, Issue 4 , pp 379-392
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Primary: 49J52
- Secondary: 58C20
- Generalized second derivatives
- nonsmooth analysis
- fully amenable functions
- integral functionals
- A. B. Levy (1)
- Author Affiliations
- 1. Department of Mathematics, University of Washington, 98195, Seattle, WA, U.S.A.