Epi-derivatives of integral functionals with applications

Loewen, Philip D.; Zheng, Harry H.

doi:10.1090/S0002-9947-1995-1282892-2

Epi-derivatives of integral functionals with applications
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by Philip D. Loewen and Harry H. Zheng

Trans. Amer. Math. Soc. 347 (1995), 443-459

DOI: https://doi.org/10.1090/S0002-9947-1995-1282892-2

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Abstract:

We study first- and second-order epi-differentiability for integral functionals defined on ${L^2}[0,T]$, and apply the results to obtain first- and second-order necessary conditions for optimality in free endpoint control problems.

References

H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
Roberto Cominetti, On pseudo-differentiability, Trans. Amer. Math. Soc. 324 (1991), no. 2, 843–865. MR 992605, DOI 10.1090/S0002-9947-1991-0992605-3
Chi Ngoc Do, Generalized second-order derivatives of convex functions in reflexive Banach spaces, Trans. Amer. Math. Soc. 334 (1992), no. 1, 281–301. MR 1088019, DOI 10.1090/S0002-9947-1992-1088019-1
A. B. Levy, Second-order epi-derivatives of integral functionals, Set-Valued Anal. 1 (1993), no. 4, 379–392. MR 1267204, DOI 10.1007/BF01027827
P. D. Loewen and H. Zheng, Generalized conjugate points for optimal control problems, Nonlinear Anal. 22 (1994), no. 6, 771–791. MR 1270169, DOI 10.1016/0362-546X(94)90226-7
Dominikus Noll, Graphical methods in first- and second-order differentiability theory of integral functionals, Set-Valued Anal. 2 (1994), no. 1-2, 241–258. Set convergence in nonlinear analysis and optimization. MR 1285832, DOI 10.1007/BF01027104
R. A. Poliquin and R. T. Rockafellar, A calculus of epi-derivatives applicable to optimization, Canad. J. Math. 45 (1993), no. 4, 879–896. MR 1227665, DOI 10.4153/CJM-1993-050-7

Amenable functions in optimization

R. Tyrrell Rockafellar, Integral functionals, normal integrands and measurable selections, Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) Lecture Notes in Math., Vol. 543, Springer, Berlin, 1976, pp. 157–207. MR 0512209
R. T. Rockafellar, First- and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), no. 1, 75–108. MR 936806, DOI 10.1090/S0002-9947-1988-0936806-9
R. Tyrrell Rockafellar, Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives, Math. Oper. Res. 14 (1989), no. 3, 462–484. MR 1008425, DOI 10.1287/moor.14.3.462
R. T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré C Anal. Non Linéaire 6 (1989), no. suppl., 449–482. Analyse non linéaire (Perpignan, 1987). MR 1019126, DOI 10.1016/S0294-1449(17)30034-3
R. T. Rockafellar, Generalized second derivatives of convex functions and saddle functions, Trans. Amer. Math. Soc. 322 (1990), no. 1, 51–77. MR 1031242, DOI 10.1090/S0002-9947-1990-1031242-0
R. T. Rockafellar, Nonsmooth analysis and parametric optimization, Methods of nonconvex analysis (Varenna, 1989) Lecture Notes in Math., vol. 1446, Springer, Berlin, 1990, pp. 137–151. MR 1079762, DOI 10.1007/BFb0084934