Epiderivatives of integral functionals with applications
Authors:
Philip D. Loewen and Harry H. Zheng
Journal:
Trans. Amer. Math. Soc. 347 (1995), 443459
MSC:
Primary 49J52; Secondary 49K15, 58C20
MathSciNet review:
1282892
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We study first and secondorder epidifferentiability for integral functionals defined on , and apply the results to obtain first and secondorder necessary conditions for optimality in free endpoint control problems.
 [1]
H.
Attouch, Variational convergence for functions and operators,
Applicable Mathematics Series, Pitman (Advanced Publishing Program),
Boston, MA, 1984. MR 773850
(86f:49002)
 [2]
JeanPierre
Aubin and Hélène
Frankowska, Setvalued analysis, Systems & Control:
Foundations & Applications, vol. 2, Birkhäuser Boston Inc.,
Boston, MA, 1990. MR 1048347
(91d:49001)
 [3]
Frank
H. Clarke, Optimization and nonsmooth analysis, Canadian
Mathematical Society Series of Monographs and Advanced Texts, John Wiley
& Sons Inc., New York, 1983. A WileyInterscience Publication. MR 709590
(85m:49002)
 [4]
Roberto
Cominetti, On pseudodifferentiability,
Trans. Amer. Math. Soc. 324 (1991),
no. 2, 843–865. MR 992605
(91h:26009), http://dx.doi.org/10.1090/S00029947199109926053
 [5]
Chi
Ngoc Do, Generalized secondorder derivatives
of convex functions in reflexive Banach spaces, Trans. Amer. Math. Soc. 334 (1992), no. 1, 281–301. MR 1088019
(93a:49011), http://dx.doi.org/10.1090/S00029947199210880191
 [6]
A.
B. Levy, Secondorder epiderivatives of integral functionals,
SetValued Anal. 1 (1993), no. 4, 379–392. MR 1267204
(95e:49025), http://dx.doi.org/10.1007/BF01027827
 [7]
P.
D. Loewen and H.
Zheng, Generalized conjugate points for optimal control
problems, Nonlinear Anal. 22 (1994), no. 6,
771–791. MR 1270169
(95d:49037), http://dx.doi.org/10.1016/0362546X(94)902267
 [8]
Dominikus
Noll, Graphical methods in first and secondorder
differentiability theory of integral functionals, SetValued Anal.
2 (1994), no. 12, 241–258. Set convergence in
nonlinear analysis and optimization. MR 1285832
(95c:49023), http://dx.doi.org/10.1007/BF01027104
 [9]
R.
A. Poliquin and R.
T. Rockafellar, A calculus of epiderivatives applicable to
optimization, Canad. J. Math. 45 (1993), no. 4,
879–896. MR 1227665
(94d:49023), http://dx.doi.org/10.4153/CJM19930507
 [10]
, Amenable functions in optimization, preprint.
 [11]
R.
Tyrrell Rockafellar, Integral functionals, normal integrands and
measurable selections, Nonlinear operators and the calculus of
variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975),
Springer, Berlin, 1976, pp. 157–207. Lecture Notes in Math.,
Vol. 543. MR
0512209 (58 #23598)
 [12]
R.
T. Rockafellar, First and secondorder
epidifferentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), no. 1, 75–108. MR 936806
(90a:90216), http://dx.doi.org/10.1090/S00029947198809368069
 [13]
R.
Tyrrell Rockafellar, Secondorder optimality conditions in
nonlinear programming obtained by way of epiderivatives, Math. Oper.
Res. 14 (1989), no. 3, 462–484. MR 1008425
(91b:49022), http://dx.doi.org/10.1287/moor.14.3.462
 [14]
R.
T. Rockafellar, Protodifferentiability of setvalued mappings and
its applications in optimization, Ann. Inst. H. Poincaré Anal.
Non Linéaire 6 (1989), no. suppl.,
449–482. Analyse non linéaire (Perpignan, 1987). MR 1019126
(90k:90140)
 [15]
R.
T. Rockafellar, Generalized second derivatives of
convex functions and saddle functions, Trans.
Amer. Math. Soc. 322 (1990), no. 1, 51–77. MR 1031242
(91b:90190), http://dx.doi.org/10.1090/S00029947199010312420
 [16]
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
(91i:49016), http://dx.doi.org/10.1007/BFb0084934
 [1]
 H. Attouch, Variational convergence for functions and operators, Pitman, Boston, 1984. MR 773850 (86f:49002)
 [2]
 J. P. Aubin and H. Frankowska, Setvalued analysis, Birkháuser, Boston, 1990. MR 1048347 (91d:49001)
 [3]
 F. H. Clarke, Optimization and nonsmooth analysis, WileyInterscience, New York, 1983. MR 709590 (85m:49002)
 [4]
 R. Cominetti, On pseudodifferentiability, Trans. Amer. Math. Soc. 324 (1988), 843865. MR 992605 (91h:26009)
 [5]
 C. Do, Generalized second derivatives of convex functions in reflexive Banach spaces, Trans. Amer. Math. Soc. (to appear). MR 1088019 (93a:49011)
 [6]
 A. Levy, Secondorder epiderivatives of integral functionals with fully amenable integrands, SetValued Analysis (to appear). MR 1267204 (95e:49025)
 [7]
 P. D. Loewen and H. Zheng, Generalized conjugate points for optimal control problems, Nonlinear Analysis, Theory Methods & Applications (to appear). MR 1270169 (95d:49037)
 [8]
 D. Noll, Second order differentiability of integral functionals on Sobolev spaces and spaces, Z. Reine Angew. Math. (to appear). MR 1285832 (95c:49023)
 [9]
 R. A. Poliquin and R. T. Rockafellar, A calculus of epiderivatives with applications to optimization, Canad. J. Math. 45 (1993), 879896. MR 1227665 (94d:49023)
 [10]
 , Amenable functions in optimization, preprint.
 [11]
 R. T. Rockafellar, Integral functionals, normal integrands and measurable selections; in Nonlinear Operators and the Calculus of Variations, Lecture Notes in Math., vol. 543, SpringerVerlag, 1976, pp. 157207. MR 0512209 (58:23598)
 [12]
 , First and secondorder epidifferentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), 75107. MR 936806 (90a:90216)
 [13]
 , Secondorder optimality conditions in nonlinear programming obtained by way of epiderivatives, Math. Oper. Res. 14 (1989), 462484. MR 1008425 (91b:49022)
 [14]
 , Protodifferentiability of setvalued mappings and its applications in optimization, Analyse Non Linéaire (H. Attouch, J.P. Aubin, F. Clarke, I. Ekeland, eds.), 1989, pp. 449482. MR 1019126 (90k:90140)
 [15]
 , Generalized second derivatives of convex functions and saddle functions, Trans. Amer. Math. Soc. 320 (1990), 810822. MR 1031242 (91b:90190)
 [16]
 , Nonsmooth analysis and parametric optimization, Methods of Nonconvex Analysis (A. Cellina, ed.), Lecture Notes in Math., vol. 1446, SpringerVerlag, 1990, pp. 137151. MR 1079762 (91i:49016)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
49J52,
49K15,
58C20
Retrieve articles in all journals
with MSC:
49J52,
49K15,
58C20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512828922
PII:
S 00029947(1995)12828922
Keywords:
Integral functionals,
epiderivative,
epiconvergence
Article copyright:
© Copyright 1995 American Mathematical Society
