×

New concepts in nondifferentiable programming. (English) Zbl 0469.90071


MSC:

90C30 Nonlinear programming
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
90C48 Programming in abstract spaces
46B99 Normed linear spaces and Banach spaces; Banach lattices
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] A. Auslender . Optimisation: Méthodes numériques . Masson et Cie, Paris ( 1976 ). MR 55 #14068 | Zbl 0326.90057 · Zbl 0326.90057
[2] A. Auslender . Minimisation sans contraintes de fonctions localement Lipschitziennes : application à la programmation mi-convexe, mi-différentiable . C. R. Acad. Sc. Paris, Série A, t. 284 ( 1977 ), 959-961, and Proceedings of the 3rd Symposium on Nonlinear Programming, University of Wisconsin, Madison (to appear). MR 55 #6275 | Zbl 0369.90095 · Zbl 0369.90095
[3] M. S. Bazaraa and J. J. Goode . Extension of optimality conditions via supporting functions . Math. Programming 5 ( 1973 ), 267-285. MR 49 #2105 | Zbl 0283.90051 · Zbl 0283.90051 · doi:10.1007/BF01580133
[4] M. S. Bazaraa , J. J. Goode and Z. Nashed . On the cones of tangents with applications to mathematical programming . J. Optimization Theory and Appl. Vol. 13, No 4 ( 1974 ), 389-426. MR 51 #2645 | Zbl 0259.90037 · Zbl 0259.90037 · doi:10.1007/BF00934938
[5] M. S. Bazaraa , C. M. Shetty , J. J. Goode and M. Z. Nashed . Nonlinear programming without differentiability in Banach spaces : necessary and sufficient constraint qualification . Applicable Anal. Vol 5 ( 1976 ), 165-173. MR 53 #12592 | Zbl 0353.90076 · Zbl 0353.90076 · doi:10.1080/00036817608839119
[6] C. Berge . Espaces topologiques, fonctions multivoques . Dunod, Paris ( 1966 ). Zbl 0164.52902 · Zbl 0164.52902
[7] H. Berliocchi and J. M. Lasry . Principe de Pontryagin pour des systemes régis par une équation différentielle multivoque . Publications Mathématiques de l’université de Bordeaux I 1973 - 1974 (2), 25-41. MR 58 #30622 | Zbl 0357.49019 · Zbl 0357.49019
[8] V. G. Bolt’yanskii . The method of tents in the theory of extremal problems . Russian Math. Surveys 30, 3, ( 1975 ), 1-54. MR 56 #3713 | Zbl 0334.49014 · Zbl 0334.49014 · doi:10.1070/RM1975v030n03ABEH001411
[9] J. M. Borwein . Fractional programming without differentiability . Math. Programming 11 ( 1976 ), 283-290. MR 56 #7965 | Zbl 0357.90054 · Zbl 0357.90054 · doi:10.1007/BF01580396
[10] N. Bourbaki . Espaces vectoriels topologiques . Fascicules XV, XVII, Hermann, Paris ( 1966 - 1967 ). Zbl 0145.37702 · Zbl 0145.37702
[11] Buì-Trong-Liêũ and P. Huard . La méthode des centres dans un espace topologique . Numer. Math. 8 ( 1966 ), 58-67. Article | MR 32 #9059 | Zbl 0171.40802 · Zbl 0171.40802 · doi:10.1007/BF02165238
[12] F. H. Clarke . Necessary conditions for nonsmooth problems in optimal control and the calculus of variations . Ph. D. Dissertation, University of Washington ( 1973 ).
[13] F. H. Clarke . Generalized gradients and applications . Transactions of the American Math. Society, Vol 205 ( 1975 ), 247-262. MR 51 #3373 | Zbl 0307.26012 · Zbl 0307.26012 · doi:10.2307/1997202
[14] F. H. Clarke . A new approach to Lagrange multipliers . Math. of Operations Res. 2 ( 1976 ), 165-174. MR 54 #2209 | Zbl 0404.90100 · Zbl 0404.90100 · doi:10.1287/moor.1.2.165
[15] F. H. Clarke . Generalized gradients of Lipschitz functionals . Madison M. R. C. Technical Summary Report, University of Wisconsin ( 1976 ). · Zbl 0463.49017
[16] F. H. Clarke , On the inverse function theorem . Pacific J. Math., Vol 64, No 1 ( 1976 ), 97-102. Article | MR 54 #13005 | Zbl 0331.26013 · Zbl 0331.26013 · doi:10.2140/pjm.1976.64.97
[17] B. D. Craven and B. Mond . Lagrangean conditions for quasi-differentiable optimization . Research Report No 7 ( 1976 ), School of Mathematical Sciences, University of Melbourne. · Zbl 0355.90058
[18] J.-P. Crouzeix . Conjugacy in quasi-convex analysis. Convex Analysis and its applications , Lecture Notes in Economics and Mathematical Systems No 144 (A. Auslender, Ed.) Springer-Verlag ( 1977 ) pp. 66-99. MR 58 #2531 | Zbl 0362.90096 · Zbl 0362.90096
[19] J.-P. Crouzeix . Contributions à l’étude des fonctions quasi-convexes . Thèse de Doctorat ès-Sciences Mathématiques, Université de Clermont II ( 1977 ). MR 58 #4332
[20] V. F. Dem’yanov and V. N. Malozemov . Introduction to minimax . John Wiley and Sons, New York ( 1974 ). MR 57 #15407b · Zbl 0781.90079
[21] J. Durdil . On the geometrical characterization of differentiability I . Commentationes Mathematicae Universitatis Carolinae 15, 3 ( 1974 ), 521-540. MR 54 #13968 | Zbl 0289.58004 · Zbl 0289.58004
[22] J. Durdil and J. Daneš . A note on geometric characterization of Fréchet-differentiability . Commentationes Mathematicae Universitatis Carolinae 17, 1 ( 1976 ), 195-204. Article | MR 54 #3743 | Zbl 0321.58009 · Zbl 0321.58009
[23] A. Feuer . Minimizing well-behaved functions . Proceedings of Twelfth Annual Allerton Conference on Circuit and System Theory. Illinois ( 1974 ), 25-34.
[24] I. V. Girsanov . Lectures on mathematical theory of extremum problems . Lecture Notes in Economics and Mathematical Systems, Springer-Verlag ( 1972 ). MR 57 #3958 | Zbl 0234.49016 · Zbl 0234.49016
[25] A. A. Goldstein . Optimization of Lipschitz continuous functions . Math. Programming, Vol 13 ( 1977 ) No 1, 14-22. MR 56 #1705 | Zbl 0394.90088 · Zbl 0394.90088 · doi:10.1007/BF01584320
[26] F. J. Gould and J. W. Tolle . Geometry of optimality conditions and constraint qualifications . Math. Programming 2 ( 1972 ), 1-18. MR 45 #6415 | Zbl 0288.90068 · Zbl 0288.90068 · doi:10.1007/BF01584534
[27] F. J. Gould and J. W. Tolle . Optimality conditions and constraint qualifications in Banach space . J. Optimization Theory Appl. Vol 15 ( 1975 ), No 6, 667-684. MR 56 #13624 | Zbl 0281.90066 · Zbl 0281.90066 · doi:10.1007/BF00935506
[28] H.J. Greenberg and W. P. Pierskalla . Quasiconjugate function and surrogate duality . Cahiers du Centre d’Etudes de Rech. Oper. Vol 15, No 4 ( 1973 ), 437-448. MR 51 #2649 | Zbl 0276.90051 · Zbl 0276.90051
[29] H. Halkin . Interior mapping theorem with set-valued derivatives . Journal d’Analyse Mathématique, Vol 30 ( 1976 ), 200-207. MR 58 #6102 | Zbl 0349.49016 · Zbl 0349.49016 · doi:10.1007/BF02786714
[30] M. R. Hestenes . Optimization Theory: the finite dimensional case . J. Wiley and Sons ( 1975 ). Zbl 0327.90015 · Zbl 0327.90015
[31] J-B. Hiriart-Urruty . Conditions nécessaires d’optimalité en programmation non différentiable . Note aux Comptes Rendus Acad. Sc. Paris t. 283, Série A ( 1976 ), 843-845. MR 54 #13679 | Zbl 0359.49009 · Zbl 0359.49009
[32] J-B. Hiriart-Urruty . On necessary optimality conditions in nondifferentiable programming . Math. Programming, Vol 14, ( 1978 ) (to appear). MR 57 #5121 | Zbl 0373.90071 · Zbl 0373.90071 · doi:10.1007/BF01588951
[33] J-B. Hiriart-Urruty . Gradients généralisés de fonctions marginales . SIAM J. on Control and Optimization ( 1978 ) (to appear). MR 58 #12599 | Zbl 0385.90099 · Zbl 0385.90099 · doi:10.1137/0316019
[34] J-B. Hiriart-Urruty . Contributions a la programmation mathématique: cas détérministe et stochastique . Thèse de Doctorat ès-Sciences Mathématiques, Université de Clermont II ( 1977 ). MR 58 #9249
[35] J-B. Hiriart-Urruty . Tangent cones, generalized gradients and mathematical programming in Banach spaces . Math. of Operations Res. ( 1978 ) (to appear). Zbl 0409.90086 · Zbl 0409.90086 · doi:10.1287/moor.4.1.79
[36] J-B. Hiriart-Urruty . Gradients généralisés de fonctions composées. Applications . Note aux Comptes Rendus Acad. Sc. Paris t. 285, Série A ( 1977 ), 781-784. MR 56 #16489 | Zbl 0385.90095 · Zbl 0385.90095
[37] Hoang Tuy . Sur les inégalités linéaires . Colloq. Math. 13 ( 1964 ), 107-123. MR 34 #8804 | Zbl 0204.19401 · Zbl 0204.19401
[38] R. Janin . Sur la dualité et la sensibilité dans les problèmes de programme mathématique . Thèse de Doctorat ès-Sciences Mathématiques, Université de Paris VI ( 1974 ).
[39] S. N. Kruzhkov . On the differentiability almost everywhere of functions of many variables . Vestnik Moskovskogo Universiteta, Matematika, Vol 31, No 6 ( 1976 ), 67-70 = Moscow University Mathematics Bulletin, Vol 31, No 5-6, ( 1976 ), 129-131. Zbl 0351.26011 · Zbl 0351.26011
[40] P-J. Laurent . Approximation et Optimisation , Hermann ( 1972 ). MR 57 #6947 | Zbl 0238.90058 · Zbl 0238.90058
[41] G. Lebourg . Valeur moyenne pour gradient généralisé . Note aux Comptes Rendus Acad. Sc. Paris, Série A, ( 1975 ), 795-797. MR 52 #8934 | Zbl 0317.46034 · Zbl 0317.46034
[42] * G. Lebourg . Generic differentiability of Lipschitzian functions . Cahiers de Mathématiques de la Décision ( 1977 ), No 7704. · Zbl 0435.46031
[43] C. Lescarret . Sous-différentiabilité de fonctions composées . Travaux du Séminaire d’Analyse Unilatérale, Vol. 1, Exposé No 2 ( 1968 ), Université de Montpellier. MR 41 #5966
[44] E. S. Levitin , A. A. Milyutin and N. P. Osmolovskii . On necessary and sufficient conditions for a local minimum in a problem with constraints . Soviet. Math. Dokl., Vol 14, No 3 ( 1973 ), 853-857. Zbl 0292.49009 · Zbl 0292.49009
[45] O. L. Mangasarian . Nonlinear programming . Mc. Graw-Hill, New York ( 1972 ). Zbl 0194.20201 · Zbl 0194.20201
[46] H. Massam and S. Zlobec . Various definitions of the derivative in mathematical programming . Math. Programming 7 ( 1974 ), 144-151. MR 50 #3066 | Zbl 0296.90042 · Zbl 0296.90042 · doi:10.1007/BF01585512
[47] P. Michel . A propos du problème des inégalités et de ses applications à la programmation et au contrôle optimal . Publications Mathématiques de l’Université de Bordeaux I, 1973 - 1974 (3).
[48] A. Nijenhuis . Strong derivatives and inverse mappings . Amer. Math. Monthly, 81 ( 1974 ), 969-980. MR 50 #13405 | Zbl 0296.58002 · Zbl 0296.58002 · doi:10.2307/2319298
[49] E. A. Nurminskii . The quasi-gradient method for the solving of the nonlinear programming problems . Kibernetika, 1, ( 1973 ), 122-125 = Cybernetics, Vol 9, No 1 ( 1974 ), 145-150. Zbl 0277.90065 · Zbl 0277.90065
[50] E. A. Nurminskii and A. A. Zhelikhovskii , e-quasi-gradient method for solving nonsmooth extremal problems . Kibernetika, 1, ( 1977 ), 109-113 = Cybernetics, Vol 13, No 1, ( 1977 ), 109-114. Zbl 0361.90070 · Zbl 0361.90070
[51] J. M. Ortega and W. C. Rheinboldt . Iterative solution of nonlinear equations in several valables . Acad. Press, New York and London ( 1970 ). Zbl 0241.65046 · Zbl 0241.65046
[52] R. Pallu De La Barrière . Fonctions sous-linéarisables et principe de Pontryagin . Publications Mathématiques de l’Université de Bordeaux I, 1973 - 1974 (3), 65-72.
[53] J. P. Penot . Sous-différentiels de fonctions numériques non convexes . Note aux Comptes Rendus Acad. Sc. Paris, t. 278, Serie A ( 1974 ), 1553-1555. MR 50 #5464 | Zbl 0318.46055 · Zbl 0318.46055
[54] J-P. Penot . Calcul sous-différentiel et Optimisation . Publications Mathématiques de l’Université de Pau ( 1974 ). MR 58 #31202 | Zbl 0363.49005 · Zbl 0363.49005
[55] J-P. Penot (to appear).
[56] B. N. Pshenichnyi . Necessary conditions for an extremum . Marcel Dekker Inc., New York ( 1971 ). MR 43 #2585 | Zbl 0212.23902 · Zbl 0212.23902
[57] B. N. Pshenichnyi . Leçons sur les jeux différentiels . Cahiers de l’I.R.I.A., No 4, ( 1971 ), 145-226. MR 43 #2585 | Zbl 0244.90045 · Zbl 0244.90045
[58] L. Rigby . Contribution to Dubovitskii and Milyutin’s optimization formalism . Optimization Techniques, Modeling and Optimization in the Service of Man. Part 2. Lecture Notes in Computer Science (J. Ceá, Ed.) Springer-Verlag ( 1976 ), 438-453. Zbl 0347.90043 · Zbl 0347.90043
[59] R. T. Rockafellar , Convex Analysis . Princeton University Press ( 1970 ). MR 43 #445 | Zbl 0193.18401 · Zbl 0193.18401
[60] N. Z. Shor . A class of almost-differentiable functions and a minimization method for functions of this class . Kibernetika, 8, ( 1972 ), 65-70 = Cybernetics, Vol 8, No 4, ( 1974 ), 509-606.
[61] E. M. Stein . Singular integrals and differentiability properties of functions . Princeton Math. Ser. No 30, Princeton Univ. Press, N. J. ( 1970 ). MR 44 #7280 | Zbl 0207.13501 · Zbl 0207.13501
[62] L. Thibault . Propriétés des sous-différentiels des fonctions localement Lipschitzcennes définies sur un espace de Banach séparable. Applications . Thèse de Doctorat de Spécialité, Université de Montpellier ( 1976 ). · Zbl 0343.46030
[63] J. Warga . Derivate containers, Inverse functions and Controllability . Calculs of Variations and Control Theory (D. L. Russell, Ed.) Acad. Press ( 1976 ). · Zbl 0355.26004
[64] Y. I. Zabotin , A. I. Korablev and R. F. Khabibullin . Conditions for an extremum of a functional in case of constraints . Cybernetics, Vol 9, No 6, ( 1975 ), 982-988.
[65] W. I. Zangwill . Nonlinear programming: a unified approach . Prentice-Hall, Inc., N. J. ( 1969 ). MR 50 #12268 | Zbl 0195.20804 · Zbl 0195.20804
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.