×

zbMATH — the first resource for mathematics

Nonconvex minimization problems. (English) Zbl 0441.49011

MSC:
49J27 Existence theories for problems in abstract spaces
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49M37 Numerical methods based on nonlinear programming
90C30 Nonlinear programming
49J50 Fréchet and Gateaux differentiability in optimization
47J05 Equations involving nonlinear operators (general)
93C15 Control/observation systems governed by ordinary differential equations
34H05 Control problems involving ordinary differential equations
35K55 Nonlinear parabolic equations
46B99 Normed linear spaces and Banach spaces; Banach lattices
47H10 Fixed-point theorems
49J15 Existence theories for optimal control problems involving ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Edgar Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31 – 47. · Zbl 0162.17501
[2] J. Aubin and J. Siegel, Fixed points and stationary points of dissipative multivalued maps, M.R. report 7712, University of Southern California, Los Angeles, 1977. · Zbl 0446.47049
[3] J. M. Borwein, Weak local supportability and applications to approximation, Pacific J. Math. 82 (1979), no. 2, 323 – 338. · Zbl 0434.46012
[4] M. Crandall, 1976, personal communication.
[5] Michael Edelstein, Farthest points of sets in uniformly convex Banach spaces, Israel J. Math. 4 (1966), 171 – 176. · Zbl 0151.17601
[6] M. Edelstein, On nearest points of sets in uniformly convex Banach spaces, J. London Math. Soc. 43 (1968), 375 – 377. · Zbl 0183.40403
[7] Errett Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97 – 98. · Zbl 0098.07905
[8] Errett Bishop and R. R. Phelps, The support functionals of a convex set, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 27 – 35.
[9] A. Brøndsted and R. T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605 – 611. · Zbl 0141.11801
[10] H. Brézis and F. E. Browder, A general principle on ordered sets in nonlinear functional analysis, Advances in Math. 21 (1976), no. 3, 355 – 364. · Zbl 0339.47030
[11] Felix E. Browder, Normal solvability for nonlinear mappings into Banach spaces, Bull. Amer. Math. Soc. 77 (1971), 73 – 77. · Zbl 0213.14704
[12] James Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241 – 251. · Zbl 0305.47029
[13] Frank H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247 – 262. · Zbl 0307.26012
[14] Frank H. Clarke, The maximum principle under minimal hypotheses, SIAM J. Control Optimization 14 (1976), no. 6, 1078 – 1091. · Zbl 0344.49009
[15] Frank H. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976), no. 2, 165 – 174. · Zbl 0404.90100
[16] Frank H. Clarke, Necessary conditions for a general control problem, Calculus of variations and control theory (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975; dedicated to Laurence Chisholm Young on the occasion of his 70th birthday), Publ. Math. Res. Center Univ. Wisconsin, No. 36, Academic Press, New York, 1976, pp. 257 – 278.
[17] F. Clarke, Generalized gradients of Lipschitz functionals, MRC Technical Report #1687, University of Wisconsin, Madison, Wis., August 1976. · Zbl 0463.49017
[18] Frank H. Clarke, Pointwise contraction criteria for the existence of fixed points, Canad. Math. Bull. 21 (1978), no. 1, 7 – 11. · Zbl 0414.54030
[19] Ivar Ekeland, Sur les problèmes variationnels, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A1057 – A1059 (French). · Zbl 0249.49004
[20] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324 – 353. · Zbl 0286.49015
[21] Ivar Ekeland, The Hopf-Rinow theorem in infinite dimension, J. Differential Geom. 13 (1978), no. 2, 287 – 301. · Zbl 0393.58004
[22] Ivar Ekeland, Le théorème de Hopf-Rinow en dimension infinie, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 3, A149 – A150. · Zbl 0345.58004
[23] Ivar Ekeland and Gérard Lebourg, Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), no. 2, 193 – 216 (1977). · Zbl 0313.46017
[24] Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. Ivar Ekeland and Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French; Studies in Mathematics and its Applications, Vol. 1.
[25] Alan J. Hoffman, On approximate solutions of systems of linear inequalities, J. Research Nat. Bur. Standards 49 (1952), 263 – 265.
[26] Alexander D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 (1979), 61 – 69. · Zbl 0427.58008
[27] J. Lasry, personal communication, 1972.
[28] G. Lebourg, Problèmes d’optimisation perturbés dans les espaces de Banach, preprint, CEREMADE, Université Paris-Dauphine, 1978.
[29] Robert H. Martin Jr., Invariant sets for evolution systems, International Conference on Differential Equations (Univ. Southern California, Los Angeles, Calif., 1974) Academic Press, New York, 1975, pp. 510 – 536.
[30] Michael Maschler and Bezalel Peleg, Stable sets and stable points of set-valued dynamic systems with applications to game theory, SIAM J. Control Optimization 14 (1976), no. 6, 985 – 995. · Zbl 0363.90145
[31] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. · Zbl 0108.10401
[32] Lucien W. Neustadt, Optimization, Princeton University Press, Princeton, N. J., 1976. A theory of necessary conditions; With a chapter by H. T. Banks. · Zbl 0353.49003
[33] Richard S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299 – 340. · Zbl 0122.10702
[34] R. Pallu de la Barrière, Optimal control theory: A course in automatic control theory, Translated from the French by Scripta Technica. Translation edited by Bernard R. Gelbaum, W. B. Saunders Co., Philadelphia, Pa.-London, 1967. · Zbl 0155.15203
[35] Математическая теория оптимал\(^{\приме}\)ных процессов, Государств. Издат. Физ.-Мат. Лит., Мосцощ, 1961 (Руссиан). Л. С. Понтрягин, В. Г. Болтянскии, Р. В. Гамкрелидзе, анд Е. Ф. Мищенко, Тхе матхематицал тхеоры оф оптимал процессес, Транслатед фром тхе Руссиан бы К. Н. Трирогофф; едитед бы Л. Щ. Неустадт, Интерсциенце Публишерс Јохн Щилеы & Сонс, Инц. Нещ Ыорк-Лондон, 1962. Л. С. Понтрјагин, В. Г. Болтјанскиј, Р. В. Гамкрелидзе, анд Е. Ф. Мисčенко, Матхематисче Тхеорие оптималер Прозессе, Р. Олденбоург, Мунич-Виенна, 1964 (Герман).
[36] Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157 – 184. · Zbl 0358.70014
[37] Stephen M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), no. 2, 130 – 143. · Zbl 0418.52005
[38] R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165 – 172. · Zbl 0119.09201
[39] I. Ekeland and M. Valadier, Representation of set-valued mappings, J. Math. Anal. Appl. 35 (1971), 621 – 629. · Zbl 0246.54018
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.