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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Second-order conditions in extremal problems. The abnormal points
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by A. V. Arutyunov PDF
Trans. Amer. Math. Soc. 350 (1998), 4341-4365 Request permission

Abstract:

In this paper we study a minimization problem with constraints and obtain first- and second-order necessary conditions for a minimum. Those conditions – as opposed to the known ones – are also informative in the abnormal case. We have introduced the class of 2-normal constraints and shown that for them the “gap" between the sufficient and the necessary conditions is as minimal as possible. It is proved that a 2-normal mapping is generic.
References
  • A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
  • R. Henrion, La théorie de la variation seconde et ses applications en commande optimale, Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8$^\textrm {o}$ (2) 41 (1974), no. 7, 208 pp. (1975). MR 432278
  • A. D. Ioffe and V. M. Tihomirov, Theorie der Extremalaufgaben, VEB Deutscher Verlag der Wissenschaften, Berlin, 1979 (German). Translated from the Russian by Bernd Luderer. MR 527119
  • N. Bourbaki, Éléments de mathématique. Première partie. (Fascicule II.) Livre III: Topologie générale. Chapitre 1: Structures topologiques. Chapitre. 2: Structures uniformes, Hermann, Paris, 1961 (French). Troisiéme édition entiérement refondue; Actualités Sci. Indust., No. 1142. MR 0141067
  • A. V. Arutyunov, Perturbations of extremal problems with constraints, and necessary conditions for optimality, Mathematical analysis, Vol. 27 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 147–235, 237 (Russian). Translated in J. Soviet Math. 54 (1991), no. 6, 1342–1400. MR 1029286
  • V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal control, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1987. Translated from the Russian by V. M. Volosov. MR 924574, DOI 10.1007/978-1-4615-7551-1
  • B. N. Pšeničnyĭ, Vypuklyĭ analiz i èkstremal′nye zadachi, Seriya “Nelineĭnyĭ Analiz i ego Prilozheniya”. [Series in Nonlinear Analysis and its Applications], “Nauka”, Moscow, 1980 (Russian). MR 581125
  • A. A. Agrachëv, The topology of quadratic mappings and Hessians of smooth mappings, Algebra. Topology. Geometry, Vol. 26 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 85–124, 162 (Russian). Translated in J. Soviet Math. 49 (1990), no. 3, 990–1013. MR 978394
  • A. V. Arutyunov, Necessary conditions for an extremum in an abnormal problem with equality constraints, Uspekhi Mat. Nauk 45 (1990), no. 5(275), 181–182 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 5, 223–224. MR 1084994, DOI 10.1070/RM1990v045n05ABEH002656
  • Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
  • A. V. Arutyunov, On necessary conditions for optimality in a problem with phase constraints, Dokl. Akad. Nauk SSSR 280 (1985), no. 5, 1033–1037 (Russian). MR 780283
  • E. R. Avakov, A. A. Agrachëv, and A. V. Arutyunov, The level set for a smooth mapping in a neighborhood of a singular point, and zeros of a quadratic mapping, Mat. Sb. 182 (1991), no. 8, 1091–1104 (Russian); English transl., Math. USSR-Sb. 73 (1992), no. 2, 455–466. MR 1128689, DOI 10.1070/SM1992v073n02ABEH002555
  • A. V. Arutyunov and N. T. Tynyanskiĭ, On necessary conditions for a local minimum in optimal control theory, Dokl. Akad. Nauk SSSR 275 (1984), no. 2, 268–272 (Russian). MR 742203
  • A. A. Agrachëv, One more condition for a conditional extremum, Uspekhi Mat. Nauk 44 (1989), no. 5(269), 153–154 (Russian); English transl., Russian Math. Surveys 44 (1989), no. 5, 189–190. MR 1040274, DOI 10.1070/RM1989v044n05ABEH002275
  • E. S. Levitin, A. A. Miljutin, and N. P. Osmolovskiĭ, Higher order conditions for local minima in problems with constraints, Uspekhi Mat. Nauk 33 (1978), no. 6(204), 85–148, 272 (Russian). MR 526013
  • A. A. Milyutin, Quadratic conditions for an extremum in smooth problems with a finite-dimensional image, Methods of the theory of extremal problems in economics, “Nauka”, Moscow, 1981, pp. 138–177 (Russian). MR 694704
  • E. R. Avakov, Conditions for an extremum for smooth problems with constraints of equality type, Zh. Vychisl. Mat. i Mat. Fiz. 25 (1985), no. 5, 680–693, 798 (Russian). MR 796117
  • R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
  • Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
  • V. I. Arnol′d, A. N. Varchenko, and S. M. Guseĭn-Zade, Osobennosti differentsiruemykh otobrazheniĭ, “Nauka”, Moscow, 1982 (Russian). Klassifikatsiya kriticheskikh tochek, kaustik i volnovykh frontov. [Classification of critical points, caustics and wave fronts]. MR 685918
  • A. V. Finkel′shteĭn, Application of Ekeland’s theorem for derivation of necessary and sufficient conditions for optimality of sequences, Optimizatsiya 31(48) (1983), 33–47 (Russian). MR 751907
  • V. V. Voevodin, Lineĭ naya algebra, Izdat. “Nauka”, Moscow, 1974 (Russian). MR 0357426
  • E. R. Avakov, Necessary conditions for an extremum for smooth abnormal problems with constraints of equality and inequality type, Mat. Zametki 45 (1989), no. 6, 3–11, 110 (Russian); English transl., Math. Notes 45 (1989), no. 5-6, 431–437. MR 1019030, DOI 10.1007/BF01158229
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Additional Information
  • A. V. Arutyunov
  • Affiliation: Department of Differentional Equations and Functional Analysis, Peoples Friendship University of Russia, Moscow 117198, Mikluka-Maklai, 6, Russia
  • Email: arutunov@sa640.cs.msu.su
  • Received by editor(s): January 25, 1995
  • Received by editor(s) in revised form: September 20, 1995
  • Additional Notes: This work was supported by Russian Fund of Fundamental Researches, project N 96-01-00800 and partially by Grant of Goskomvuz of Russia, project N 95-0-1.9-48
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 4341-4365
  • MSC (1991): Primary 49B27
  • DOI: https://doi.org/10.1090/S0002-9947-98-01775-9
  • MathSciNet review: 1390966