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Uspekhi Mat. Nauk, 2012, Volume 67, Issue 3(405), Pages 3–62 (Mi umn9478)  

This article is cited in 13 scientific papers (total in 13 papers)

Smooth abnormal problems in extremum theory and analysis

A. V. Arutyunov

Peoples Friendship University of Russia

Abstract: A survey is given of results related to the inverse function theorem and to necessary and sufficient first- and second-order conditions for extrema in smooth extremal problems with constraints. The main difference between the results here and the classical ones is that the former are valid and meaningful without a priori normality assumptions.
Bibliography: 48 titles.

Keywords: abnormal point, Lagrange principle, necessary and sufficient second-order conditions for an extremum, 2-regularity, inverse function theorem, 2-normality, quadratic map.

DOI: https://doi.org/10.4213/rm9478

Full text: PDF file (1002 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2012, 67:3, 403–457

Bibliographic databases:

Document Type: Article
UDC: 519.7
MSC: Primary 47J07, 49K27; Secondary 15A63
Received: 14.06.2011

Citation: A. V. Arutyunov, “Smooth abnormal problems in extremum theory and analysis”, Uspekhi Mat. Nauk, 67:3(405) (2012), 3–62; Russian Math. Surveys, 67:3 (2012), 403–457

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Arutyunov, F. L. Pereira, S. E. Zhukovskiy, “Application of covering mappings to constrained dynamic systems and differential inclusions”, European Control Conference, IEEE, 2014, 1456–1461  isi
    2. K. V. Storozhuk, “O verkhnem topologicheskom predele semeistva vektornykh podprostranstv korazmernosti $k$”, Sib. elektron. matem. izv., 12 (2015), 432–435  mathnet  crossref
    3. D. Yu. Karamzin, “The Dines theorem and some other properties of quadratic mappings”, Comput. Math. Math. Phys., 55:10 (2015), 1633–1641  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. A. V. Arutyunov, S. E. Zhukovskii, “On Surjective Quadratic Mappings”, Math. Notes, 99:2 (2016), 192–195  mathnet  crossref  crossref  mathscinet  isi  elib
    5. A. V. Arutyunov, S. E. Zhukovskiy, “Properties of surjective real quadratic maps”, Sb. Math., 207:9 (2016), 1187–1214  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. A. V. Arutyunov, I. A. Shvartsman, Z. T. Zhukovskaya, “Second order conditions in optimal control problems with equality constraints”, 2016 IEEE 55Th Conference on Decision and Control (CDC), IEEE Conference on Decision and Control, IEEE, 2016, 1069–1073  isi
    7. J. A. Becerril, J. F. Rosenblueth, “The importance of being normal, regular and proper in the calculus of variations”, J. Optim. Theory Appl., 172:3 (2017), 759–773  crossref  mathscinet  zmath  isi  scopus
    8. A. V. Arutyunov, S. E. Zhukovskiy, D. Yu. Karamzin, “Some properties of two-dimensional surjective $p$-homogeneous maps”, Comput. Math. Math. Phys., 57:7 (2017), 1081–1089  mathnet  crossref  crossref  mathscinet  isi  elib
    9. A. V. Arutyunov, I. A. Shvartsman, Z. T. Zhukovskaya, “Second-order optimality conditions for singular extremals in optimal control problems with equality endpoint constraints”, Nonlinear Anal., 157 (2017), 20–43  crossref  mathscinet  zmath  isi  scopus
    10. A. V. Arutyunov, A. F. Izmailov, “Stability of possibly nonisolated solutions of constrained equations, with applications to complementarity and equilibrium problems”, Set-Valued Var. Anal., 26:2 (2018), 327–352  crossref  mathscinet  zmath  isi  scopus
    11. Karzhemanov I., Zhdanovskiy I., “Some Properties of Surjective Rational Maps”, Eur. J. Math., 4:1, 1, SI (2018), 326–329  crossref  mathscinet  zmath  isi  scopus
    12. A. V. Arutyunov, K. Yu. Osipenko, “Recovering linear operators and Lagrange function minimality condition”, Siberian Math. J., 59:1 (2018), 11–21  mathnet  crossref  crossref  isi  elib
    13. E. R. Avakov, G. G. Magaril-Il'yaev, “Generalized Needles and Second-Order Conditions in Optimal Control”, Proc. Steklov Inst. Math., 304 (2019), 8–25  mathnet  crossref  crossref  elib
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