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Mat. Sb., 2000, Volume 191, Number 1, Pages 3–26 (Mi msb446)  

This article is cited in 20 scientific papers (total in 20 papers)

Implicit function theorem as a realization of the Lagrange principle. Abnormal points

A. V. Arutyunov

Peoples Friendship University of Russia

Abstract: A smooth non-linear map is studied in a neighbourhood of an abnormal (degenerate) point. Inverse function and implicit function theorems are proved. The proof is based on the examination of a family of constrained extremal problems; second-order necessary conditions, which make sense also in the abnormal case, are used in the process. If the point under consideration is normal, then these conditions turn into the classical ones.


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English version:
Sbornik: Mathematics, 2000, 191:1, 1–24

Bibliographic databases:

UDC: 517.9
MSC: Primary 46A99, 58C15; Secondary 49K27
Received: 20.08.1998

Citation: A. V. Arutyunov, “Implicit function theorem as a realization of the Lagrange principle. Abnormal points”, Mat. Sb., 191:1 (2000), 3–26; Sb. Math., 191:1 (2000), 1–24

Citation in format AMSBIB
\by A.~V.~Arutyunov
\paper Implicit function theorem as a~realization of the~Lagrange principle. Abnormal points
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 1
\pages 3--26
\jour Sb. Math.
\yr 2000
\vol 191
\issue 1
\pages 1--24

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    This publication is cited in the following articles:
    1. Arutyunov A.V., “Calculation of the cone tangent to the image of a mapping at a singular point”, Dokl. Math., 62:3 (2000), 351–352  mathnet  mathscinet  mathscinet  zmath  isi  isi  elib
    2. A. V. Arutyunov, “Necessary Extremum Conditions and an Inverse Function Theorem without a priori Normality Assumptions”, Proc. Steklov Inst. Math., 236 (2002), 25–36  mathnet  mathscinet  zmath
    3. Arutyunov A.V., Yachimovich V., “2-normal processes in controlled dynamical systems”, Differ. Equ.–1094, 38:8 (2002), 1081–1094  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    4. Arutyunov A.V., “Inverse function theorem on a cone in the neighborhood of an abnormal point”, Dokl. Math., 67:2 (2003), 149–152  mathnet  mathscinet  zmath  isi  elib
    5. A. V. Arutyunov, A. F. Izmailov, “The sensitivity theory for abnormal optimization problems with equality constraints”, Comput. Math. Math. Phys., 43:2 (2003), 178–193  mathnet  mathscinet  zmath  elib
    6. Avakov E.R., Arutyunov A.V., “Abnormal problems with a nonclosed image”, Dokl. Math., 70:3 (2004), 924–927  mathnet  mathscinet  isi  elib
    7. Arutyunov A.V., Izmailov A.F., “Abnormal equality-constrained optimization problems: sensitivity theory”, Math. Program., 100:3, Ser. A (2004), 485–515  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    8. A. V. Arutyunov, “Covering of nonlinear maps on a cone in neighborhoods of irregular points”, Math. Notes, 77:4 (2005), 447–460  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. A. V. Arutyunov, “On real quadratic forms annihilating an intersection of quadrics”, Russian Math. Surveys, 60:1 (2005), 157–158  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. E. R. Avakov, A. V. Arutyunov, “Inverse function theorem and conditions of extremum for abnormal problems with non-closed range”, Sb. Math., 196:9 (2005), 1251–1269  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. A. F. Izmailov, “On the analytical and numerical stability of critical Lagrange multipliers”, Comput. Math. Math. Phys., 45:6 (2005), 930–946  mathnet  mathscinet  zmath  elib  elib
    12. A. V. Arutyunov, “An implicit function theorem without a priori assumptions about normality”, Comput. Math. Math. Phys., 46:2 (2006), 195–205  mathnet  crossref  mathscinet  zmath  elib  elib
    13. A. F. Izmailov, “Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications”, Comput. Math. Math. Phys., 47:4 (2007), 533–554  mathnet  crossref  mathscinet  zmath  elib  elib
    14. A. V. Arutyunov, “On implicit function theorems at abnormal points”, Proc. Steklov Inst. Math. (Suppl.), 271, suppl. 1 (2010), S18–S27  mathnet  crossref  isi  elib
    15. A. V. Arutyunov, S. E. Zhukovskiy, “Existence and properties of inverse mappings”, Proc. Steklov Inst. Math., 271 (2010), 12–22  mathnet  crossref  mathscinet  zmath  isi  elib
    16. A. V. Arutyunov, D. Yu. Karamzin, “Regular zeros of quadratic maps and their application”, Sb. Math., 202:6 (2011), 783–806  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    17. E. R. Avakov, A. V. Arutyunov, D. Yu. Karamzin, “Inverse function in the neighborhood of an abnormal point of a smooth map”, Dokl. Math, 85:3 (2012), 305  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    18. A. V. Arutyunov, “Smooth abnormal problems in extremum theory and analysis”, Russian Math. Surveys, 67:3 (2012), 403–457  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    19. E. R. Avakov, A. V. Arutyunov, D. Yu. Karamzin, “An investigation of smooth maps in a neighbourhood of an abnormal point”, Izv. Math., 78:2 (2014), 213–250  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    20. A. V. Arutyunov, S. E. Zhukovskii, “On Surjective Quadratic Mappings”, Math. Notes, 99:2 (2016), 192–195  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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