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Izv. RAN. Ser. Mat., 1996, Volume 60, Issue 1, Pages 37–62 (Mi izv61)  

This article is cited in 7 scientific papers (total in 7 papers)

Second-order conditions in extremal problems with finite-dimensional range. 2-normal maps

A. V. Arutyunov


Abstract: A minimization problem with constraints that includes problems for which the constraints are of equality and inequality type is considered. First- and second-order necessary conditions in the Lagrangian form are obtained for this problem. The main difference between these conditions and most of the previously known ones is the fact that they also remain meaningful for abnormal problems, in both the finite-dimensional and infinite-dimensional cases. The notion of 2-normal map is introduced. It is proved that if the map that defines a constraint is 2-normal, then the necessary conditions obtained turn into second-order sufficient conditions after an arbitrarily small perturbation of the problem by terms of second order of smallness. It is also proved that in the space of smooth maps, the set of 2-normal maps is everywhere dense in the Whitney topology.

DOI: https://doi.org/10.4213/im61

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English version:
Izvestiya: Mathematics, 1996, 60:1, 39–65

Bibliographic databases:

MSC: 49K27
Received: 09.09.1994

Citation: A. V. Arutyunov, “Second-order conditions in extremal problems with finite-dimensional range. 2-normal maps”, Izv. RAN. Ser. Mat., 60:1 (1996), 37–62; Izv. Math., 60:1 (1996), 39–65

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Ledzewicz U., Schattler H., “High-order approximations and generalized necessary conditions for optimality”, SIAM Journal on Control and Optimization, 37:1 (1998), 33–53  crossref  mathscinet  isi
    2. A. V. Arutyunov, “Implicit function theorem as a realization of the Lagrange principle. Abnormal points”, Sb. Math., 191:1 (2000), 1–24  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Marinkovic B., “Optimality conditions in discrete optimal control problems with state constraints”, Numerical Functional Analysis and Optimization, 28:7–8 (2007), 945–955  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    4. Marinkovic B., “Second-order optimality conditions in a discrete optimal control problem”, Optimization, 57:4 (2008), 539–548  crossref  mathscinet  zmath  isi  scopus  scopus
    5. 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
    6. 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
    7. K. V. Storozhuk, “O verkhnem topologicheskom predele semeistva vektornykh podprostranstv korazmernosti $k$”, Sib. elektron. matem. izv., 12 (2015), 432–435  mathnet  crossref
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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