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

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

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
\Bibitem{Aru96} \by A.~V.~Arutyunov \paper Second-order conditions in extremal problems with finite-dimensional range. 2-normal maps \jour Izv. RAN. Ser. Mat. \yr 1996 \vol 60 \issue 1 \pages 37--62 \mathnet{http://mi.mathnet.ru/izv61} \crossref{https://doi.org/10.4213/im61} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1391117} \zmath{https://zbmath.org/?q=an:0884.49018} \transl \jour Izv. Math. \yr 1996 \vol 60 \issue 1 \pages 39--65 \crossref{https://doi.org/10.1070/IM1996v060n01ABEH000061} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1996VE15400002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746962348} 

• http://mi.mathnet.ru/eng/izv61
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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. Ledzewicz U., Schattler H., “High-order approximations and generalized necessary conditions for optimality”, SIAM Journal on Control and Optimization, 37:1 (1998), 33–53
2. A. V. Arutyunov, “Implicit function theorem as a realization of the Lagrange principle. Abnormal points”, Sb. Math., 191:1 (2000), 1–24
3. Marinkovic B., “Optimality conditions in discrete optimal control problems with state constraints”, Numerical Functional Analysis and Optimization, 28:7–8 (2007), 945–955
4. Marinkovic B., “Second-order optimality conditions in a discrete optimal control problem”, Optimization, 57:4 (2008), 539–548
5. A. V. Arutyunov, S. E. Zhukovskiy, “Existence and properties of inverse mappings”, Proc. Steklov Inst. Math., 271 (2010), 12–22
6. A. V. Arutyunov, “Smooth abnormal problems in extremum theory and analysis”, Russian Math. Surveys, 67:3 (2012), 403–457
7. K. V. Storozhuk, “O verkhnem topologicheskom predele semeistva vektornykh podprostranstv korazmernosti $k$”, Sib. elektron. matem. izv., 12 (2015), 432–435
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