Geodesic
Smooth abnormal problems in~extremum theory and analysis
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 3, pp. 403-457

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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. 
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     author = {A. V. Arutyunov},
     title = {Smooth abnormal problems in~extremum theory and analysis},
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A. V. Arutyunov. Smooth abnormal problems in~extremum theory and analysis. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 3, pp. 403-457. http://geodesic.mathdoc.fr/item/RM_2012_67_3_a0/