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  Presented at WCNA'96, Athens, Greece, July 1996
  Published in "Nonlinear Analysis", 1997, 30:4, 2439-2448


            Abstract
 
  It is well known in the theory of extremal problems that the abnormal case, i.e. the case when equality constraints
  are degenerate at the examined point, is a difficult subject to obtain higher order conditions of a local minimum.
  Especially it is true for necessary conditions. The matter is that "standard" necessary conditions, relevant to the
  general case, are always trivially fulfilled in the abnormal case and do not provide any information about the presence
  or absence of a local minimum at the given point. Here we present a method of treatment extremal problems with
  degenerate equality constraints, originally proposed by A.A.Milyutin. It consists of the passing from the given problem
  to another one, in which the equality constraints are nondegenerate. Application of this method and of its refinement
  allows one to obtain informative quadratic order necessary conditions for local minima in some classes of problems.}
 

Key words and phrases:
   equality constraints, Lyusternik condition, Lagrange multipliers,
   second and third variations of Lagrange function, weak and Pontryagin minimum,
   quadratic order conditions, finite codimension, Legendre type conditions.

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