ISSN 1064-5624, Doklady Mathematics, 2012, Vol. 85, No. 3, pp. 328–330. © Pleiades Publishing, Ltd., 2012.
Original Russian Text © E.R. Avakov, A.V. Arutyunov, D.Yu. Karamzin, 2012, published in Doklady Akademii Nauk, 2012, Vol. 444, No. 2, pp. 127–130.
328
In this paper, we give second-order necessary opti-
mality conditions for abnormal finite-dimensional
optimization problems. Given a smooth mapping F :
X → Y, where X =
n
and Y =
k
are arithmetic
spaces, and a smooth function f : X → , consider the
optimization problem
(1)
Let x
0
be a local minimizer in problem (1).
Define the Lagrangian
where = (λ
0
, λ), λ
0
∈ , and λ
0
≥ 0 and λ ∈ Y are
Lagrange multipliers. Define the set of Lagrange mul-
tipliers
Consider the following two cases. First, let x
0
be a
normal point; i.e., imF '(x
0
) = Y, where im denotes the
image of the linear operator. In this case, the first- and
second-order necessary conditions are well known [1].
Specifically, the exists a Lagrange multiplier ∈
(x
0
) such that the second derivative
xx
(x
0
, ) of the
Lagrangian is positive semidefinite on the kernel
kerF '(x
0
) of F '(x
0
), which in this case coincides with
fx () min, Fx () → 0 . =
x λ , ( ) λ
0
fx () λ Fx () , 〈 〉 , + =
λ
Λ x
0
( )
:= λ λ
0
λ , ( ) , λ 1 , λ
0
0 ,
x
x
0
λ , ( ) ≥ 0 = = = { } .
λ
Λ λ
the tangent subspace to the set {x: F (x) = 0} at the
point x
0
.
Now let x
0
be an abnormal point; i.e., imF '(x
0
) ≠ Y.
Then the Lagrange principle holds automatically; i.e.,
it holds for λ
0
= 0 irrespective of the minimized func-
tion, while, according to [2], the above second-order
necessary conditions may be violated. At the same
time, the following second-order necessary conditions
were obtained in [2], which remain valid without using
the a priori assumption that the point x
0
is normal.
For an arbitrary positive integer r, consider the set
of Lagrange multipliers ∈ (x
0
) for each of which
there exists a linear subspace Π
r
⊆ X (depending on )
such that
The set of such Lagrange multipliers is denoted by
(x
0
).
The following result was proved in [2].
Theorem 1. Let x
0
be a local minimizer in problem (1).
Then (x
0
) ≠ . Moreover,
For an abnormal point x
0
, Theorem 1 was strength-
ened in [3]. Specifically, it was proved that, if x
0
is an
abnormal point, then the set (x
0
) in Theorem 1 can
be replaced with a generally smaller set (x
0
). For
a quadratic problem, this assertion was proved in [4].
Naturally, the approach to the study of abnormality
used in Theorem 1 can be called an index approach,
since it is based on estimating the index of the second
derivative of the Lagrangian. Note also that another
approach to the study of necessary optimality condi-
tions at abnormal points was proposed in [5]. It is
λ Λ
λ
codimΠ
r
r , Π
r
kerF ' x
0
( ), ⊆ ≤
xx
x
0
λ , ( ) xx , [ ] 0 x ∀ Π
r
. ∈ ≥
Λr
Λk
xx
x
0
λ , ( ) hh , [ ]
λ Λ
k
x
0
( ) ∈
max 0 h ∀ kerF ' x
0
( ). ∈ ≥
Λk
Λk 1 –
On Second-Order Necessary Optimality Conditions
in Finite-Dimensional Abnormal Optimization Problems
E. R. Avakov
a
, A. V. Arutyunov
b
, and D. Yu. Karamzin
c
Presented by Academician A.B. Kurzhanskii December 24, 2011
Received December 26, 2011
DOI: 10.1134/S1064562412030076
a
Trapeznikov Institute of Control Sciences,
Russian Academy of Sciences,
Profsoyuznaya ul. 65, Moscow, 117997 Russia
e-mail: eramag@mail.ru
b
Russian University of Peoples’ Friendship,
ul. Miklukho-Maklaya 6, Moscow, 117198 Russia
e-mail: arutun@orc.ru
c
Dorodnicyn Computing Center, Russian Academy of Sciences,
ul. Vavilova 40, Moscow, 119333 Russia
e-mail: dmitry_karamzin@mail.ru
MATHEMATICS