e use the notation of the problem
(
Smooth optimization problem
).
Note that (1) implies that the proposition
(
Fritz John conditions
) cannot take
place with
.
The conditions (2),(3) imply that the components of
are "informative" in the sense that the set
of the proposition (
Fritz John
conditions
) is nonempty and the nonzero components of
mark those conditions
and
that are "active"
(
of the proposition (
Fritz John
conditions
)'s proof violates these conditions and the
lies on the boundary set by such conditions).
We introduce the
notation
The condition 2 of the above definition may be equivalently written as
Here the
sign
after the brackets
indicates that the summation of the scalar product is applied to the
components of the gradient
.
Proof
We assume that all the conditions of the definition
(
Pseudonormality
) hold and reach a
contradiction.
We introduce the
notation
According to the condition 4 of this proposition and condition 2 of the
definition (
Pseudonormality
), there exists a
such
that
The condition 1 of the definition
(
Pseudonormality
) requires
that
thus
Hence, we already proven the statement for the case
.
It remains to consider the case
under the assumption that the conditions 1,2,3 of of the definition
(
Pseudonormality
) and the conditions 1,2,3,4
of this proposition are true and arrive to contradiction. By the assumption
,
we have
and by condition 2 of the definition
(
Pseudonormality
) we have
The condition 1 of the definition
(
Pseudonormality
) implies
Hence, by the condition 2 of the proposition, all
are
zero:
By the condition 3 there is a
from the interior of
such that
Hence,
Hence, we have found a point
and an interior point
of
such
that
This is a contradiction. For an interior point of a cone
we must
have
