In this section, we state firstorder necessary conditions for
x
∗
to be a local minimizer
and show how these conditions are satisfied on a small example. The proof of the result is
presented in subsequent sections.
Asa preliminarytostatingthenecessaryconditions, wedefinetheLagrangianfunction
for the general problem (12.1).
L
(
x
, λ
)
f
(
x
)
−
i
∈
E
∪
I
λ
i
c
i
(
x
)
.
(12.33)
(We had previously defined special cases of this function for the examples of Section 1.)
FIRSTORDER OPTIMALITY CONDITIONS
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The necessary conditions defined in the following theorem are called
firstorder con
ditions
because they are concerned with properties of the gradients (firstderivative vectors)
of the objective and constraint functions. These conditions are the foundation for many of
the algorithms described in the remaining chapters of the book.
Theorem 12.1
(FirstOrder Necessary Conditions).
Suppose that
x
∗
is a local solution of (12.1), that the functions
f
and
c
i
in (12.1) are
continuously differentiable, and that the LICQ holds at
x
∗
. Then there is a Lagrange multiplier
vector
λ
∗
, with components
λ
∗
i
,
i
∈
E
∪
I
, such that the following conditions are satisfied at
(
x
∗
, λ
∗
)
∇
x
L
(
x
∗
, λ
∗
)
0
,
(12.34a)
c
i
(
x
∗
)
0
,
for all
i
∈
E
,
(12.34b)
c
i
(
x
∗
)
≥
0
,
for all
i
∈
I
,
(12.34c)
λ
∗
i
≥
0
,
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 Fall '10
 Staff
 Math, Optimization, Mathematical optimization, Constraint, lagrange multipliers

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