... the
direction finding mapping of the simple gradient projection method. At a given point let
13 .1 Penalty Methods 403
c = 1
cP
(x)
c
= 1
c
= 10 c = 10
c = 10 0
a
b
c
= 10 0
x
Fig. 13 .1 Plot of cPx
The ... examples.
Example 1. Define
Bx =
p
i =1
−
1
g
i
x
( 31)
The barrier objective
rc
k
x = fx −
1
c
k
p
i =1
1
g
i
x
has its minimum at a point...
... indicated
by drawing partitioning lines through the matrix, as for example,
A =
⎡
⎣
a
11
a
12
a
13
a
14
a
21
a
22
a
23
a
24
a
31
a
32
a
33
a
34
⎤
⎦
=
A
11
A
12
A
21
A
22
The resulting submatrices ... [B 11] . The
SOLVER method was proposed by Wilson [W2] for convex programming problems and
was later interpreted by Beale [B7]. Garcia-Palomares and Mangasarian [G3...
... hypothesis both g
k
and Qd
k
belong to g
0
Qg
0
Q
k +1
g
0
, the
first by (a) and the second by (b). Thus g
k +1
∈ g
0
Qg
0
Q
k +1
g
0
. Furthermore
g
k +1
g
0
Qg
0
Q
k
g
0
=d
0
... have
g
T
k
d
k
=g
T
k
g
k
−
k 1
g
T
k
d
k 1
and the second term is zero by the Expanding Subspace Theorem.
9.7 Parallel Tangents 2 81
d
k 1
x
k...
... determining the eigen-
values of E
T
LE. These eigenvalues are independent of the particular orthonormal
basis E.
Example 1. In the last section we considered
L =
⎡
⎣
011
10 1
11 0
⎤
⎦
Ly
λy
y
M
Fig. 11 .5 ... 2n
(13 )
We begin by ignoring the nonnegativity constraints, believing that they may
be inactive. Introducing two Lagrange multipliers, and , the Lagrangian is
l =
n
...
... find
A
q
A
T
q
1
=
1
11
⎡
⎣
6 −5 19
−5 614
19 14 73
⎤
⎦
and finally
P =
1
11
⎡
⎢
⎢
⎣
1 − 310
−39−30
1 − 310
0000
⎤
⎥
⎥
⎦
(22)
The gradient at the point (2, 2, 1, 0) is g =2 4 2 −3 and hence ... and therefore g
2
=0 is adjoined to the set of working constraints.
g
1
= 0
∇f
T
g
2
= 0
x
Feasible region
g
1
T
Fig. 12 .4 Constraint to be dropped...
... Suppose
1
,
2
are in the finite region, and let 0 ≤ ≤ 1. Then
1
+ 1
2
= inf fx +
1
+ 1
2
T
g xx ∈
≥inf fx
1
+
T
1
g x
1
x
1
∈
+inf 1 −fx
2
+ 1
T
2
g x
2
x
2
∈
=
1
... often very easy to implement.
14 .1 GLOBAL DUALITY
Duality in nonlinear programming takes its most elegant form when it is formu-
late...
... =Q
1
A
T
AQ
1
A
T
1
AQ
1
c +b −Q
1
c
=−Q
1
I −A
T
AQ
1
A
T
1
AQ
1
c (8)
+Q
1
A
T
AQ
1
A
T
1
b
15 .2 STRATEGIES
There are some general strategies that guide the development ... −6654653 y
9
=− 216 5239
9 −6654653 y
10
=−0736802
14 .5 AUGMENTED LAGRANGIANS
One of the most effective general classes of nonlinear programming methods is
the augmente...