David G Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 12 pptx
David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 12 pptx
... 150–151,
21 2, 523 , 526
Gauss-Southwell method, 25 3 25 5, 395
Generalized reduced gradient method, 385
Geodesics, 374–380
Geometric convergence, 20 9
Global Convergence, 20 1 20 8, 22 6 22 8,
25 3, 27 9, 29 6, ... type methods, 25 0
Limit superior, 5 12
Line search, 1 32, 21 5 23 3, 24 0 24 1, 24 8,
25 3 25 4, 27 8 27 9, 29 1, 3 02, 304–305,
3 12, 360, 3 62, 395, 466
Lin...
David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 1 Part 6 pptx
... chapters.
Not only have nonlinear methods improved linear programming, but interior-
point methods for linear programming have been extended to provide new
approaches to nonlinear programming. This chapter ... The Central Path 121
5.5 THE CENTRAL PATH
The concept underlying interior-point methods for linear programming is to use
nonlinear programming techniques of analys...
David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 1 pot
... 2 149690 2 06 023 4
6 2 17 027 2 2 149693 2 06 023 7
7 2 1 727 86 2 167983 2 165641
8 2
17 427 9 2 173169 2 165704
9 2 174583 2 1743 92 2 168440
10 2 174638 2 174397 2 173981
11 2 174651 2 1745 82 ... 2 1745 82 2 174048
12 2 174655 2 174643 2 174054
13 2 174658 2 174656 2 174608
14 2 174659 2 174656 2 174608
15 2 174659 2 174658 2 174 622
16 2 174659 2 174655
17 2 174659 2 174656
18...
David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 2 ppt
... 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
... g
0
g
1
g
k
= g
0
Qg
0
Q
k
g
0
b) d
0
d
1
d
k
= g
0
Qg
0
Q
k
g
0
c) d
T
k
Qd
i
=0 for i k −1
d)
k
=g
T
k
g
k
/d
T...
David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 3 pot
... Self-scaling
1 20 0.333 20 0.333 20 0.333 20 0.333
2 2.7 327 89 93.65457 93.65457 2. 811061
33836899×10
2
56. 929 99 56. 929 99 35 627 69×10
2
46376461×10
−4
1. 620 688 1. 620 688 4 20 0600 ×10
−4
51 21 9515×10
−5
5 25 1115×10
−1
5 25 1115×10
−1
4 726 918×10
−6
62 457944 ... 6994 023 ×10
−1
6994 023 ×10
−1
69 020 72 ×10
−1
32 939804×10
2
1 22 5501×10
2
1 22 5501×10
2...
David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 4 pps
... problem
minimize 2x
2
1
+2x
1
x
2
+x
2
2
−10x
1
−10x
2
subject to x
2
1
+x
2
2
5
3x
1
+x
2
6
The first-order necessary conditions, in addition to the constraints, are
4x
1
+2x
2
−10 +2
1
x
1
+3
2
=0
2x
1
+2x
2
−10 +2
1
x
2
+
2
=0
1
... the problem
extremize x
1
+x
2
2
+x
2
x
3
+2x
2
3
subject to
1
2
x
2
1
+x
2
2
+x
2
3
=1
The first-order...
David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 5 potx
... 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
11.9 Zero-Order Conditions and Lagrange ... problem
minimize x
2
1
+x
2
2
+x
2
3
+x
2
4
−2x
1
−3x
4
subject to 2x
1
+x
2
+x
3
+4x
4
=7 (20 )
x
1
+x
2
+2x
3
+x
4
=6
x
i
0i=1 2 3 4
Suppose that given the feasi...
David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 6 pps
... −3756 423
20 −6656377 20 −3759 123
40 −6658443 50 −3765 128
60 −6659191 100 −3771 625
80 −6659514 20 0 −3778983
100 −6659656 500 −3787989
120 −6659 825 1000 −37930 12
121 −6659 827 1500 ... the projected
negative gradient was computed:
minimize x
2
1
+x
2
2
+x
2
3
+x
2
4
−2x
1
−3x
4
subject to 2x
1
+x
2
+x
3
+4x
4
=7
x
1
+x
2
+2x
3
+x
4
=6
x
i
0i=1...
David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 7 doc
... set
strategy. See Gill, Murray, and Wright [G7 ] for a discussion of working sets and active set
strategies.
12. 5 This material is taken from Luenberger [L14].
12. 6– 12. 7 The reduced gradient method ... study of the global convergence properties of feasible
direction methods was begun by Topkis and Veinott [T8] and by Zangwill [Z2].
12. 3– 12. 4 The gradient projection meth...
David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 8 pot
... absolute-value penalty function. We minimize the function
2x
2
+2xy +y
2
−2y +cx (66)
We rewrite (66) as
2x
2
+2xy +y
2
−2y +cx
=2x
2
+2xy +cx+y −1
2
−1
=2x
2
+2x +cx+y −1
2
+2xy −1 ... problem
minimize x
2
1
+2x
2
2
subject to x
1
+x
2
≥1 /2
x
1
x
2
nonnegative integers
It is clear that the solution is x
1
= 1x
2
= 0, with objective value f
∗
= 1....
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