X
Minimize
2
2
X 12 1 X 1 2
subject to
X1 X 2 t 4
X 2 t 0.25 X 12 2
i.e.,
Minimize
s.t.
X 22 2 X 12 X 2 X 14 1 2 X 1 X 12
4 X 11 X 21 d 1
0.25 X 12 X 21 2 X 21 d 1
© Dennis L. Bricker
Dept of Mechanical & Industrial Engineering
The University of Iowa
4/11/2006
page 1 of 51
page 2 of 51
Next, rewrite the signomial constraint:
Introduce a new variable X3
X 22 X 14 1 X 12 d X 3 2 X 12 X 2 2 X 1 d
so that the signomial now appears in a constraint:
Minimize
4/11/2006
X3
Ÿ
subject to
X 22 2 X 12 X 2 X 14 1 2 X 1 X 12 d X 3
1
1
4X X
1
2
X 22 X 14 1 X 12
d1
X 3 2 X 12 X 2 2 X 1
If we condense the denominator
d1
X 3 2 X 12 X 2 2 X 1
0.25 X 12 X 21 2 X 21 d 1
into a monomial, using the Arithmetic-Geometric Mean
Inequality, the result will be a posynomial approximation!
4/11/2006
page 3 of 51
4/11/2006
page 4 of 51
Condensing the numerator:
G1
i
i
n
for all G satisfying
¦G
i
§ ui ·
¸
1 © i ¹
for G such that
1, G i t 0
G1 G 2 G 3 1, G i t 0, i 1,2,3
i 1
with equality if & only if
u1
u2
G1
G2
G1
"
G3
2
1
¦u t –¨ G
i 1
G2
§ X · § 2X 2X · § 2X ·
X 3 2 X X 2 2 X1 t ¨ 3 ¸ ¨ 1 2 ¸ ¨ 1 ¸
© G1 ¹ © G 2 ¹ © G 3 ¹
Gi
n
n
G2
G3
§1· § 2· §2·
X 3 2 X X 2 2 X 1 t ¨ ¸ ¨ ¸ ¨ ¸ u X 12G 2 G3 X 2G 2 X 3G1
G 2 ¹ © G 2 ¹ © G3 ¹
©
un
2
1
Gn
coefficient C G 4/11/2006
page 5 of 51
4/11/2006
page 6 of 51
We choose G so that, for a given X ,
so that
G1
X3
X 3 2 X 12 X 2 2 X 1
G2
2
1
2
1
2X X2
X 3 2 X X 2 2 X1
G3
2 X1
X 3 2 X 12 X 2 2 X 1
Minimize
X3
subject to
X 22 X 14 1 X 12
d1
X 3 2 X 12 X 2 2 X 1
4 X 11 X 21 d 1
0.25 X 12 X 21 2 X 21 d 1
G1 G 2 G 3 1
and the approximation is exact at X .
4/11/2006
page 7 of 51
4/11/2006
page 8 of 51
2,2,1 as the initial point.
We will choose X
X3 1
Ÿ G1
2 X 12 X 2
16 Ÿ G 2
2 X1
4
Ÿ G3
G1
So we get the posynomial approximation
X 22 X 14 1 X 12
d1
3.77409X 11.7143 X 20.7619 X 30.04762
1
0.047619
21
16
0.761905
21
4
0.190476
21
which is the posynomial constraint:
0.264965 X 11.7143 X 21.2381 X 30.04762
G3
G2
§1· § 2· §2·
3.77409
¨ ¸ ¨ ¸ ¨ ¸
G
G
G
© 2¹ © 2¹ © 3¹
X 12G 2 G3 X 2G 2 X 3G1 X 11.7143 X 20.7619 X 30.04762
4/11/2006
0.264965 X 12.28571 X 20.7619 X 30.04762
0.264965 X 11.7143 X 20.7619 X 30.04762
0.264965 X 10.28571 X 20.7619 X 30.04762 d 1
page 9 of 51
4/11/2006
page 10 of 51
4/11/2006
page 12 of 51
Posynomial GP approximation of the signomial GP:
Minimize
X3
subject to
0.264965 X 11.7143 X 21.2381 X 30.04762
0.264965 X 12.28571 X 20.7619 X 30.04762
0.264965 X 11.7143 X 20.7619 X 30.04762
0.264965 X 10.28571 X 20.7619 X 30.04762 d 1
4 X 11 X 21 d 1
0.25 X 12 X 21 2 X 21 d 1
4/11/2006
page 11 of 51
Rosenbrock et al.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Coefficients and exponent matrix:
Number of variables: 3
Number of polynomials: 4
Total number of terms: 10
Degrees of difficulty: 6
Terms per polynomial: 1 6 1 2
t p
Ct
¯¯¯¯¯¯¯¯¯¯
1 1
1
-----------2 2
1
3 2 ¯2
4 2
1
5 2
1
6 2 ¯2
7 2
1
-----------8 3
4
-----------9 4
0.25
10 4
2
------------
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
exponents
¯¯¯¯¯¯¯¯¯
0 0 1
--------0 2 ¯1
2 1 ¯1
4 0 ¯1
0 0 ¯1
1 0 ¯1
2 0 ¯1
--------¯1 ¯1 0
--------2 ¯1 0
0 ¯1 0
---------
t = term number, p = polynomial
Ct = coefficient
4/11/2006
page 13 of 51
4/11/2006
Current Parameters
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Tolerances for duality gap: 0.0001
Tolerances for constraints
(maximum allowable infeasibility)
Bounds on variables
# var
LB
UB
1 X[1] 0.00001 100000
2 X[2] 0.00001 100000
3 X[3] 0.00001 100000
page 14 of 51
= 0.0001
Tolerance "epsilon" for stopping criterion:
epsilon > sum |d_rho|,
where
d_rho is the vector of changes in the weights for the
terms of the polynomials: epsilon = 0.0005
Maximum # Posynomial subproblems to be solved = 5
Maximum # LPs to be solved per posynomial subproblem = 5
4/11/2006
page 15 of 51
4/11/2006
page 16 of 51
User-specified Grid Point
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i
variable
1
X[1]
2
X[2]
User-specified Grid Point
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
value
2
2
i
¯¯
1
2
3
The objective function value for this point is 5
The values of the constraint polynomials are:
k
P(k)
2
1
3
1.5
variable
X[1]
X[2]
X[3]
value
¯¯¯¯¯
2
2
1
The objective function value for this point is 1
The values of the constraint polynomials are:
k
P(k)
2
5
3
1
4
1.5
4/11/2006
page 17 of 51
<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>
Major Iteration # 1
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1
X[1]
2
2
X[2]
2
3
X[3]
1
4/11/2006
constraint Value
2
5.0
3
1.0
4
1.5
page 18 of 51
Infeasibility
4.0
0.0
0.5
Weights of negative terms, used for condensation:
poly term
value
2
2 0.761905
2
5 0.190476
term# poly# value
1
1
1.0
2
2
4.0
3
2 ¯16.0
4
2 16.0
5
2
1.0
6
2 ¯4.0
7
2
4.0
8
3
1.0
9
4
0.5
10
4
1.0
Objective function = 1
4/11/2006
page 19 of 51
4/11/2006
page 20 of 51
Condensation of Signomial GP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Number of variables: 2
Number of posynomials: 10
Total number of terms: 14
Degrees of difficulty: 11
Terms per posynomial: 1 4 1 2 1 1 1 1 1 1
(includes bounds on variables to ensure dual feasibility)
Coefficients and exponent matrix:
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
p
1
2
2
2
2
3
4
4
5
6
7
8
9
10
Ct
1
0.264965
0.264965
0.264965
0.264965
4
0.25
2
0.00001
0.00001
0.00001
0.00001
0.00001
0.00001
exponents
0
¯1.71429
2.28571
¯1.71429
0.285714
¯1
2
0
¯1
0
0
1
0
0
0
1.2381
¯0.761905
¯0.761905
¯0.761905
¯1
¯1
¯1
0
¯1
0
0
1
0
1
¯0.047619
¯0.047619
¯0.047619
¯0.047619
0
0
0
0
0
¯1
0
0
1
t = term number, p = posynomial, Ct = coefficient
4/11/2006
page 21 of 51
4/11/2006
page 22 of 51
<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>
Posynomial GP via Generalized LP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1 X[1]
1.597846
2 X[2]
2.636614
3 X[3]
0.365643
Major Iteration # 2
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1
X[1]
1.597846
2
X[2]
2.636614
3
X[3]
0.365643
Posynomial objective functions:
Primal: 0.365643 Dual: 0.365643
Duality Gap: 0 = 0 percent
term# poly#
value
1
1
0.365643
2
2 19.012371
3
2 ¯36.820482
4
2 17.827182
5
2
2.734911
6
2 ¯8.739933
7
2
6.982533
8
3
0.949464
9
4
0.242082
10
4
0.758549
Objective function = 0.365643
4/11/2006
page 23 of 51
4/11/2006
page 24 of 51
constraint
2
3
4
Value
0.996581
0.949464
1.000631
Infeasibility Lambda
0.000000000 24.0059
0.000000000 0.0000
0.000631135 0.0000 small infeasibility!
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Weights of negative terms, used for condensation:
poly term
2
2
2
5
value
0.790811
0.187712
Condensation of Signomial GP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Coefficients and exponent matrix:
change
0.02890610
0.00276453
p
1
2
2
2
2
3
4
4
5
6
7
8
9
10
Ct
1
0.283551
0.283551
0.283551
0.283551
4
0.25
2
0.00001
0.00001
0.00001
0.00001
0.00001
0.00001
exponents
0
¯1.76933
2.23067
¯1.76933
0.230667
¯1
2
0
¯1
0
0
1
0
0
0
1.20919
¯0.790811
¯0.790811
¯0.790811
¯1
¯1
¯1
0
¯1
0
0
1
0
1
¯0.0214775
¯0.0214775
¯0.0214775
¯0.0214775
0
0
0
0
0
¯1
0
0
1
t = term number, p = posynomial, Ct = coefficient
4/11/2006
page 25 of 51
4/11/2006
page 26 of 51
<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>
Posynomial GP via Generalized LP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1 X[1]
1.561004
2 X[2]
2.605930
3 X[3]
0.289182
Major Iteration # 3
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1 X[1]
1.561004
2 X[2]
2.605930
3 X[3]
0.289182
term# poly#
value
1
1
0.289182
2
2 23.483061
3
2 ¯43.916735
4
2 20.532668
5
2
3.458033
6
2 ¯10.796007
7
2
8.426306
8
3
0.983316
9
4
0.233768
10
4
0.767480
Posynomial objective functions:
Primal: 0.289182 Dual: 0.289182
Duality Gap: 0 = 0 percent
Objective function = 0.289182
4/11/2006
page 27 of 51
4/11/2006
page 28 of 51
constraint
2
3
4
Value
1.187326
0.983316
1.001248
Infeasibility Lambda
0.18732621 3.4648 large infeasibility!
0.00000000 0.0000
0.00124837 46.5604
Weights of negative terms, used for condensation:
poly term
2
2
2
5
value
0.788271
0.193780
change
0.00253994
0.00606819
Condensation of Signomial GP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Coefficients and exponent matrix:
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
p
1
2
2
2
2
3
4
4
5
6
7
8
9
10
Ct
1
0.284107
0.284107
0.284107
0.284107
4
0.25
2
0.00001
0.00001
0.00001
0.00001
0.00001
0.00001
exponents
0
¯1.77032
2.22968
¯1.77032
0.229678
¯1
2
0
¯1
0
0
1
0
0
0
1.21173
¯0.788271
¯0.788271
¯0.788271
¯1
¯1
¯1
0
¯1
0
0
1
0
1
¯0.0179492
¯0.0179492
¯0.0179492
¯0.0179492
0
0
0
0
0
¯1
0
0
1
t = term number, p = posynomial, Ct = coefficient
4/11/2006
page 29 of 51
Posynomial GP via Generalized LP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i
1
2
3
4/11/2006
page 30 of 51
<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>
Major Iteration # 4
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1
X[1]
1.549938
2
X[2]
2.600550
3
X[3]
0.347202
variable
value
X[1]
1.549938
X[2]
2.600550
X[3]
0.347202
Posynomial objective functions:
Primal: 0.347202 Dual: 0.347202
Duality Gap: 0 = 0 percent
term# poly#
value
1
1
0.347202
2
2 19.478183
3
2 ¯35.986660
4
2 16.621670
5
2
2.880168
6
2 ¯8.928162
7
2
6.919047
8
3
0.992386
9
4
0.230942
10
4
0.769068
Objective function = 0.347202
4/11/2006
page 31 of 51
4/11/2006
page 32 of 51
constraint
Value Infeasibility Lambda
2
0.984246 0.0000000000 11.9815
3
0.992386 0.0000000000 0.0000
4
1.000010 0.0000100608 47.3088
Condensation of Signomial GP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Coefficients and exponent matrix:
small infeasibility!
Weights of negative terms, used for condensation:
poly term
value
change
2
2 0.783770 0.004500932
2
5 0.194451 0.000670688
Notice that the solution seems to alternate between
one with small infeasibility and objective approximately 0.347,
and
one with large infeasibility and objective approximately 0.284
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
p
1
2
2
2
2
3
4
4
5
6
7
8
9
10
Ct
1
0.280612
0.280612
0.280612
0.280612
4
0.25
2
0.00001
0.00001
0.00001
0.00001
0.00001
0.00001
exponents
0
¯1.76199
2.23801
¯1.76199
0.238009
¯1
2
0
¯1
0
0
1
0
0
0
1.21623
¯0.78377
¯0.78377
¯0.78377
¯1
¯1
¯1
0
¯1
0
0
1
0
1
¯0.0217795
¯0.0217795
¯0.0217795
¯0.0217795
0
0
0
0
0
¯1
0
0
1
t = term number, p = posynomial, Ct = coefficient
4/11/2006
page 33 of 51
4/11/2006
page 34 of 51
<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>
Posynomial GP via Generalized LP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1 X[1]
1.548186
2 X[2]
2.595916
3 X[3]
0.284417
Major Iteration # 5
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1 X[1]
1.548186
2 X[2]
2.595916
3 X[3]
0.284417
Posynomial objective functions:
Primal: 0.284417 Dual: 0.284417
Duality Gap: 0 = 0 percent
term# poly#
value
1
1
0.284417
2
2 23.693282
3
2 ¯43.753293
4
2 20.199298
5
2
3.515961
6
2 ¯10.886721
7
2
8.427333
8
3
0.995282
9
4
0.230832
10
4
0.770441
Objective function = 0.284417
4/11/2006
page 35 of 51
4/11/2006
page 36 of 51
constraint
Value
2
1.195860
3
0.995282
4
1.001273
Infeasibility
Lambda
0.19585984 4.05055 large infeasibility!
0.00000000 0.00000
0.00127279 45.91482
Weights of negative terms, used for condensation:
poly term
value
change
2
2 0.786364 0.00259380
2
5 0.195664 0.00121298
Condensation of Signomial GP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Coefficients and exponent matrix:
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
p
1
2
2
2
2
3
4
4
5
6
7
8
9
10
Ct
1
0.28334
0.28334
0.28334
0.28334
4
0.25
2
0.00001
0.00001
0.00001
0.00001
0.00001
0.00001
exponents
0
¯1.76839
2.23161
¯1.76839
0.231609
¯1
2
0
¯1
0
0
1
0
0
0
1.21364
¯0.786364
¯0.786364
¯0.786364
¯1
¯1
¯1
0
¯1
0
0
1
0
1
¯0.0179727
¯0.0179727
¯0.0179727
¯0.0179727
0
0
0
0
0
¯1
0
0
1
t = term number, p = posynomial, Ct = coefficient
4/11/2006
page 37 of 51
4/11/2006
page 38 of 51
<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>
Posynomial GP via Generalized LP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1 X[1]
1.549449
2 X[2]
2.600175
3 X[3]
0.347004
Major Iteration # 6
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1 X[1]
1.549449
2 X[2]
2.600175
3 X[3]
0.347004
Posynomial objective functions:
Primal: 0.347004 Dual: 0.347004
Duality Gap: 0 = 0 percent
term# poly#
value
1
1
0.347004
2
2 19.483638
3
2 ¯35.979230
4
2 16.610155
5
2
2.881808
6
2 ¯8.930426
7
2
6.918618
8
3
0.992842
9
4
0.230830
10
4
0.769179
Objective function = 0.347004
4/11/2006
page 39 of 51
4/11/2006
page 40 of 51
constraint
2
3
4
Value
0.984563
0.992842
1.000009
Infeasibility
Lambda
0.0000000000 5.39064
0.0000000000 0.00000
0.0000088935 54.00238 small infeasibility!
Condensation of Signomial GP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Coefficients and exponent matrix:
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Weights of negative terms, used for condensation:
poly term
value
change
2
2 0.783696 0.00266745
2
5 0.194522 0.00114178
p
1
2
2
2
2
3
4
4
5
6
7
8
9
10
Ct
1
0.280582
0.280582
0.280582
0.280582
4
0.25
2
0.00001
0.00001
0.00001
0.00001
0.00001
0.00001
exponents
0
¯1.76191
2.23809
¯1.76191
0.238086
¯1
2
0
¯1
0
0
1
0
0
0
1.2163
¯0.783696
¯0.783696
¯0.783696
¯1
¯1
¯1
0
¯1
0
0
1
0
1
¯0.0217819
¯0.0217819
¯0.0217819
¯0.0217819
0
0
0
0
0
¯1
0
0
1
t = term number, p = posynomial, Ct = coefficient
4/11/2006
page 41 of 51
4/11/2006
page 42 of 51
<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>
Posynomial GP via Generalized LP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable value
1 X[1]
1.54807
2 X[2]
2.59582
3 X[3]
0.28439
Major Iteration # 7
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable value
1 X[1]
1.54807
2 X[2]
2.59582
3 X[3]
0.28439
Posynomial objective functions:
Primal: 0.28439 Dual: 0.28439
Duality Gap: 0 = 0 percent
term# poly#
value
1
1
0.284390
2
2 23.693892
3
2 ¯43.749274
4
2 20.195068
5
2
3.516301
6
2 ¯10.886941
7
2
8.426858
8
3
0.995394
9
4
0.230805
10
4
0.770468
Objective function = 0.28439
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4/11/2006
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constraint
Value
2
1.195904
3
0.995394
4
1.001273
Infeasibility
Lambda
0.19590358 4.05662
0.00000000 0.00000
0.00127302 45.90966
large infeasibility!
Weights of negative terms, used for condensation:
poly term
value
change
2
2 0.786345 0.00264891
2
5 0.195681 0.00115910
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
p
1
2
2
2
2
3
4
4
5
6
7
8
9
10
Condensation of Signomial GP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Ct
exponents
1
0
0
0.283332
¯1.76837
1.21365
0.283332
2.23163 ¯0.786345
0.283332
¯1.76837 ¯0.786345
0.283332
0.231629 ¯0.786345
4
¯1
¯1
0.25
2
¯1
2
0
¯1
0.00001
¯1
0
0.00001
0
¯1
0.00001
0
0
0.00001
1
0
0.00001
0
1
0.00001
0
0
1
¯0.0179739
¯0.0179739
¯0.0179739
¯0.0179739
0
0
0
0
0
¯1
0
0
1
t = term number, p = posynomial, Ct = coefficient
4/11/2006
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4/11/2006
page 46 of 51
<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>
Posynomial GP via Generalized LP
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1 X[1]
1.549436
2 X[2]
2.600165
3 X[3]
0.347001
Major Iteration # 8
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
i variable
value
1 X[1]
1.549436
2 X[2]
2.600165
3 X[3]
0.347001
term# poly#
value
1
1
0.347001
2
2 19.483698
3
2 ¯35.978894
4
2 16.609793
5
2
2.881838
6
2 ¯8.930448
7
2
6.918579
8
3
0.992854
9
4
0.230827
10
4
0.769182
Posynomial objective functions:
Primal: 0.347001 Dual: 0.347001
Duality Gap: 0 = 0 percent
Objective function = 0.347001
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4/11/2006
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constraint
2
3
4
Value
0.984567
0.992854
1.000009
Infeasibility
Lambda
0.00000000000 5.32287
0.00000000000 0.00000
0.00000886453 54.06786
small infeasibility!
<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>
4/11/2006
page 49 of 51
Dropping two initial points from the plot:
“zig-zagging”
behavior is
apparent!
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4/11/2006
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