CHAPTER
13
More on Numerical Methods for
Constrained Optimum Design
Section 13.3 Approximate Step Size Determination
13.1 ________________________________________________________________________________
Refer to Exercise 12.3 for detailed formulation.
Iteration 1:
2. Compute cost and constraint functions:
fo = 75000, g1 = 0.0667, g2 = 0.5, g3 = − 0.4, g4 = − 24, g5 = − 0.75, g6 = − 29, g7 = − 0.7;
∇f = (300, 250)
3. QP subproblem defined using the data given in Step 2 gives the search direction as
4. ||d(0)|| >
ε
2; Convergence criteria are not satisfied.
6. Step size at the 4th trial (
α
o = 0.125) satisfies the descent condition. Design is updated as
7. R1 = 58196.7, k = 1, go to Step 2.
13.2 ________________________________________________________________________________
Refer to Exercise 12.4 for detailed formulation.
Iteration 1:
2. Compute cost and constraint functions:
fo = 1183.752, g1 = − 0.99337, g2 = − 89.01498, g3 = − 1.4, g4 = − 0.88, g5 = − 7, g6 = − 0.2;
∇f = (98.646, 295.938)
3. QP subproblem defined using the data given in Step 2 gives the search direction as
4. ||d(0)|| >
ε
2; Convergence criteria are not satisfied.
6. Step size at the lst trial (
α
Chapter 13 More on Numerical Methods for Constrained Optimum Design
13.3 ________________________________________________________________________________
Refer to Exercise 12.5 for detailed formulation.
Iteration 1:
2. Compute cost and constraint functions:
fo = 13500, g1 = − 0.16667, g2 = − 0.33333, g3 = − 150, g4 = − 150
3. QP subproblem defined using the data given in Step 2 gives the search direction as
4. ||d(0)|| > ε2; Convergence criteria are not satisfied.
6. Step size at the lst trial (
α
A(1)
1 = 119.99694, A(1)
2 = 110
Chapter 13 More on Numerical Methods for Constrained Optimum Design
13.4 ________________________________________________________________________________
Refer to Exercise 12.6 for detailed formulation.
Iteration 1:
2. Compute cost and constraint functions:
fo = 11.2, g1 = 0.3143, g2 = − 0.25714, g3 = − 2.42857, g4 = − 0.42857, g5 = − 4000
∇f = (0.6, 0.001)
d(0) = (5.49918, 0.0173) with Lagrange multipliers as u = (106.7, 0, 0, 0, 0)
5. ro = 106.7; R = max (Ro, ro) = 106.7
o = 1) satisfies the descent condition. Design is updated as
7. R1 = 106.7, k = 1, go to Step 2.
Arora, Introduction to Optimum Design, 4e
13-5
13.5 ________________________________________________________________________________
Refer to Exercise 12.7 for detailed formulation.
Iteration 1:
1. Initial design is given as (R(0), H (0)) = (6, 15); set Ro = 1,
γ
= 0.5,
ε
1,
ε
2 = 0.001
2. Compute cost and constraint functions:f
o
= −1696.46, g
1
= − 0.3717, g
2
= − 0.2, g
3
= − 0.7,
g4 = − 15, g5 = − 0.25,
∇f = (− 565.487, − 113.097); ∇g1 = (0.10472, 0.04189); ∇g2 = (− 0.2, 0);
∇g3 = (0.05, 0); ∇g4 = (0, − 1); ∇g5 = (0, 0.05);
Vo = max {0, − 0.3717, − 0.2, − 0.7, − 15, − 0.25} = 0
2. QP subproblem defined using the data given in Step 2 gives the search direction as
d(0) = (9.54975, − 15) with Lagrange multipliers as u = (5308.8, 0, 0, 94.3, 0)
4. ||d(0)|| >
ε
2; Convergence criteria are not satisfied.
5. ro = 5403.1; R = max (Ro, ro) = 5403.1
6. Step size at the 2nd trial (
α
o = 0.5) satisfies the descent condition. Design is updated as
R(1) = 10.774875, H (1) = 7.5
7. R
1
= 5403.1, k = 1, go to Step 2.
13.6 ________________________________________________________________________________
Refer to Exercise 12.8 for detailed formulation.
Iteration 1:
2. Compute cost and constraint functions: fo = − 1256.64, g1 = − 0.3717, g2 = − 3, g3 = −100;
3. QP subproblem defined using the data given in Step 2 gives the search direction as
4. ||d(0)|| > ε2; Convergence criteria are not satisfied.
6. Step size at the lst trial (
α
N (1) = 106, R(1) = 2.53158
Chapter 13 More on Numerical Methods for Constrained Optimum Design
13.7 ________________________________________________________________________________
Refer to Exercise 12.9 for detailed formulation.
Iteration 1:
2. Compute cost and constraint functions: f
o
= 30000, g
1
= 0, g
2
= − 0.5, g
3
= 0, g
4
= − 0.5,
3. QP subproblem defined using the data given in Step 2 gives the search direction as
4. ||d(0)|| >
ε
2; Convergence criteria are not satisfied.
5. ro = 15000; R = max (Ro, ro) = 15000
6. Step size at the 2nd trial (
α
W (1) = 75, D(1) = 125
7. R
1
= 15000, k = 1, go to Step 2.
13.8 ________________________________________________________________________________
Refer to Exercise 12.10 for detailed formulation.
Iteration 1:
1. Initial design is given as (r (0), h(0)) = (6, 16); set Ro = 1,
γ
= 0.5,
ε
ε
3. QP subproblem defined using the data given in Step 2 gives the search direction as
5. ro = 201.8; R = max (Ro, ro) = 201.8
o= 1) satisfies the descent condition. Design is updated as
7. R1 = 201.8, k = 1, go to Step 2.
13-6
13.9 _______________________________________________________________________________
Refer to Exercise 12.11 for detailed formulation.
Iteration 1:
1. Initial design is given as (b(0), h(0)) = (5, 10); set Ro = 1,
γ
= 0.5,
ε
1,
ε
2 = 0.001
2. Compute cost and constraint functions: fo = 1.06667, g1 = − 0.5, g2 = − 0.44444, g3 = − 5,
3. QP subproblem defined using the data given in Step 2 gives the search direction as
d(0) = (0.17067, 0.02133) with Lagrange multipliers as u = (0, 0, 0, 0)
5. ro = 0; R = max (Ro, ro) = 1
o= 1) satisfies the descent condition. Design is updated as
7. R1 = 1, k = 1, go to Step 2.
13.10 _______________________________________________________________________________
Refer to Exercise 12.12 for detailed formulation.
Iteration 1:
1. Initial design is given as (b(0), d (0), h(0)) = (5, 5, 5); set Ro = 1,
γ
= 0.5,
ε
1,
ε
2 = 0.001
2. QP subproblem defined using the data given in Step 2 gives the search direction as
5. ro = 14329.93; R = max (Ro, ro) = 14329.93
6. Step size at the 3rd trial (
α
o= 0.25) satisfies the descent condition. Design is updated as
7. R1 = 14329.93, k = 1, go to Step 2.
13-8
13.11 _______________________________________________________________________________
Refer to Exercise 12.13 for detailed formulation.
Iteration 1:
1. Initial design is given as (D (0), H (0)) = (4, 8); set Ro = 1,
γ
= 0.5,
ε
1,
ε
2 = 0.001
2. Compute cost and constraint functions: fo = 50265.482, h1 = − 0.32979, g1 = 0, g2 = − 4,
d(0) = (2.98427, − 8) with Lagrange multipliers as u = (− 45009.3, 0, 0, 1247.7)
5. ro = 46257; R = max (Ro, ro) = 46257
o= 0.5) satisfies the descent condition. Design is updated as
7. R1 = 46257, k = 1, go to Step 2.
13.12 _______________________________________________________________________________
Refer to Exercise 12.14 for detailed formulation.
Iteration 1:
2. Compute cost and constraint functions: f
= 2800, g
= 0.33333, g
= − 10, g
= −10, g
= − 4;
∇
f = (80, 120, 200);
∇
g1 = (− 0.06667, − 0.06667, − 0.16667);
∇
g2 = (− 1, 0, 0);
∇g3 = (0, − 1, 0); ∇g4 = (0, 0, − 1); Vo = max {0, 0.33333, − 10, − 10, − 4} = 0.33333
3. QP subproblem defined using the data given in Step 2 gives the search direction as
d(0) = (2.0711, − 10, 5.17159) with Lagrange multipliers as u = (1231, 0, 27.9, 0)
4. ||d(0)|| >
ε
2; Convergence criteria are not satisfied.
5. ro = 1258.9; R = max (Ro, ro) = 1258.9
6. Step size at the 3rd trial (
α
7. R
1
= 1258.9, k = 1, go to Step 2.
Arora, Introduction to Optimum Design, 4e
13-9
13.13 ______________________________________________________________________________
Refer to Exercise 12.15 for detailed formulation.
Iteration 1:
1. Initial design is given as (P
(0)
, P
(0)
) = (2, 1); set Ro = 1,
γ
= 0.5,
ε
ε
2. Compute cost and constraint functions: fo = 5.6, g1 = 0.95, g2 = − 2, g3 = − 1; ∇f = (3, 2.6);
d(0) = (28.29943, 28.69943) with Lagrange multipliers as u = (1877. 9, 0, 0)
5. ro = 1877.9; R = max (Ro, ro) = 1877.9
o= 0.25) satisfies the descent condition. Design is updated as
P
(1)
1
= 9.0748575, P
(1)
2
= 8.1748575
Chapter 13 More on Numerical Methods for Constrained Optimum Design
Arora, Introduction to Optimum Design, 4e
13-10
Section 13.4 Constrained Quasi–Newton Methods
13.14 _______________________________________________________________________________
Refer to Exercise 12.3 for detailed formulation.
Iteration 1: Refer to Exercise 13.1.
Iteration 2:
2. Computed cost and constraint functions and their gradients are given in 13.1 as:
0.001952); ∇g4 = (− 0.1, 0); ∇g5 = (0.001, 0); ∇g6 = (0, − 0.1); ∇g7 = (0, 0.001);
V1 = max {0; 0.00640156, 0.4394197, − 0.4044, − 24.61475, − 0.74385, − 29.5123, − 0.69488}
S(0) = (6.1475413, 5.12295); Z(0) = (6.1475413, 5.12295); y(0) = (27.266794, 22.588108);
2.173721 1.800734
0.491803 0.409836
1.681918 2.390898
3. QP subproblem defined using the data given in Step 2 gives the search direction as
4. ||d(1)|| >
ε
2; Convergence criteria are not satisfied.
5. r1 = 90407.47; R = max (R1, r1) = 90407.47
6. Step size at the 1st trial (to = 1) satisfies the descent condition. Design is updated as
Chapter 13 More on Numerical Methods for Constrained Optimum Design
13.15 _______________________________________________________________________________
Refer to Exercise 12.4 for detailed formulation
Iteration 1: Refer to Exercise 13.2.
Iteration 2:
2. Computed cost and constraint functions and their gradients are given in 13.2 as:
fo = 142.3516, g1 = − 0.944855, g2 = − 9.01843, g3 = − 1.30889, g4 = − 0.88455, g5 = 0,
S(0) = (-0.45556, -3.5); Z(0) = (-0.45556, -3.5); y(0) = (1.24607, -0.380518);
ξ
-0.335891 0.2841
0.127992 0.983341
-0.463883 0.300759
QP subproblem defined using the data given in Step 2 gives the search direction as
4. ||d(1)|| >
ε
2; Convergence criteria are not satisfied.
6. Step size at the 1st trial (to = 1) satisfies the descent condition. Design is updated as
7. R2 = 117, k = 2, go to Step 2.
Arora, Introduction to Optimum Design, 4e
13-12
13.16 ______________________________________________________________________________
Refer to Exercise 12.5 for detailed formulation
Iteration 1: Refer to Exercise 13.3.
Iteration 2:
2. Computed cost and constraint functions and their gradients are given in 13.3 as:
S(0) = (− 30.00306, − 40); Z(0) = (− 30.00306, − 40); y(0) = (− 11.252975, 0);
ξ
5. r1 = 10904.98; R = max (R1, r1) = 10904.98
6. Step size at the 1st trial (to = 1) satisfies the descent condition. Design is updated as
7. R2 = 10904.98, k = 2, go to Step 2.