Problems 4837.14 Consider the problem:
Minimizef=(x 1 − 1 )^2 +(x 2 − 5 )^2subject to
g 1 = −x^21 +x 2 − 4 ≤ 0g 2 = −(x 1 − 2 )^2 +x 2 − 3 ≤ 0Formulate the direction-finding problem atXi={− 1
5}
as a linear programming problem
(in Zoutendijk method).7.15 Minimizef (X)=(x 1 − 1 )^2 +(x 2 − 5 )^2
subject to
−x 12 +x 2 ≤ 4
−(x 1 − 2 )^2 +x 2 ≤ 3
starting from the point X 1 ={ 1
1}
and using Zoutendijk’s method. Complete two
one-dimensional minimization steps.7.16 Minimizef (X)=(x 1 − 1 )^2 +(x 2 − 2 )^2 − 4
subject to
x 1 + 2 x 2 ≤ 5
4 x 1 + 3 x 2 ≤ 10
6 x 1 +x 2 ≤ 7
xi≥ 0 , i= 1 , 2
by using Zoutendijk’s method from the starting point X 1 ={ 1
1}. Perform two
one-dimensional minimization steps of the process.
7.17 Complete one iteration of Rosen’s gradient projection method for the following problem:
Minimizef=(x 1 − 1 )^2 +(x 2 − 2 )^2 − 4subject to
x 1 + 2 x 2 ≤ 5
4 x 1 + 3 x 2 ≤ 10
6 x 1 +x 2 ≤ 7
xi≥ 0 , i= 1 , 2Use the starting point,X 1 ={ 1
1}
.7.18 Complete one iteration of the GRG method for the problem:
Minimizef=x^21 +x^22subject to
x 1 x 2 − 9 = 0starting fromX 1 ={
2. 0
4. 5}
.