Problems 3776.35 Consider the minimization of the function
f=
1
x 12 +x 22 + 2Perform one iteration of Newton’s method from the starting point X 1 ={ 4
0}
using
Eq. (6.86). How much improvement is achieved withX 2?6.36 Consider the problem:
Minimizef= 2 (x 1 −x^21 )^2 +( 1 −x 1 )^2If a base simplex is defined by the verticesX 1 ={
0
0}
, X 2 ={
1
0}
, X 3 ={
0
1}find a sequence of four improved vectors using reflection, expansion, and/or contraction.6.37 Consider the problem:
Minimizef=(x 1 + 2 x 2 − 7 )^2 +( 2 x 1 +x 2 − 5 )^2If a base simplex is defined by the verticesX 1 ={
− 2
− 2}
, X 2 ={
− 3
0}
, X 3 ={
− 1
− 1}find a sequence of four improved vectors using reflection, expansion, and/or contraction.6.38 Consider the problem:
f= 100 (x 2 −x^21 )^2 +( 1 −x 1 )^2Find the solution of the problem using grid search with a step size
xi= 0 .1 in the range
− 3 ≤xi≤3,i= 1 ,2.6.39 Show that the property of quadratic convergence of conjugate directions is independent
of the order in which the one-dimensional minimizations are performed by considering
the minimization of
f= 6 x^21 + 2 x^22 − 6 x 1 x 2 −x 1 − 2 x 2using the conjugate directionsS 1 ={ 1
2}
andS 2 ={ 1
0}
and the starting pointX 1 ={ 0
0}
.6.40 Show that the optimal step lengthλ∗i that minimizesf (X)along the search direction
Si= −∇fiis given by Eq. (6.75).
6.41 Show thatβ 2 in Eq. (6.76) is given by Eq. (6.77).
6.42 Minimizef= 2 x^21 +x 22 from the starting point (1,2) using the univariate method (two
iterations only).
6.43 Minimizef= 2 x^21 +x^22 by using the steepest descent method with the starting point
(1,2) (two iterations only).
6.44 Minimizef=x^21 + 3 x^22 + 6 x^23 by the Newton’s method using the starting point as
(2,− 1 ,1).