Minimization

Minimization

Usage notes

We want to find a local minimum of the function

$F(x,y)=(5-x^{2}+y^{2})\div(1+x^{4}+10*y^{2})$

given a starting point

X* = (x*, y*) = (1,0).

a: Brent's Algorithm

Assume that the function F has a minimum on the half line X* + kd, where d is the direction-vector (3/5,-2/5), and k is any positive real number. The following graph shows a cross-section through F in direction d, i.e. the function g(t) = F(t*3/5, -t*2/5).

You may use the following constants:

gamma=(sqrt(5)-1)/2
phi=1/gamma=1+gamma

The first part of your task is to perform an exponential search in direction d. Make as many steps as it takes to bracket the minimum. Then press Terminate exponential search to start the Brent Minimization.
Watch the plot to see what happens!
Note: You have to enter 1 and phi as the first two "steps" of the exponential search to initialize it.
If you did a wrong step, please type in RESET and start anew.

The next step of exponential search will be t = ___________________

Please use Maple Syntax (Vector, Matrix) to specify vectors and matrices.

b: Compute the gradient of the function F at X*

grad F = ___________________

c: Compute the Hessian of the function F at X*

H(F) = ___________________

d: Perform one step of the steepest decent method

Express F as a function f(t) of one variable:

f(t) = ___________________

In practice we would start Brent's method in this situation, but here we're going to find the minimum analytically.

Enter the equation to solve ___________________

e: Perform one step of Newton's method

Compute the increment vector h

h = ___________________

Given the starting point X* and the increment vector h from above you can compute the next point X**.

X** = ___________________

f: Finally please perform one step of the spectral method

First try to find out which of the following statements are true in this special case (let U, Lambda be the results of the eigenvalue/eigenvector decomposition)
U = I (Identity matrix)

z = F'

d = F'

z = d

Lambda=I

Lambda=F''

The Hessian is diagonal.

The Gradient has norm || 1 ||.

z1 = ___________________

z2 = ___________________

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