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C&O 367, 2001
Assignment 1
Due on Thursday, Jan. 18 (at start of class)
Instructor H. Wolkowicz
-----------------
Notes:
- Questions and comments can be posed on the newsgroup uw.co.co367.
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- MATLAB
- Matlab appears to be on all the Solaris systems in the undergrad
environment, e.g. hermite and magnus.
Enter the command matlab to start up the matlab session. Then enter
help optim
You will see a list of the optimization functions that are available
with the optimization toolkit.
- Now try the command
optdemo
This will bring up a menu. You can try the four choices. The first
choice is a tutorial.
This demonstrates several of the optimization functions.
- The second choice in the optdemo is the minimization of the banana
function or Rosenbrock's function. This is a classical example of a
function with a narrow valley that exhibits ``very'' slow convergence for
steepest descent type methods. Try out several methods to minimize this
function.
- (5 marks)
Submit a contour plot of the banana function for variable values between
![$-4,+4.$](img1.gif)
- (10 marks)
How many function evaluations
did the minimization take for: steepest descent; simplex
search; Broyden-Fletcher-Golfarb-Shanno; Davidon-Fletcher-Powell;
Levenberg-Marquardt? (Please specify the line search you used.)
Note: You can modify the matlab programs.
You can see what the matlab routines are doing by looking
at the matlab m-files. To do this change directory using: cd /software;
cd matlab; cd distribution;
cd toolbox; cd optim. In particular, there is an m-file
called bandemo.m.
- (15 marks)
Classify the following matrices according to whether they are positive
or negative definite or semidefinite or indefinite:
-
-
-
-
-
-
-
- (20 marks)
(Text: Problem 7, page 32)
Use the principal minor criteria to determine (if possible) the nature
of the critical points of the following functions:
-
-
-
-
-
- (10 marks)
(Text: Problem 16, page 33)
- Show that no matter what value of
is chosen, the function
has no global maximizers.
- Determine the nature of the critical points of this function for all
values of
.
- (15 marks)
(Text: Problem 3, page 77)
A quadratic function in
variables is any function defined on
which can be expressed in the form
where
and
is an
symmetric
matrix.
- Show that the function
defined on
by
is a quadratic function of two variables by finding the appropriate
![$a,b,A.$](img25.gif)
- Compute the gradient
and Hessian
of the
quadratic function in 5a and express these in terms of ![$a,b,A.$](img25.gif)
- If
is a quadratic function of
variables such that the
corresponding matrix
is positive definite, show that
has a unique solution and that this solution is the strict global
minimizer of ![$f(x).$](img29.gif)
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Henry Wolkowicz
2001-01-10