Homework #2
Assigned: Jan. 13th, 2009
Due: Jan. 20th, 2009
Points: 20

1. 10 points
   Find or formulate a two-variable minimization problem with one equality constraint or one active inequality constraint such that there is a minimum for the problem. Your problem should also satisfy another condition: the Hessian of the Lagrangian should not be positive definite at the minimizing point; instead it should be positive definite only for those perturbations for which the equality (or the active inequality) condition is satisfied.
   If you find this problem in a book or in the Internet, please state the source. More points will be given to those who formulate a problem on their own.
   Also solve the problem and state the solution.
2. 10 points
   Solve your problem numerically using fmincon routine of Matlab's optimization toolbox. Print the Matlab script (the .m file) and the output and submit.
   Plot the countours of the objective function and the constraint. Mark the minimizer. Mark the gradients of the objective function and show that you have understood the geometric interpretation of the Karush-Kuhn-Tucker (KKT) conditions.
   Print the plot and submit.