Homework #2
Assigned: Jan. 12th, 2012
Due: Jan. 24th, 2012
Points: 50
Additional points for work that is beyond instructor's expectation!

1. 20 points
   Take any constrained minimization problem in two variables with one objective function and two inequality constraints and one equality constraint. Solve it to find all its solutions. Plot the contours and locate the stationary points and label them as minima/maxima/saddle points as the case may be.
   Verify all of the conditions contained in the Karush-Kuhn-Tucker conditions for all the solutions. Do not forget to write a clear statement of the problem.
   Extra points for those who choose or create interesting and new problems.
2. 15 points
   Search in the library and the Internet, choose, and write a pair of "Theorems of the Alternative". Write about the significance of those theorems, and particularly the one you have chosen, in the context of optimization with a finite set of variables.
3. 15 points
   Given that f = x₁ + x²₂ + x₂x₃ + 2 x²₃ and h = 0.5*(x²₁ + x²₂ + x²₃) - 0.5 we want to minimize f with respect to {x₁, x₂, x₃} subject to h = 0.
   verify if {1,0,0}^T with mu = -1, satisfies the necessary and sufficiency conditions.
   This is a problem given by Luenberger (1986) and used by Papalambros and Wilde (2008) to illustrate a subtle issue. Make up a new problem that has the same feature.