Homework #2
Assigned: Jan. 22nd, 2013
Due: Jan. 29th, 2013
Points: 40
Additional points for work that is beyond instructor's expectation!
Look up Homework #1 of 2010 offering of this course and the solution posted there to know what is meant by "beyond instructor's expectation".

1. 20 points
   Consider the following constrained minimnization problem.

   Min f(x1, x2) = (x1^2 + x2 -11)^2 + (x1 +x2^2 -7)^2
   subject to
   (x1-5)^2 + x2^2 - 26 lessThanOrEqualTo 0, 4*x1 + x2 - 20 LessThanOrEqualTo 0,
   and x1 and x2 are both non-negative.

   Check if (0,1) and (3,2) are also KKT points of this problem. Please check the sufficient conditions and decide if they are minima or not. Show details.
   Find all KKT points of this problem.
   Plot the contours and the inequality constraints and identify the KKT points.

2. 20 points
   
Write the dual of the following constrained minimization problem and solve it.

Min f(x1, x2, x3, x4, x4) = 0.0624*(x1 + x2 + x3 + x4 + x5)
Subject to
61/x1^3 + 37/x2^3 + 19/x3^3 + 7/x4^3 + 1/x5^3 - 1 LessThanOrEqualTo 0

Write the KKT conditions of the primal (i.e., the one given above) and check if the solution that you found satidfies those conditions.