Homework #2
Assigned: Jan. 11th, 2007
Due: Jan. 18th, 2007
Points: 30

1. 15 points
   Consider a loop of a wire bent into the shape of a 3D curve whose projection onto the xy-plane is a circle of radius R with its center at the origin. The height along the z of the curve at each point on the circle is given by 0.5Rsin(2*theta)) where theta is the angle joining the point on the circle with the origin with the x axis. If you dip this dip wire into soap solution and take it out, what surface does the soap film enclosed by the wire-loop take? Provide a numerical solution to the problem by discretizing the circular area and minimizing the are of the surface enclosed by the wire-loop. Ten extra points to those who do the soap-film experiment to verify the solution and bring it to the class to demonstrate.
2. 8 points
   Minimize {x1 + x2^2 + x2*x3 + 2*x3^2} subject to h = 0.5*(x1^2 + x2^2+x3^2)-0.5 = 0. Find the minimum for this problem using KKT conditions and then show that it satisfies the sufficient conditions.
4. 7 points
   Minimize {x1^2 + x2^2 - 4*x1 + 4} subject to g1 = -x1 <=0, g2 = -x2 <=0, and g3 = x2 - (1-x1)^3 <=0. Plot the contours of the objective function and the constraints. Check if {x1 = 1, and x2 = 0} satisfies the KKT conditions. Is this point a minimum for this problem?