Wednesday, July 21, 2010

A small excercise in conic quadratic optimization.

Assume we have the small conic quadartic problem

min c'x

st. ax=b
     x_1 = l_1
     x_2 = l_2
     x_1 >= sqrt(x_2^2 + x_3^2)

c and a are arbitrary 3 dimensional vectors.  l_1 and l_2 are scalar constants.

The above problem is in fact a 1 dimensional LP.  The exercise is:
  1. Write the 1 dimensional LP?
  2. State the corresponding dual problem.
  3. Show how to recover the dual solution to conic problem from the dual solution to the the LP.
  4. What if l_1 - |l_2| = 0? Does it give rise to problems?