Tuesday, April 20, 2010

A question about the conic representation of the power function.

My previous blog item discussed how to represent the set

x^5/3 <= t, x>=0

using linear and quadratic cones. Now the book by Ben-Tal and Nemirovski shows how to represent the set

x^p/q <= t , x>=0       (1)

using conic quadratic cones where p and q are integers and p>=q>=1 which much more general than my example of course. Now the method suggested by Ben-Tal and Nemirovsky is not optimal in a complexity sense. Hence, I do not think it leads to the minimal number of quadratic cones. Therefore, a natural question  is: What is the optimal conic quadratic representation of (1)?

Friday, April 16, 2010

Is x raised to the power 5/3 conic quadratic representable?

The set

   x^5/3 <= t, x,t>=0.0 

can be represented by

x^2    <= 2 s t,  s,t >= 0,
u         =     x,
v         =     s, 
z         =     v,
z^2   <= 2fg,  f,g > = 0,
f         =     0.5,
4 g      =     h,
h^2  <=  2uv,  u,v  >= 0.

I will leave it to reader to verify it is correct.

this particular set I have come up over again and again in financial applications. I suppose it has something to do with modeling of transactions costs.

An obvious question is why replace a simple problem but something that looks quite a bit complicated.
However, both in theory and practice the conic quadratic optimization problems is easier to solve then general convex problems.