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)?

This blog is about my work at MOSEK ApS where I am the CEO, a computer programmer and tea maker.

## Tuesday, April 20, 2010

## 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.

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.

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