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?

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)?

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.

A conic quadratic representable set.

Assume you have the problem

min v
st.   |x|^3 / y^2 <= v, y>=0

A customer asked me if that can be formulated as conic quadratic problem. Indeed it can and my solution is

z^2  <=  2ty
2t      <= a
a^2   <= 2 uv
z         =  2u
x       <= z
-x      <= z
0       <= y,t,u,v
 

Effective cuts for mixed integer goal programs

Assume you have the problem

min c'x + ||Fx-f||_1
st.   A x = b
       x binary

This is sort of a goal programming model I have seen frequently.  Now the typical LP formulation is


min c'x +e'(s+t)
st.   F x - f =  s-t
      A x = b
      s,t >= 0
       x binary

Note this model includes continuous variables too.  At least for an application I have seen then

1. It easy to find a feasible solution.
2. There is huge gap between the LP relaxation and the feasible solution.
3. It is hard to close the gap. A huge number of nodes is required and the standard cuts works poorly.

Then leads to the question. What is good cuts for this type of problems?

Windows PowerShell

I have played a bit around with the Windows PowerShell but is now going back to the DOS command line. What I was looking for was something like Linux bash for Windows. PowerShell is not that and in my opinion it sucks.

Here are some problems

• It has a history but works nowhere nearly as well on bash. For instance it does not save the history automatically.
• The terminal is still lousy with respect to copy and paste.
• If you C program sends a lot of output to the terminal then t is so slow.
• If you press ctrl break then it closes down the whole console. That is not what the DOS command does and hence the close down too frequently.

So a good advice is. If you are looking for a replacement for Linux bash then do not waste time on Windows PowerShell. Maybe the Hotwire Shell (http://code.google.com/p/hotwire-shell/) is worth looking at. Let me know if you a better suggestion.

From Eudora to Gmail.

I have used the email client Eudora since about 1997 when I did a post doc at TU Delft.  However, now Eudora is no longer really support so I thought I should switch to another email client and decided to go for Googles Gmail.

I exported all my old emails Eudora to Gmail using the IMAP mode. It was fairly easy. However, it seems that Eudora adds custom headers to emails so not all old emails shows up properly in Gmail.  However, installing the script http://userscripts.org/scripts/show/62058 that works in Firefox, then most old emails shows up properly.

To summarize I am very happy about my switch from Eudora to Gmail. It is so nice to have all your emails available online everywhere.