## Thursday, March 19, 2009

### MSF Solver Foundation API Question

Hi,
I am trying to port an Excel sheet that uses Solver the add-in from Frontline Systems to a C# program that uses the Solver Fondation API.
The sheet uses solver to find the combination of stocks in a portfolio for a given variance.
I have followed the Markowitz sample that comes with the express installation but one of the constraints that have to be added is that the portfolio's variance should be a certain value. The variance of the portfolio is calculated the following way:
SQRT(MMULT(MMULT(D38:L38;\$B\$10:\$J\$18);TRANSPOSE(D38:L38)))*SQRT(252)
where D38:L38 is the allocation vector (the result) and \$B\$10:\$J\$18 is the covariance matrix of the available stocks.
thank,

I believe this ends up as a nonlinear constraint. In OML-like notation (I invent the Sqrt function here):

`  Sqrt[Sum[{i,I},{j,I},x[i]*covar[i,j]*x[j]]] * Sqrt == SomeValue`

This shows an obvious advantage of using MSF/OML compared to Excel formulas: it is much easier to write readable constraints. Excel's Solver and its bigger brother from Frontline Systems can handle such nonlinear constraints through its GRG2 based NLP solver, but MSF does not support this. MSF supports convex QP's, but I don't think this can be reformulated into a linearly constrained QP.

Notes:

1. the lack of Sqrt is not a show-stopping problem, as we could write:

Sum[{i,I},{j,I},x[i]*covar[i,j]*x[j]] == SomeOtherValue

2. For large problems this construct can be improved somewhat, first by using symmetry of Covar, and depending on the modeling system/solver by reformulating the quadratic form. As in your case the number of instruments is small, there is no need for that.

3. For very large problems sometimes a linear approximation is used (easy if minimizing variance, actually not so easy in this case).