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How I can solve the following Quadratic Programming Problem:

Min[X’VX+X’GX]

In this case, X is a list of coefficient to be solved for, V is a square matrix of Price returns, and G is a square matrix of Income returns.

Also, in this case, both V and G should normally be uncorrelated, but in cases where they are correlated, is there a straight forward way to make them orthogonal?

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I assume that $V$ and $G$ are variance matrices (of returns, etc.): in particular, they are positive semi-definite. If there are no constraints, then $X=0$ is a solution. If you have equality constraints, you can use Lagrange multipliers. If you also have inequality constraints, you can use a quadratic solver. I am not sure what you mean by "both V and G should be uncorrelated" (diagonal variance matrices?). –  Vincent Zoonekynd Jun 21 '12 at 14:36
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Other than your mention of what V and G represent, which is irrelevant to your actual question (and FYI non-sensical, what is a matrix of returns?), what does this have to do with finance? –  Tal Fishman Jun 21 '12 at 14:36
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2 Answers 2

Just add V and G together and treat like an ordinary quadratic programming problem.

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$$ M=X^{T} VX+X^{T} GX= X^{T} (V+G)X $$ $$ \frac{\partial M}{\partial X}=(V+V^{T}+G+G^{T} )X$$ $$ \frac{\partial^{2} M}{\partial X^{2}}=(V+V^{T}+G+G^{T} ) > 0 $$

If V and G are positive definite the under such conditions the minimum exists and it is a matter of solving the linear system $$(V+V^{T}+G+G^{T} )X=0 $$

One of the solutions for this equation is a trivial solution and the other requires you to reduce the matrix to row-echolon form

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