Combining Quadratic Forms in a Portfolio Objective

Summary

The document considers minimizing an objective formed from two quadratic terms, one associated with price returns and another with income returns. The central algebraic observation is that both terms share the same coefficient vector, so they combine into a single quadratic form using the sum of their matrices. This means the objective can be treated as an ordinary quadratic optimization problem; separate orthogonalization is not required merely to add the terms.

The response differentiates the combined expression and gives a stationarity condition involving the symmetric parts of the matrices. It then suggests solving the resulting linear system and notes positive definiteness as a condition associated with a minimum. However, the treatment is incomplete: the displayed homogeneous condition admits the zero vector, and a meaningful portfolio problem usually requires constraints or a normalization to avoid that trivial solution. It also does not explain how to make correlated return components orthogonal, or specify whether the matrices represent covariance, return, or other quantities.

Key ideas

• Quadratic terms with the same coefficient vector combine by adding their matrices.
• The gradient depends on the symmetric parts of the matrices in the quadratic objective.
• Positive definiteness supports convexity and existence of a minimum under suitable constraints.
• A homogeneous stationarity equation can yield a trivial zero solution, so portfolio constraints may be essential.
• Combining matrices does not itself orthogonalize correlated return components.

Tags

# Quadratic Programming Problem


# Quadratic Programming Problem












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?

## Answer by Tal Fishman (score 2)

https://quant.stackexchange.com/a/3650

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

## Answer by Kumar (score 1)

https://quant.stackexchange.com/a/3659

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

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.