Maximum return portfolio using linear programming with quadratic constraints

Question

In the maximum return portfolio problem formulation above,

1. is $A=\mu^\top \Sigma^{-1} \mu$?
2. What is $b$ equal to, and
3. is the second constraint required? An inequality constraint for target portfolio volatility doesn't seem to have anything to do with maximizing portfolio return.

Source

Answer 1

In words, the formulation implies to maximize expected portfolio returns for a given maximum portfolio variance, $\sigma$ (Commonly noted as $\sigma^2$, though).

The linear restrictions $Aw\leq b$ imply the 'usual' conditions, i.e. $w_i\geq 0$, $\sum_i w_i \leq 1$ etc. Please note that your stated formulation does not include a restriction on 'full investment', i.e. you might come up with an optimal portfolio that is not fully invested. Hence, $A$ and $b$ encapsulate linear inequality conditions. Standard quadratic programming solvers let you specify equality and inequality constraints, though.

Edit: You finally construct the matrices by combining your investment constraints, each represented by a line in $A$ and a corresponding entry in $b$. Say you want to say $\sum_i w_i\leq 1$ then this is equivalent to $w_1 + w_2 + \ldots + w_n \leq 1$ which equals $\mathbb{1}^Tw\leq 1$. Hence, you add a row of ones to $A$ and a $1$ to $b$.

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Example

Say you want to invest $\geq0\%$ in each asset, but no asset should be invested at more than, say, $10\%$. this would imply

$$Ax\leq b = \begin{pmatrix}-1&0&\ldots & 0\\ 0&-1&\ldots&0\\ \ldots &\ldots &\ldots &\ldots &\\ 0 & 0 & \ldots & -1\\ 1 & 1 & \ldots & 1\\ 1 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0\\ \ldots&\ldots&\ldots&\ldots\\ 0 & 0 &\ldots & 1\end{pmatrix}x\leq\begin{pmatrix}0\\0\\ \ldots\\0\\1\\0.1\\0.1\\ \ldots\\ 0.1\end{pmatrix}$$

HTH?