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Add non-negativity constraint to a reverse optimization function
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I am working with a model for portfolio management (Black-Litterman specifically) in finance. I have problems trying to impose a constraint where I want the values of a vector ($w_i$) to be non-negative.
And this is my reverse optimization function.
$$mu = delta Sigma w, $$ where $delta$ is a constant
$$ Omega = tau (P' Sigma P), $$ where $P$ is a non-singular $3times n$ matrix and $Omega$ is a diagonal matrix with $0$'s in the off-diagonal positions
$$ Pi = [(tau Sigma)^-1 + P' Omega P ]^-1 [(tau Sigma)^-1 mu + P'Omega^-1 Q],$$ where $Q$ is a $1 times 3$ vector
$$
w_i = frac1 tau Sigma ^-1 Pi
$$
$w_i$ is a $n times 1$ vector
$Sigma$ is a $n times n$ non-singular matrix
$Pi$ is a $n times 1$ vector
- $tau$ is a constant
Is it possible to add a constraint to this function such that all values in $w_i$ is non-negative?
matrices finance bayesian reverse-math
asked Jul 18 at 13:43
Mataunited17
217
add a comment |Â
up vote
0
down vote
favorite
I am working with a model for portfolio management (Black-Litterman specifically) in finance. I have problems trying to impose a constraint where I want the values of a vector ($w_i$) to be non-negative.
And this is my reverse optimization function.
$$mu = delta Sigma w, $$ where $delta$ is a constant
$$ Omega = tau (P' Sigma P), $$ where $P$ is a non-singular $3times n$ matrix and $Omega$ is a diagonal matrix with $0$'s in the off-diagonal positions
$$ Pi = [(tau Sigma)^-1 + P' Omega P ]^-1 [(tau Sigma)^-1 mu + P'Omega^-1 Q],$$ where $Q$ is a $1 times 3$ vector
$$
w_i = frac1 tau Sigma ^-1 Pi
$$
$w_i$ is a $n times 1$ vector
$Sigma$ is a $n times n$ non-singular matrix
$Pi$ is a $n times 1$ vector
- $tau$ is a constant
Is it possible to add a constraint to this function such that all values in $w_i$ is non-negative?
matrices finance bayesian reverse-math
asked Jul 18 at 13:43
Mataunited17
217
While your problem's setup is surely clear to you, some important details need to be shared with your Readers. By "reverse optimization" it appears that you are working backward from a value of some (unstated) objective function to get values $w_r$. It is unclear how if the $w_r$ are determined by your formula, you would propose to "add a constraint". Perhaps you have in mind conditions on the data $Pi$ and $Sigma$ which force a strictly positive solution $vec w$?
– hardmath
Jul 18 at 13:55
Thank you. I thought the final formula was sufficient for my problem. I will add all of the formulas to the original post so it would be clear for everyone as well.
– Mataunited17
Jul 18 at 14:00
I have now edited the original post. Hope it is more clear now, and thanks for your patience. I have been tried to find a solution. A lot of papers mention the non-negativity constraint, but no formulas or solution are given for the problem.
– Mataunited17
Jul 18 at 14:25
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am working with a model for portfolio management (Black-Litterman specifically) in finance. I have problems trying to impose a constraint where I want the values of a vector ($w_i$) to be non-negative.
And this is my reverse optimization function.
$$mu = delta Sigma w, $$ where $delta$ is a constant
$$ Omega = tau (P' Sigma P), $$ where $P$ is a non-singular $3times n$ matrix and $Omega$ is a diagonal matrix with $0$'s in the off-diagonal positions
$$ Pi = [(tau Sigma)^-1 + P' Omega P ]^-1 [(tau Sigma)^-1 mu + P'Omega^-1 Q],$$ where $Q$ is a $1 times 3$ vector
$$
w_i = frac1 tau Sigma ^-1 Pi
$$
$w_i$ is a $n times 1$ vector
$Sigma$ is a $n times n$ non-singular matrix
$Pi$ is a $n times 1$ vector
- $tau$ is a constant
Is it possible to add a constraint to this function such that all values in $w_i$ is non-negative?
matrices finance bayesian reverse-math
asked Jul 18 at 13:43
Mataunited17
217
I am working with a model for portfolio management (Black-Litterman specifically) in finance. I have problems trying to impose a constraint where I want the values of a vector ($w_i$) to be non-negative.
And this is my reverse optimization function.
$$mu = delta Sigma w, $$ where $delta$ is a constant
$$ Omega = tau (P' Sigma P), $$ where $P$ is a non-singular $3times n$ matrix and $Omega$ is a diagonal matrix with $0$'s in the off-diagonal positions
$$ Pi = [(tau Sigma)^-1 + P' Omega P ]^-1 [(tau Sigma)^-1 mu + P'Omega^-1 Q],$$ where $Q$ is a $1 times 3$ vector
$$
w_i = frac1 tau Sigma ^-1 Pi
$$
$w_i$ is a $n times 1$ vector
$Sigma$ is a $n times n$ non-singular matrix
$Pi$ is a $n times 1$ vector
- $tau$ is a constant
Is it possible to add a constraint to this function such that all values in $w_i$ is non-negative?
matrices finance bayesian reverse-math
asked Jul 18 at 13:43
Mataunited17
217
edited Jul 18 at 14:15
asked Jul 18 at 13:43
Mataunited17
217
asked Jul 18 at 13:43
Mataunited17
217
asked Jul 18 at 13:43
Mataunited17
217
217
While your problem's setup is surely clear to you, some important details need to be shared with your Readers. By "reverse optimization" it appears that you are working backward from a value of some (unstated) objective function to get values $w_r$. It is unclear how if the $w_r$ are determined by your formula, you would propose to "add a constraint". Perhaps you have in mind conditions on the data $Pi$ and $Sigma$ which force a strictly positive solution $vec w$?
– hardmath
Jul 18 at 13:55
Thank you. I thought the final formula was sufficient for my problem. I will add all of the formulas to the original post so it would be clear for everyone as well.
– Mataunited17
Jul 18 at 14:00
I have now edited the original post. Hope it is more clear now, and thanks for your patience. I have been tried to find a solution. A lot of papers mention the non-negativity constraint, but no formulas or solution are given for the problem.
– Mataunited17
Jul 18 at 14:25
add a comment |Â
While your problem's setup is surely clear to you, some important details need to be shared with your Readers. By "reverse optimization" it appears that you are working backward from a value of some (unstated) objective function to get values $w_r$. It is unclear how if the $w_r$ are determined by your formula, you would propose to "add a constraint". Perhaps you have in mind conditions on the data $Pi$ and $Sigma$ which force a strictly positive solution $vec w$?
– hardmath
Jul 18 at 13:55
Thank you. I thought the final formula was sufficient for my problem. I will add all of the formulas to the original post so it would be clear for everyone as well.
– Mataunited17
Jul 18 at 14:00
I have now edited the original post. Hope it is more clear now, and thanks for your patience. I have been tried to find a solution. A lot of papers mention the non-negativity constraint, but no formulas or solution are given for the problem.
– Mataunited17
Jul 18 at 14:25
While your problem's setup is surely clear to you, some important details need to be shared with your Readers. By "reverse optimization" it appears that you are working backward from a value of some (unstated) objective function to get values $w_r$. It is unclear how if the $w_r$ are determined by your formula, you would propose to "add a constraint". Perhaps you have in mind conditions on the data $Pi$ and $Sigma$ which force a strictly positive solution $vec w$?
– hardmath
Jul 18 at 13:55
While your problem's setup is surely clear to you, some important details need to be shared with your Readers. By "reverse optimization" it appears that you are working backward from a value of some (unstated) objective function to get values $w_r$. It is unclear how if the $w_r$ are determined by your formula, you would propose to "add a constraint". Perhaps you have in mind conditions on the data $Pi$ and $Sigma$ which force a strictly positive solution $vec w$?
– hardmath
Jul 18 at 13:55
Thank you. I thought the final formula was sufficient for my problem. I will add all of the formulas to the original post so it would be clear for everyone as well.
– Mataunited17
Jul 18 at 14:00
Thank you. I thought the final formula was sufficient for my problem. I will add all of the formulas to the original post so it would be clear for everyone as well.
– Mataunited17
Jul 18 at 14:00
I have now edited the original post. Hope it is more clear now, and thanks for your patience. I have been tried to find a solution. A lot of papers mention the non-negativity constraint, but no formulas or solution are given for the problem.
– Mataunited17
Jul 18 at 14:25
I have now edited the original post. Hope it is more clear now, and thanks for your patience. I have been tried to find a solution. A lot of papers mention the non-negativity constraint, but no formulas or solution are given for the problem.
– Mataunited17
Jul 18 at 14:25
add a comment |Â
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While your problem's setup is surely clear to you, some important details need to be shared with your Readers. By "reverse optimization" it appears that you are working backward from a value of some (unstated) objective function to get values $w_r$. It is unclear how if the $w_r$ are determined by your formula, you would propose to "add a constraint". Perhaps you have in mind conditions on the data $Pi$ and $Sigma$ which force a strictly positive solution $vec w$?
– hardmath
Jul 18 at 13:55
Thank you. I thought the final formula was sufficient for my problem. I will add all of the formulas to the original post so it would be clear for everyone as well.
– Mataunited17
Jul 18 at 14:00
I have now edited the original post. Hope it is more clear now, and thanks for your patience. I have been tried to find a solution. A lot of papers mention the non-negativity constraint, but no formulas or solution are given for the problem.
– Mataunited17
Jul 18 at 14:25