Formulating Linear Constraints on a Matrix

[JeSuisConf]

Hi everyone. I have this problem which I am trying to formulate. Basically, I have the following linear constraints:

$$p_{11} = 2$$
$$p_{22} = 5$$
$$p_{33}+2p_{12}=-1$$
$$2p_{13} =2$$
$$2p_{23} = 0$$

And these are for the symmetric matrix

$$\mathbf{P} = \left( \begin{array}{ccc} p_{11} & p_{12} & p_{13} \\ p_{12} & p_{22} & p_{23} \\ p_{13} & p_{23} & p_{33} \end{array} \right)$$

I would like to formulate a way to represent the linear constraints and $$\mathbf{P}$$ as a matrix at the same time.

I can do this using $$\mathbf{P}$$ or a vector of the entries of $$\mathbf{P}$$. The linear constraints are easy if I use a vector ($$\mathbf{Ap}=\mathbf{b}$$, but then I don't know how to represent $$\mathbf{P}$$ as a matrix from the vector! And if I leave $$\mathbf{P}$$ as a matrix, all the constraints are easy to formulate except $$p_{33}+2p_{12}=-1$$. Can anyone help me figure this out?

If anyone's curious, I'm trying to solve for $$\mathbf{P}$$ over the cone of PSD matrices using SDP. But I am entirely new to SDP and I'm scratching my head formulating this problem. I feel stupid right now :'(

 

[JeSuisConf]

I answered my own question... and the answer has to do with algebraic dependence in the structure of the problem, and how SDP problems are formulated. I just thought I was missing something obvious ... hence my convoluted search.

 

[HallsofIvy]

Well, I'm glad you answered it. It looks simple to me: I you let $p_{12}= p$ then $p_{33}= -1- 2p$ and all other values are essentially given:
$$\begin{pmatrix}2 & p & 1 \\ p & 5 & 0 \\1 & 0 & -1-2p\end{pmatrix}$$

 

[JeSuisConf]

Well, I'm glad you answered it. It looks simple to me: I you let $p_{12}= p$ then $p_{33}= -1- 2p$ and all other values are essentially given:
$$\begin{pmatrix}2 & p & 1 \\ p & 5 & 0 \\1 & 0 & -1-2p\end{pmatrix}$$

That's exactly what came out in the end. I'm not sure why I didn't see it at first... I always seem to manage to do things in the most roundabout way.

 

[blue_raver22]

Hello there,

I can understand your frustration with formulating linear constraints on a matrix. It can be a daunting task, especially when working with SDP and PSD matrices. However, with some guidance and understanding, I am confident that you will be able to solve this problem.

Firstly, let's take a look at the linear constraints you have provided. These constraints can be represented in matrix form as follows:

\mathbf{A}\mathbf{P} = \mathbf{b}

where \mathbf{A} is a matrix with the coefficients of the linear constraints and \mathbf{b} is a vector with the values on the right-hand side of the constraints. This is a common way to represent linear constraints in matrix form.

Now, to incorporate the symmetric matrix \mathbf{P} into this formulation, we can use a vector of the entries of \mathbf{P} as you have mentioned. This can be represented as:

\mathbf{p} = \begin{pmatrix}p_{11} \\ p_{12} \\ p_{13} \\ p_{22} \\ p_{23} \\ p_{33}\end{pmatrix}

Using this vector, we can then represent \mathbf{P} as a matrix using the following formula:

\mathbf{P} = \begin{pmatrix}p_{11} & p_{12} & p_{13} \\ p_{12} & p_{22} & p_{23} \\ p_{13} & p_{23} & p_{33}\end{pmatrix}

Now, let's take a look at the constraint p_{33}+2p_{12}=-1. We can rewrite this constraint as:

p_{33}+2p_{12}+0p_{13}+0p_{22}+0p_{23}+0p_{33} = -1

This can then be incorporated into the vector \mathbf{b} as the last entry. So, our final formulation would be:

\mathbf{A}\mathbf{p} = \mathbf{b}

where \mathbf{A} is a 6x6 matrix and \mathbf{b} is a 6x1 vector.

I hope this helps you in formulating your problem. Please don't feel stupid, as SDP can be a complex topic and it