CVX optimisation prroblem

1

I am unable to code the following optimisation problem in cvx.

Here, W is an nxn weight matrix and x is an nx1 vector. I want to find a weight matrix W that minimises the following objective function.

\begin{array}{r}{\varepsilon(W)=\sum_{i=1}^{n}\left\|x_{i}-\sum_{j \ne i} W_{(i, j)} x_{j}\right\|} \\ {\text { s.t. } \sum_{j} W_{(i, j)}=1}\end{array}
1 Like
2

Have you read the CVX Users’ Guide http://cvxr.com/cvx/doc/? This problem is quite straightforward to enter in CVX. After you have read the CVX Users’ Guide, if you are still stuck on something, please describe where you need help.

3

Thanks for your reply.

Yes I read the documentation for a long time but still couldn’t identify a case similar to mine where I’m iterating over the elements of a vector and elements of matrix and has sum of norms in it.

I am not that familiar with high level optimisation theory as well.

It would be great help if you direct me to a similar example in the documentation, or provide me some hints to represent my objective function as a product of matrices and then I could maybe use sum and norms functions.

4

Each argument of the norm is a scalar, so the norm (regardless of “p’”) just amounts to absolute value of its argument. You can compute the objective in this case the same as you would if W were a numerical matrix in MATLAB.

You are allowed in CVX to do the following:

Objective = 0
for i=1:n
 Objective = Objective + cvx_expression_this_time_through_for_loop
end
minimize(Objective)

Vectorization (avoiding for loops) is faster in CVX model processing time when you can do it, but for loops are always valid.