Mosek Vectorization, Slow Adding Constraints

Question

I am trying to solve a semi-definite program in Mosek with C++ Fusion with constraints of the form

$$ \sum_j A_{i,j} M_j - \vec{c}_i^T \vec{y} + x \le b_i \forall i, $$ (links to picture of equation)

where each M_j is a positive semi-definite matrix. Currently I am creating the constraints using a for loop to sum over A_{i,j} M_j and another for loop to add all of the constraints. However, the number of constraints is fairly large and using M->constraint(...) that many times is by far the slowest part of my program. I was looking through the Mosek documentation, and it seems like vectorization could speed up the program. However, I'm struggling with the vectorization for summing A_{i,j} M_j. The data type for each variable are:

A: vector < vector < Matrix::t > > 
M: vector < Variable::t >
c: vector < shared_ptr < ndarray < int,1 > > >
y: Variable::t
x: Variable::t
b: vector < int >  

I've tried using new_array_ptr on both A and M, and then using Expr::add and Expr::dot, but neither of those worked. Any help either with the vectorization or speeding up M->constraint(...) would be greatly appreciated!

Answer 1

It is a bit tricky. Since each of your constraints involves a product of matrices, vectorizing it further would require the ability to multiply higher-dimensional objects (tensors?) and Fusion does not have an interface for that.

One thing you could do is to make sure your A are represented as sparse matrices, if they indeed are sparse.

Another solution is to rewrite everything using the C API. It seems your data is already in a format that would allow to do the translation relatively easily.

Last but not least, if you could package your code into a reproducible example and send to MOSEK support (email here https://www.mosek.com/support/) then we [I work there] can check more concretely if there is some inefficiency in Mosek or if anything can be improved in your code.