Hi everyone, I’m experiencing some numerical problems

when using the SDPT3 solver for a (apparently) fairly simple SDP problem.

The problem under consideration is

$$ \min_X tr(MX), ; s.t. X\succcurlyeq0, X(end,end) = 1 $$

This problem is just the lifted relaxation of

$$ \min_\tilde{x} \tilde{x}^\top M \hat{x}, ; s.t. \tilde{x}(end)^2=1 $$

which in turn corresponds to a *simple* Least Squares (linear) problem:

$$ min_x ||A x - b|| $$

Obviously, this is just a testing problem before proceeding to the relaxation

of a more complex one which is not a Least Squares problem.

The curious fact is that Sedumi is working well, but SDPT3 fails to converge.

I know I should just stick with the solver working (it’s what I usually do)

but since this seems (to me) like a very simple problem

I was a little surprised that the usual SDP solvers could have any difficulty with it.

Furthermore, I’m concerned that I might be missing part of the complexity of the problem

since this could become the source of many headaches when things don’t work

in the more complex problem extending this one.

I’ve tried looking for some theoretical background that may

explain in which situations a SDP problem could get ill-conditioned

or something related, but couldn’t find anything straightforward

in the context of this specific model.

I would appreciate if any of the experts here

could share their impressions about this

Just to do some simple tests, I’m providing a simple test code:

```
% generate positive semidefinite matrix
m = 10;
n = 3;
A = randn(m,n);
b = randn(m,1);
M = [A,-b]'*[A,-b];
cvx_solver SDPT3
% cvx_solver sedumi
% cvx_precision low
cvx_begin SDP
variable X(n+1,n+1) semidefinite
X(end,end) == 1
minimize(trace(M*X))
cvx_end
```

It seems sometimes (rarely) the SDPT3 solver worked,

but in general any random problem will fail.

You can easily check that apparently there are no conditioning

issues with the data matrix M.

I tried different precisions, by the way.