# Cvx is unable to recover the true minimizer?

I am solving a convex program that is exactly the same as the one presented at this link. Namely, let A be a symmetric positive semi-definite matrix. I want to minimize ||H vec(A) - Y|| where vec(A) is the vectorized version of A and Y is a column vector.

I am experiencing 2 problems:

1. In my situation, I know the true value for A (and it satisfies the symmetric positive semidefinite constraint). When I plug this A into the objective function, I get a lower objective value than what is returned by CVX.

2. the optimal value by cvx, namely, cvx_optval is not exactly the same as what I would get by computing norm(H*A(:)-Y) in matlab. cvx_optval is 2.5175e-10 while Matlab provides me with 1.3138e-10. Any insights on the issue?

The difference between 2.5175e-10 and 1.3138e-10 is just solver tolerance “noise”; perhaps the exact optimal value is zero.

The solvers called by CVX only solve, even when reporting optimal solution found, to within a solver tolerance.

And on some problems, the argmin is not unique, even though the optimal objective value is unique o within solver tolerance.

If that doesn’t address your concern, then please provide more details, including exact output.

Hi Mark,

Thanks for the quick reply. I agree that a convex program does not imply a unique solution.
The thing is, when I compute norm(H*A(:)-Y) in matlab where A is the true matrix, the objective value is 1e-15 which is much much lower than the objective value of cvx. Can this difference in the objective value magnitude still be attributed to solver tolerance noise?

1e-15 is not much smaller than 2e-10 in this context. Both are essentially zero, so it might just be a roundoff and transformation thing.