[Csdp] Csdp Digest, Vol 10, Issue 1

[Imre Polik]

Brock,

your numbers don't add up. As far as I can see the objective function
(F0) coefficients corresponding to the extra numbers are zero, so they
can't possible influence the objective value. Besides, once the
duality gap is zero, and the solutions are primal-dual feasible, you
have optimal solutions. There is no way to come up with a better
optimal solution (unless of course you shake the foundations of
mathematics).

Your problem has multiple optimal solutions. In that case
interior-point methods converge to somewhere in the middle of the
optimal set. Unfortunately, you're probably looking for the extreme
points of the optimal face. Probably due to the symmetry in the
problem you always get the midpoint. you can try something that breaks
this symmetry, like slightly perturbing the objective.

Another way to put this is that interior-point methods return
maximally complementary solutions. It means that the rank of X+Y will
be as large as possible. CSDP does do that, the rank of the 3x3 block
is 3, the rank of the 4x4 is 4. Your solution however only gives rank
two for the 3x3 block, it is not maximally complementary. In fact,
sqrt(23)/12 is the largest number you can plug in there to still get a
positive semidefinite Y.

Re: equalities and strict feasibility. Strict feasibility means the
existence of a feasible point that satisfies the inequalities
strictly. In other words, it lives in the relative interior of the
feasible set.

I don't know how you could get a better SDP model for your problem,
but this is what I can tell you about the SDP problem you have.

Best,
Imre