Affine SDP constraint

[Chen Chen]

I'm coding with MOSEK Toolbox for MATLAB. My problem is a SDP, which is the same as Section 6.7 Affine conic constraints in the MOSEK MATLAB Manual, but instead with

the affine conic constraint being

    Fx + g is positive semi-definite.

I wonder how can I could up this affine SDP constraint? I don't find a cone type parameter from res.symbcon.

Thanks a ton!

 

[Michal Adamaszek]

You have to make a semidefinite variable X (section 6.6) and add linear constraints Fx+g-X=0 for the upper triangular part.

The PSD cone is not available in the same way as "ordinary" cones, it can only be represented by introducing a PSD variable. In the future it may change.

Best,

Michal

[Michal Adamaszek]

By the way, it depends on the rest of the problem, but often if you want to input an LMI this way you will get a much more efficient model if you instead dualize. Compare example 8.7 in https://docs.mosek.com/modeling-cookbook/duality.html#semidefinite-duality-and-lmis

[Chen Chen]

Thanks a lot, Michal! Both your answer and example 8.7 you mentioned
are very helpful.

Just a quick question: I read Section 6.6 Semidefinite Optimization,
and there it says
"only the lower triangular part should be specified explicitly"
I guess specifying either lower triangular and upper triangular (but
not both) should work?

Also, I guess I need to specify X= Fx+g element-wisely, like how you
handle (\bar{X}_2)_{01} <= k in Section 6.6.2 Example SDO2?

Thanks a lot again,

Chen

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[Michal Adamaszek]

There must be k>=l (lower triangular) or you will get an error. It's a matter of convention, but that's how it is.

Yes, you will need to write Fx+g=X as n*(n+1)/2 constraints, one containing each X_{k,l} as in SDO2.

Michal