Semi definite programming and direct support for trace of matrix?
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I have two questions.
1. Does Gurobi support semi definite programming?
2. Is there direct support to include trace of matrix in the objective function?
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In its current form, no, but ...
Although Gurobi does not implement barrier or any other algorithms for SDP, if you have a toy-sized SDP model, we can still do something with it starting from the upcoming release 9.0 (by the end of Nov 2019). Note with 9.0 we allow non-convex quadratics, e.g., equality constraints like x^2 == y, which in turn permits building multi-variate polynomial constraints. Now, in all the likelihood you will not be able to deal with polynomial-equality constraints for large degrees, however, for "small" matrices I would certainly try re-expressing SDP constraints as a set of, say, polynomial inequalities on determinants of the principal minors, e.g., see here, https://en.wikipedia.org/wiki/Sylvester%27s_criterion
It may be worth a try, so hope it helps.
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