Semi definite programming and direct support for trace of matrix?
AnsweredI have two questions.
1. Does Gurobi support semi definite programming?
2. Is there direct support to include trace of matrix in the objective function?

In its current form, no, but ...
Although Gurobi does not implement barrier or any other algorithms for SDP, if you have a toysized 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 nonconvex quadratics, e.g., equality constraints like x^2 == y, which in turn permits building multivariate polynomial constraints. Now, in all the likelihood you will not be able to deal with polynomialequality constraints for large degrees, however, for "small" matrices I would certainly try reexpressing 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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