The solution I get is not feasible but violated by, say, -1e-6


Most solvers actually use infeasible/exterior algorithms, so slightly infeasible solutions are common. If you really need a strictly feasible point (assuming one really exists), simply define a slightly tighter constraint than wanted (e.g., change your constraint from X>=0 to X>=eye(n)*smallconstant in the case of semidefinite constraints etc.)

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