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To me, your Linear Matrix Inequalities really sound like Linear Programming... And if it actually is linear programming, you may be glad to learn that GLPK, Coin, and CPLEX can be used through Sage to solve LP (GLPK being installed as a standard package).

Here is an introduction to LP using Sage

Here is the documentation page for the MixedIntegerLinearProgram class (your second best friend when dealing with LP)

Nathann