Linear matrix inequalities are powerful tools for solving numerous analysis and design control problems. The numerical technique for their solution is based on interior-point methods and barrier functions. We propose an alternative approach exploiting random search. It is based on new methods for convex optimization which use “boundary oracle” and Monte Carlo methods for finding center of gravity of a convex body. The theoretical validation of the approach relies on Radon’s theorem relating center of gravity. Results of numerical simulation will be provided.