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In this work, we propose a new local optimization method to solve a class of nonconvex semidefinite programming (SDP) problems. The basic idea is to approximate the feasible set of the nonconvex SDP problem by inner positive semidefinite convex approximations via a parameterization technique. This leads to an iterative procedure to search a local optimum of the nonconvex problem. The convergence of the algorithm is analyzed under mild assumptions. Applications to optimization problems with bilinear matrix inequality (BMI) constraints in static output feedback control are benchmarked and numerical tests are implemented based on the data from the COMPL<inf>e</inf>ib library.

An inner convex approximation algorithm for BMI optimization and applications in control