We consider the problem of minimizing rank of a matrix under linear and nonlinear matrix inequality constraints. This problem arises in diverse applications such as estimation, control and signal processing and it is known to be computationally NP-hard even when constraints are linear . In this paper, we first formulize the RMP as an optimization problem with linear objective and simple nonlinear semialgebraic constraints. We then proceed to solve the problem with augmented Lagrangian method known in nonlinear optimization. Despite of other heuristic and approximate methods in the subject, this method guarantees to find the global optimum in the sense that it does not depends on the choice of initial point for convergence. Several numerical examples demonstrate the effectiveness of the considered algorithm.
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