Ela Exact Relaxation for the Semidefinite Matrix Rank Minimization Problem with Extended Lyapunov Equation Constraint

The semidefinite matrix rank minimization, which has a broad range of applications in system control, statistics, network localization, econometrics and so on, is computationally NPhard in general due to the noncontinuous and non-convex rank function. A natural way to handle this type of problems is to substitute the rank function into some tractable surrogates, most popular ones of which include the convex trace norm and the non-convex Schatten p-norm relaxations with p ∈ (0, 1). The corresponding exactness of these relaxations have absorbed great attention and interest from researchers both in mathematics and engineering fields. In this paper, a special semidefinite matrix rank minimization problem with the extended Lyapunov equation constraint arising from low-order optimal control is considered and shown to possess the desired exact relaxation properties by exploiting the special structures of the involved linear transformation and by developing some essential properties and features on rank function and the semidefinite matrix cone.