A sequential SDP/Gauss-Newton algorithm for rank-constrained LMI problems

The paper develops a second-order Newton algorithm for finding local solutions of rank-constrained LMI problems in robust synthesis. The algorithm is based on a quadratic approximation of a suitably defined merit function and generates sequences of LMI feasible iterates. The main thrust of the algorithm is that it inherits the good local convergence properties of Newton methods and thus overcomes the difficulties encountered with earlier methods such as the Frank and Wolfe or conditional gradient methods which tend to be very slow in the neighborhood of a local solution. Moreover, it is easily implemented using available Semi-Definite Programming (SDP) codes. Proposed algorithms have proven global and local convergence properties and thus represent improvements over classically used D-K iteration schemes but also outperform earlier conditional gradient algorithms. Reported computational results demonstrate these facts