A Sequential SDP/Gauss-Newton Algorithm for Rank-Constrained LMI Problems

This paper develops a second-order Newton algorithm for nding local solutions of rank-constrained LMI problems in robust synthesis. The algorithm is based on a quadratic approximation of a suitably deened merit function and generates sequences of LMI feasible iterates. The main trust of the algorithm is that it inherits the good local convergence properties of Newton methods and thus overcome the diiculties encountered with earlier methods such as the Frank & 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-Deenite 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.