A randomized approximation algorithm for the minimal-norm static-output-feedback problem

A new randomized algorithm is suggested, for extracting static-output-stabilizing-feedbacks, with approximately minimal-norm, for LTI systems. The algorithm has two similar stages, where in the first one the feasibility problem is solved, and in the second one the optimization problem is solved. The formulation is unified for the feasibility and for the optimization problems, as well as for continuous-time or discrete-time systems. The method is demonstrated by applying it to the hard (conjectured to be NP-hard) problem of the minimal-gain static-output-stabilizing-feedback, and to the hard (conjectured to be NP-hard) problem of regional pole-placement via static-output-feedback in non-convex or unconnected regions. A proof of convergence (in probability) that captures the two rounds of the algorithm is given, and complexity analysis is provided, under some mild assumptions.