Special backtracking proximal bundle method for nonconvex maximum eigenvalue optimization

We present a proximal bundle method for minimizing the nonconvex maximum eigenvalue function based on a real time control system. The oracle used in our proximal bundle method is able to compute separately the value and subgradient of the outer convex function. Besides, it can also calculate the value and derivatives of the smooth inner mapping. In each iteration, we solve a certain quadratic programming problem in which the smooth inner mapping is replaced by its Taylor-series linearization around the current serious step. By using the backtracking test, we can make a better approximation of the objective function. With no additional assumption, we prove the global convergence of our special bundle method. We present numerical examples demonstrating the efficiency of our algorithm on several feedback control syntheses.