A Trust Region Spectral Bundle Method for Nonconvex Eigenvalue Optimization

Abstract

Abstract. We present a nonsmooth optimization technique for nonconvex maximum eigenvalue functions and for nonsmooth functions which are infinite maxima of eigenvalue functions. We prove global convergence of our method in the sense that for an arbitrary starting point, every accumulation point of the sequence of iterates is critical. The method is tested on several problems in feedback control synthesis. Here our interest is in nonconvex eigenvalue programs, which arise frequently in automatic control applications and especially in controller synthesis. In particular, solving bilinear matrix inequalities (BMIs) is a prominent application, which may be addressed via nonconvex eigenvalue optimization. In [45,46] we have shown how to adapt the approach of [41,48] to handle nonconvex situations. Applications and extensions of these ideas are presented in [4,1,57,10,5,6].

Control design in the time and frequency domain using nonsmooth techniques

“…This leads to a variation of the present algorithm discussed in [6,24,7], where a trust region strategy replaces the present line search method.…”

“…We have used the proposed nonsmooth technique to minimize the worst case objective in (30) over the set of PID feedback controllers with parametrization given in (6). The following parameter values were obtained:…”

“…See [10,8] for stabilization problems, [4,5,20,3,11] for H ∞ synthesis, and [3,6] for design with IQCs. These techniques are in our opinion a valuable addition to the designer's toolkit:…”

An inexact spectral bundle method for convex quadratic semidefinite programming

“…This updating rule of the metric originates from [27], and it is employed in [23]. At the same time, to the best of our knowledge, most of the literature about bundle methods exploit the special matrix M y,k = I and exclude the matrix variable X, for instance, see [2,4,13,14,16,24,34,44]. However, in order to ensure convergence, we should use variable metric at both descent steps and null steps in Algorithm 3.1.…”

“…Owing to the fact that interior-point algorithms perform poorly with large scale problems because of their high demand for storage and being time-consuming, there has been a recent, renewed interest in bundle methods. There are very recent related works on this subject presenting in, for instance, [2,4,21,23,25,36,45] and the references therein. We emphasize that a bundle method has been employed to solve a quite general problem-the equilibrium problem in [36], which covers a wide range, such as the optimization problem, the variational inequality problem, the Nash equilibrium problem in noncooperative games, the fixed point problem, the nonlinear complementarity problem and the vector optimization problem.…”