- Two Numerical Methods for Optimizing Matrix Stability James V. Burke (burkemath.washington.edu) Adrian S. Lewis (aslewiscecm.sfu.ca) Michael L. Overton (overtoncs.nyu.edu) Abstract: Consider the affine matrix family $A(x) = A_0 + \sum_{k=1}^m x_k A_k$, mapping a design vector $x\in\Rl^m$ into the space of $n \times n$ real matrices. Consider the affine matrix family $A(x) = A_0 + \sum_{k=1}^m x_k A_k$, mapping a design vector $x\in\Rl^m$ into the space of $n \times n$ real matrices. We are interested in the question of how to choose $x$ to optimize the stability of the dynamical system $\dot z = A(x)z$. A classic example in control is stabilization by output feedback. We take two approaches. The first is to directly minimize $\alpha(A(x))$, the spectral abscissa (the largest real part of the eigenvalues) of $A(x)$, since this quantity bounds the asymptotic decay rate of the trajectories of the dynamical system. The spectral abscissa $\alpha(X)$ is a continuous but nonsmooth, in fact non-Lipschitz, function of the matrix argument $X$, and finding a global minimizer of $\alpha(A(x))$ is difficult. We introduce a novel random gradient bundle method for approximating \emph{local} minimizers, motivated by recent work on nonsmooth analysis of the function $\alpha(X)$. Our second approach is to minimize a related function $\rsa(A(x))$, where $\delta$ is a \emph{robustness} parameter in (0,1). The motivation for the definition of the robust spectral abscissa Keywords: Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization ) Citation: Submitted to Lin. Alg. Appl. Download: [Postscript]Entry Submitted: 05/31/2001Entry Accepted: 05/31/2001Entry Last Modified: 05/31/2001Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.