Abstract
Consider the linear space ℘n of polynomials of degree n or less over the complex field. The abscissa mapping on ℘n is the mapping that takes a polynomial to the maximum real part of its roots. This mapping plays a key role in the study of stability properties for linear systems. Burke and Overton have shown that the abscissa mapping is everywhere subdifferentially regular in the sense of Clarke on the manifold ℳn of polynomials of degree n. In addition, they provide a formula for the subdifferential. The result is surprising since the abscissa mapping is not Lipschitzian on ℳn. A key supporting lemma uses a proof technique due to Levantovskii for determining the tangent cone to the set of stable polynomials. This proof is arduous and opaque. It is a major obstacle to extending the variational theory to other functions of the roots of polynomials. In this note, we provide an alternative proof based on the Gauss-Lucas Theorem. This new proof is both insightful and elementary.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burke, J.V., Lewis, A.S. and Overton, M.L. (2000), Optimizing matrix stability, Proc. of the American Math. Soc. 129, 1635–1642.
Burke, J.V., Lewis, A.S. and Overton, M.L. (2001), Optimal stability and eigenvalue multiplicity, Foundations of Computational Mathematics 1, 205–225.
Burke, J.V. and Overton, M.L. (1994), Differential properties of the spectral abscissa and the spectral radius for analytic matrix-valued mappings, Journal of Nonlinear Analysis, Theory, Methods, and Applications 23, 467–488.
Burke, J.V. and Overton, M.L. (2000), Variational analysis of the abscissa mapping for polynomials, SIAM Journal on Control and Optimization 39, 1651–1676.
Burke, J.V. and Overton, M.L. (2001), Variational analysis of non-Lipschitz spectral functions, Math. Programming 90, 317–351.
Clarke, F.H. (1973), Necessary conditions for nonsmooth problems in optimal control and the calculus of variations, PhD thesis, University of Washington.
Clarke, F.H., Ledyaev, Yu.S., Stern, R.J. and Wolenski P.R. (1998), Nonsmooth Analysis and Control Theory, Springer, Berlin.
Levantovskii, L.V. (1980), On singularities of the boundary of a stability region, Moscow Univ. Math. Bull. 35, 19–22.
Lucas, F. (1879), Sur une application de la méchanique rationnelle á la théorie des équations, C.R. Acad. Sci. Paris 89, 224–226.
Marden, M. (1966), Geometry of Polynomials, American Mathematical Society.
Rockafellar, R.T. and Wets, R.J.B. (1998), Variational Analysis, Springer, Berlin.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Burke, J.V., Lewis, A.S. & Overton, M.L. Variational Analysis of the Abscissa Mapping for Polynomials via the Gauss-Lucas Theorem. Journal of Global Optimization 28, 259–268 (2004). https://doi.org/10.1023/B:JOGO.0000026448.63457.51
Issue Date:
DOI: https://doi.org/10.1023/B:JOGO.0000026448.63457.51