Abstract
The spectral abscissa is a fundamental map from the set of complex matrices to the real numbers. Denoted α and defined as the maximum of the real parts of the eigenvalues of a matrix X, it has many applications in stability analysis of dynamical systems. The function α is nonconvex and is non-Lipschitz near matrices with multiple eigenvalues. Variational analysis of this function was presented in Burke and Overton (Math Program 90:317–352, 2001), including a complete characterization of its regular subgradients and necessary conditions which must be satisfied by all its subgradients. A complete characterization of all subgradients of α at a matrix X was also given for the case that all active eigenvalues of X (those whose real part equals α(X)) are nonderogatory (their geometric multiplicity is one) and also for the case that they are all nondefective (their geometric multiplicity equals their algebraic multiplicity). However, necessary and sufficient conditions for all subgradients in all cases remain unknown. In this paper we present necessary and sufficient conditions for the simplest example of a matrix X with a derogatory, defective multiple eigenvalue.
Similar content being viewed by others
References
Arnold, V.I.: On matrices depending on parameters. Russ. Math. Surv. 26, 29–43 (1971)
Burke, J.V., Henrion, D., Lewis, A.S., Overton, M.L.: Stabilization via nonsmooth, nonconvex optimization. IEEE Trans. Autom. Control 51, 1760–1769 (2006)
Burke, J.V., Lewis, A.S., Overton, M.L.: Optimizing matrix stability. Proc. Am. Math. Soc. 129, 1635–1642 (2001)
Burke, J.V., Overton, M.L.: Variational analysis of non-Lipschitz spectral functions. Mathe. Program. 90, 317–352 (2001)
Gade, K.K., Overton, M.L.: Optimizing the asymptotic convergence rate of the Diaconis-Holmes-Neal sampler. Adv. Appl. Math. 38(3), 382–403 (2007)
Grundel, S.: Eigenvalue Optimization in C 2 Subdivision and Boundary Subdivision. PhD thesis, New York University (2011). http://cs.nyu.edu/overton/phdtheses/sara.pdf
Henrion, D., Overton, M.L.: Maximizing the closed loop asymptotic decay rate for the two-mass-spring control problem. Technical Report 06342, LAAS-CNRS, March (2006). http://homepages.laas.fr/henrion/Papers/massspring.pdf
Mordukhovich, B.S.: Maximum principle in the problem of time optimal response with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969 (1976)
Overton, M.L., Womersley, R.S.: On minimizing the spectral radius of a nonsymmetric matrix function – optimality conditions and duality theory. SIAM J. Matrix Anal. Appl. 9, 473–498 (1988)
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, New York (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by National Science Foundation Grant DMS-0714321 and DMS-1016325 at Courant Institute of Mathematical Sciences, New York University.
Rights and permissions
About this article
Cite this article
Grundel, S., Overton, M.L. Variational Analysis of the Spectral Abscissa at a Matrix with a Nongeneric Multiple Eigenvalue. Set-Valued Var. Anal 22, 19–43 (2014). https://doi.org/10.1007/s11228-013-0234-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-013-0234-7