### Abstract

We consider spectral functions f ○ λ, where f is any permutation-invariant mapping from C^{n} to R, and λ is the eigenvalue map from the set of n x n complex matrices to C^{n}, ordering the eigenvalues lexicographically. For example, if f is the function "maximum real part", then f ○ λ is the spectral abscissa, while if f is "maximum modulus", then f ○ λ is the spectral radius. Both these spectral functions are continuous, but they are neither convex nor Lipschitz. For our analysis, we use the notion of subgradient extensively analyzed in Variational Analysis, R.T. Rockafellar and R. J.-B. Wets (Springer, 1998). We show that a necessary condition for Y to be a subgradient of an eigenvalue function f ○ λ at X is that Y* commutes with X. We also give a number of other necessary conditions for Y based on the Schur form and the Jordan form of X In the case of the spectral abscissa, we refine these conditions, and we precisely identify the case where subdifferential regularity holds. We conclude by introducing the notion of a semistable program: maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non-positive. Semistable programming is a nonconvex generalization of semidefinite programming. Using our analysis, we derive a necessary condition for a local maximizer of a semistable program, and we give a generalization of the complementarity condition familiar from semidefinite programming.

Original language | English (US) |
---|---|

Pages (from-to) | 317-351 |

Number of pages | 35 |

Journal | Mathematical Programming |

Volume | 90 |

Issue number | 2 |

State | Published - Apr 2001 |

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### Keywords

- Eigenvalue function
- Nonsmooth analysis
- Semistable program
- Spectral abscissa
- Spectral radius
- Stability

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics
- Safety, Risk, Reliability and Quality
- Management Science and Operations Research

### Cite this

*Mathematical Programming*,

*90*(2), 317-351.

**Variational analysis of non-Lipschitz spectral functions.** / Burke, James V.; Overton, Michael L.

Research output: Contribution to journal › Article

*Mathematical Programming*, vol. 90, no. 2, pp. 317-351.

}

TY - JOUR

T1 - Variational analysis of non-Lipschitz spectral functions

AU - Burke, James V.

AU - Overton, Michael L.

PY - 2001/4

Y1 - 2001/4

N2 - We consider spectral functions f ○ λ, where f is any permutation-invariant mapping from Cn to R, and λ is the eigenvalue map from the set of n x n complex matrices to Cn, ordering the eigenvalues lexicographically. For example, if f is the function "maximum real part", then f ○ λ is the spectral abscissa, while if f is "maximum modulus", then f ○ λ is the spectral radius. Both these spectral functions are continuous, but they are neither convex nor Lipschitz. For our analysis, we use the notion of subgradient extensively analyzed in Variational Analysis, R.T. Rockafellar and R. J.-B. Wets (Springer, 1998). We show that a necessary condition for Y to be a subgradient of an eigenvalue function f ○ λ at X is that Y* commutes with X. We also give a number of other necessary conditions for Y based on the Schur form and the Jordan form of X In the case of the spectral abscissa, we refine these conditions, and we precisely identify the case where subdifferential regularity holds. We conclude by introducing the notion of a semistable program: maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non-positive. Semistable programming is a nonconvex generalization of semidefinite programming. Using our analysis, we derive a necessary condition for a local maximizer of a semistable program, and we give a generalization of the complementarity condition familiar from semidefinite programming.

AB - We consider spectral functions f ○ λ, where f is any permutation-invariant mapping from Cn to R, and λ is the eigenvalue map from the set of n x n complex matrices to Cn, ordering the eigenvalues lexicographically. For example, if f is the function "maximum real part", then f ○ λ is the spectral abscissa, while if f is "maximum modulus", then f ○ λ is the spectral radius. Both these spectral functions are continuous, but they are neither convex nor Lipschitz. For our analysis, we use the notion of subgradient extensively analyzed in Variational Analysis, R.T. Rockafellar and R. J.-B. Wets (Springer, 1998). We show that a necessary condition for Y to be a subgradient of an eigenvalue function f ○ λ at X is that Y* commutes with X. We also give a number of other necessary conditions for Y based on the Schur form and the Jordan form of X In the case of the spectral abscissa, we refine these conditions, and we precisely identify the case where subdifferential regularity holds. We conclude by introducing the notion of a semistable program: maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non-positive. Semistable programming is a nonconvex generalization of semidefinite programming. Using our analysis, we derive a necessary condition for a local maximizer of a semistable program, and we give a generalization of the complementarity condition familiar from semidefinite programming.

KW - Eigenvalue function

KW - Nonsmooth analysis

KW - Semistable program

KW - Spectral abscissa

KW - Spectral radius

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=0001823671&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001823671&partnerID=8YFLogxK

M3 - Article

VL - 90

SP - 317

EP - 351

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 2

ER -