## Convex analysis on the Hermitian matrices (1996)

Venue: | SIAM Journal on Optimization |

Citations: | 45 - 20 self |

### BibTeX

@ARTICLE{Lewis96convexanalysis,

author = {A. S. Lewis},

title = {Convex analysis on the Hermitian matrices},

journal = {SIAM Journal on Optimization},

year = {1996},

volume = {6},

pages = {164--177}

}

### Years of Citing Articles

### OpenURL

### Abstract

There is growing interest in optimization problems with real symmetric matrices as variables. Generally the matrix functions involved are spectral: they depend only on the eigenvalues of the matrix. It is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues. A new approach to this characterization is given, via a simple Fenchel conjugacy formula. We then apply this formula to derive expressions for subdifferentials, and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables. Analogous results hold for Hermitian matrices. Key Words: convexity, matrix function, Schur convexity, Fenchel duality, subdifferential, unitarily invariant, spectral function, positive semidefinite programming, quasi-Newton update. AMS 1991 Subject Classification: Primary 15A45 49N15 Secondary 90C25 65K10 1 Introduction A matrix norm on the n \Theta n complex matrices is called unitarily inv...