Convex Analysis of Spectrally Defined Matrix Functions
Alberto Seeger · SIAM Journal on Optimization · 1997
The purpose of this work is to carry out a systematic study of a special class of convex functions defined over the space $S_n$ of symmetric matrices of order $n\ti n$. The functions under consideration $(\Phi: S_n \ra R\cup \{+\infty\})$ are spectrally defined in the sense that the value $\Phi(A)$ depends only on the spectrum $\{\lam_1(A), \ldots, \lam_n(A)\}$ of the matrix $A\in S_n$. Fenchel--Legendre conjugation, first- and second-order subdifferentiability, asymptotic behavior, and other concepts of convex analysis are the main ingredients of our exposition.