Minimal compact operators, subdifferential of the maximum eigenvalue and semi-definite programming

TL;DR

This paper explores the minimality of self-adjoint operators via semi-definite programming, linking eigenvalue subdifferentials to optimization problems, and provides new formulas for eigenvalue subdifferentials and minimizing diagonals.

Contribution

It introduces new formulas for subdifferentials of maximum eigenvalues of compact operators and characterizes minimal self-adjoint operators within a semi-definite programming framework.

Findings

New formulas for subdifferentials of maximum eigenvalues of compact operators

Characterization of minimal self-adjoint operators using eigenvalue subdifferentials

Formulas for minimizing diagonals of rank one self-adjoint operators

Abstract

We formulate the issue of minimality of self-adjoint operators on a Hilbert space as a semi-definite problem, linking the work by Overton in [1] to the characterization of minimal hermitian matrices. This motivates us to investigate the relationship between minimal self-adjoint operators and the subdifferential of the maximum eigenvalue, initially for matrices and subsequently for compact operators. In order to do it we obtain new formulas of subdifferentials of maximum eigenvalues of compact operators that become useful in these optimization problems. Additionally, we provide formulas for the minimizing diagonals of rank one self-adjoint operators, a result that might be applied for numerical large-scale eigenvalue optimization. [1] On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appl.9 (1988), no 4, 905-918