Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

This paper gives max characterizations for the sum of the largest eigen-values of a symmetric matrix. The elements which achieve the maximum provide a concise characterization of the generalized gradient of the eigenvalue sum in terms of a dual matrix. The dual matrix provides the information required to either verify rst-order optimality conditions at a point or to generate a descent direction for the eigenvalue sum from that point, splitting a multiple eigenvalue if necessary. A model minimization algorithm is outlined, and connections with the classical literature on sums of eigenvalues are explained. Sums of the largest eigenvalues in absolute value are also addressed.