566 OPTIMALITY CONDITIONS AND DUALITY THEORY FOR MINIMIZING SUMS OF THE LARGEST EIGENVALUES OF SYMMETRIC MATRICES M. L. Overton, R. S. Womersley, June 1991 This paper gives max characterizations for the sum of the largest eigenvalues 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 first-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. Keywords: symmetric matrix, maximum eigenvalues, spectral radius, minimax problem, max characterization, generalized gradient.