On Maximizing the Minimum Eigenvalue of a Linear Combination of Symmetric Matrices

The problem considered is that of maximizing, with respect to the weights, the minimum eigenvalue of a weighted sum of symmetric matrices when the Euclidean norm of the vector of weights is constrained to be unity. A procedure is given for determining the sign of the maximum of the minimum eigenvalue and for approximating the optimal weights arbitrarily accurately when that sign is positive or zero. Linear algebra, a conical hull representation of the set of n × n symmetric positive semidefinite matrices and convex programming are employed.