Yuan's alternative theorem and the maximization of the minimum eigenvalue function

Let A1 and A2 be two symmetric matrices of order n×n. According to Yuan, there exists a convex combination of these matrices which is positive semidefinite, if and only if the function x∈Rn {mapping} max {xTA1x, xTA2x} is nonnegative. We study the case in which more than two matrices are involved. We study also a related question concerning the maximization of the minimum eigenvalue of a convex combination of symmetric matrices. © 1994 Plenum Publishing Corporation.