  We study properties and algorithms of a minimization problem of the maximum
generalized eigenvalue of symmetric-matrix-valued affine functions, which is
nonsmooth and quasiconvex, and has application to eigenfrequency optimization
of truss structures. We derive an explicit formula of the Clarke
subdifferential of the maximum generalized eigenvalue and prove the maximum
generalized eigenvalue is a pseudoconvex function, which is a subclass of a
quasiconvex function, under suitable assumptions. Then, we consider smoothing
methods to solve the problem. We introduce a smooth approximation of the
maximum generalized eigenvalue and prove the convergence rate of the smoothing
projected gradient method to a global optimal solution in the considered
problem. Also, some heuristic techniques to reduce the computational costs,
acceleration and inexact smoothing, are proposed and evaluated by numerical
experiments.
