  In this paper we analyze and solve eigenvalue programs, which consist of the
task of minimizing a function subject to constraints on the "eigenvalues" of
the decision variable. Here, by making use of the FTvN systems framework
introduced by Gowda, we interpret "eigenvalues" in a broad fashion going beyond
the usual eigenvalues of matrices. This allows us to shed new light on
classical problems such as inverse eigenvalue problems and also leads to new
applications. In particular, after analyzing and developing a simple projected
gradient algorithm for general eigenvalue programs, we show that eigenvalue
programs can be used to express what we call vanishing quadratic constraints. A
vanishing quadratic constraint requires that a given system of convex quadratic
inequalities be satisfied and at least a certain number of those inequalities
must be tight. As a particular case, this includes the problem of finding a
point $x$ in the intersection of $m$ ellipsoids in such a way that $x$ is also
in the boundary of at least $\ell$ of the ellipsoids, for some fixed $\ell >
0$. At the end, we also present some numerical experiments.
