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 eigenvalues problems and also leads to new applications. In particular,
after analyzing and developing a simple projected gradient algorithm for general eigenvalue pro-
grams, 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 l of the ellipsoids, for some fixed l > 0.
At the end, we also present some numerical experiments.

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