  A new approach to solving eigenvalue optimization problems for large
structured matrices is proposed and studied. The class of optimization problems
considered is related to computing structured pseudospectra and their extremal
points, and to structured matrix nearness problems such as computing the
structured distance to instability or to singularity. The structure can be a
general linear structure and includes, for example, large matrices with a given
sparsity pattern, matrices with given range and co-range, and Hamiltonian
matrices. Remarkably, the eigenvalue optimization can be performed on the
manifold of complex (or real) rank-1 matrices, which yields a significant
reduction of storage and in some cases of the computational cost. The method
relies on a constrained gradient system and the projection of the gradient onto
the tangent space of the manifold of complex rank-$1$ matrices. It is shown
that near a local minimizer this projection is very close to the identity map,
and so the computationally favorable rank-1 projected system behaves locally
like the %computationally expensive gradient system.
