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A new approach to solving eigenvalue optimization problems for largestructured matrices is proposed and studied. The class of optimization problemsconsidered is related to computing structured pseudospectra and their extremalpoints, and to structured matrix nearness problems such as computing thestructured distance to instability or to singularity. The structure can be ageneral linear structure and includes, for example, large matrices with a givensparsity pattern, matrices with given range and co-range, and Hamiltonianmatrices. Remarkably, the eigenvalue optimization can be performed on themanifold of complex (or real) rank-1 matrices, which yields a significantreduction of storage and in some cases of the computational cost. The methodrelies on a constrained gradient system and the projection of the gradient ontothe tangent space of the manifold of complex rank-$1$ matrices. It is shownthat near a local minimizer this projection is very close to the identity map,and so the computationally favorable rank-1 projected system behaves locallylike the %computationally expensive gradient system.

Rank-$1$ matrix differential equations for structured eigenvalue optimization