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We consider the minimization or maximization of the $J$th largest eigenvalueof an analytic and Hermitian matrix-valued function, and build on Mengi et al.(2014, SIAM J. Matrix Anal. Appl., 35, 699-724). This work addresses thesetting when the matrix-valued function involved is very large. We describesubspace procedures that convert the original problem into a small-scale one bymeans of orthogonal projections and restrictions to certain subspaces, and thatgradually expand these subspaces based on the optimal solutions of small-scaleproblems. Global convergence and superlinear rate-of-convergence results withrespect to the dimensions of the subspaces are presented in the infinitedimensional setting, where the matrix-valued function is replaced by a compactoperator depending on parameters. In practice, it suffices to solve eigenvalueoptimization problems involving matrices with sizes on the scale of tens,instead of the original problem involving matrices with sizes on the scale ofthousands.

A Subspace Method for Large Scale Eigenvalue Optimization