Spectrally Constrained Optimization

We investigate how to solve smooth matrix optimization problems with general linear inequality constraints on the eigenvalues of a symmetric matrix. We present solution methods to obtain exact global minima for linear objective functions, i.e., \(F(\varvec{X}) = \langle \varvec{C}, \varvec{X}\rangle \) $F(\mathit{X})=⟨\mathit{C},\mathit{X}⟩$ , and perform exact projections onto the eigenvalue constraint set. Two first-order algorithms are developed to obtain first-order stationary points for general non-convex objective functions. Both methods are proven to converge sublinearly when the constraint set is convex. Numerical experiments demonstrate the applicability of both the model and the methods.