Trace minimization method via penalty for linear response eigenvalue problems

<p style='text-indent:20px;'>In various applications, such as the computation of energy excitation states electrons and molecules, analysis interstellar clouds, linear response eigenvalue problem, which is a special type Hamiltonian frequently encountered. However, traditional eigensolvers may not be applicable to this problem owing its inherently large scale. In fact, we are usually more interested in computing some smallest positive eigenvalues. To end, trace minimum principle optimization model with orthogonality constraint has been proposed. On basis, propose an unconstrained surrogate called minimization via penalty, establish equivalence original constrained model, provided that penalty parameter larger than certain threshold. By avoiding constraint, can use gradient-type method solve model. Specifically, gradient descent Barzilai–Borwein step size. Moreover, develop restarting strategy for proposed algorithm whereby higher accuracy faster convergence achieved. This verified by preliminary experimental results.</p>