Trace minimization method via penalty for linear response eigenvalue problems

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