We consider a nonlinear eigenvalue problem related to the trace minimization of §188.8.131.52: minimize where is some nonlinear term.
This sort of minimization occurs in electronic structures computations such as local density approximations (LDAs), where the columns of represent electron wave functions, is a fixed Hamiltonian, and represents the energy of the electron-electron interactions (see, for example, [151,428]). In this case, the optimal represents the state of lowest energy of the system.
The only additions to the differentials of the trace minimization are terms from which vary from problem to problem. For our example, we have taken for some coupling constant , where (the charge density).
In this case