In this talk, we consider the problem of computing several extreme eigenpairs of real symmetric matrices. Based on the variational principles, we put forward some new unconstrained optimization models for this classical problem and further analyze their fundamental properties. It is shown that the extreme eigenpairs of any real symmetric matrix can be recovered from the global minimizers of our unconstrained models. The alternate Barzilai-Borwein method with the adaptive nonmonotone line search is then utilized for solving the unconstrained models. The preliminary numerical results indicate that our approach is quite promising.