Diagonal BFGS updates and applications to the limited memory BFGS method

We propose two diagonal BFGS-type updates. One is the diagonal part of the ordinary BFGS update on a diagonal matrix. The other is its inverse version. Both diagonal updates preserve the positive definiteness as the ordinary BFGS update. The related diagonal BFGS methods can be regarded as extensions of the well-known Barzilai-Borwein method. Under appropriate conditions, we prove that both diagonal BFGS methods are globally convergent when applied to minimizing a convex or non-convex function. In addition, the diagonal quasi-Newton method with inverse diagonal BFGS update can be even superlinearly convergent if the function to be minimized is uniformly convex and completely separable. We apply the proposed diagonal BFGS updates to the limited memory BFGS (L-BFGS) method using the diagonal BFGS matrix as initial matrix. Our numerical results show the efficiency of the L-BFGS methods with diagonal BFGS updates.