  A merger of two optimization frameworks is introduced: SEquential Subspace
OPtimization (SESOP) with MultiGrid (MG) optimization. At each iteration of the
algorithm, the search direction implied by the coarse-grid correction process
of MG is added to the low dimensional search-space of SESOP, which includes the
preconditioned gradient and search directions involving the previous iterates,
called {\em history}. Numerical experiments demonstrate the effectiveness of
this approach. We then study the asymptotic convergence factor of the two-level
version of SESOP-MG (dubbed SESOP-TG) for optimization of quadratic functions,
and derive approximately optimal fixed parameters, which may reduce the
computational overhead for such problems significantly.
