Merging Multigrid Optimization with SESOP
For optimization practitioners, this hybrid approach improves convergence by combining subspace optimization with multigrid, though it is incremental.
The paper merges SESOP with multigrid optimization, adding coarse-grid correction to SESOP's search space. Numerical experiments show effectiveness, and asymptotic convergence analysis yields optimal parameters for quadratic problems.
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.