Optimization Problems

In this chapter, we describe extensions of the Schwarz subspace methods from Chap. 2 to iteratively solve minimization problems [TA4, TA5]. Such methods correspond to block generalizations of the Gauss-Seidel and Jacobi relaxation methods for minimization problems. In general terms, domain decomposition and multilevel methodology can be applied to minimization problems in two alternative ways. In the first approach, domain decomposition methods can be employed within an inner iteration, to solve the quadratic minimization problem occurring during each iteration of a traditional Newton or trust region method. Such an approach requires a global quadratic approximation of the underlying functional whose minimum is sought. In the second approach, the divide and conquer Schwarz subspace methodology seeks the global minimum using lower dimensional minimization problems on subspaces. This approach requires only local quadratic approximations.