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Description: |
We consider the solution of large scale optimization problems,
subject to bound constaints, arising in the discretization
of infinite dimensional problems. In particular, we suppose that
we have access to a collection of functions representing
the problem for different value of a discretization parameter, ranging
from the coarsest level, where computations are cheap, to
the finest level. As in multigrid methods, our idea is to
combine approximate solutions of the problems at all available levels,
to obtain a fast algorithm for solving the problem at the finest level.
Starting from an existing multi-level trust-region algorithm
for unconstrained optimization, where the trust region
is based on the Euclidean norm, we design an infinity norm variant,
that is better suited to bound constrained problems and exploit
the multi-level structure. The resulting solution method, that
depends on some algorithmic parameters, is then shown to be
convergent to first order critical point, from an
arbitrarily chosen starting point. Using a set of test-cases
arising from the calculus of variations, we determine reasonable
values for the algorithmic parameters and compare the obtained
algorithm with both a traditional trust-region technique, and
with a mesh refinement strategy.
We conclude with a discussion on stopping criteria for bound-constrained
discretized problems that naturally take into account
the discretization errors as does the backward error approach in
linear algebra.
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