Место издания:Peoples' Friendship University of Russia Moscow
Первая страница:390
Аннотация:Mathematical programs with vanishing constraints constitute a new
class of difficult optimization problems with important
applications in optimal topology design of mechanical structures.
Vanishing constraints usually violate standard constraint
qualifications, which gives rise to serious difficulties in
theoretical and numerical treatment of these problems.
In this work, we suggest several globalization strategies for the
active-set Newton-type methods developed earlier by the authors
for this problem class, preserving superlinear convergence rate of
these methods under weak assumptions. Globalization strategies
discussed in this work are of a hybrid nature: the active-set
method is combined with some outer-phase algorithm with good
global convergence properties, but otherwise it can be rather
arbitrary, and in particular, unrelated to the active-set local
phase. The role of a generic outer-phase method is to drive the
iterative sequence towards a stationary point of the problem and
an associated Lagrange multiplier. Each step of the outer-phase
method is followed by an attempt to switch to the active-set steps
which are accepted if the linear decrease test for a residual of
the Karush--Kuhn--Tucker system of the problem turns satisfied. In
the first suggested globalization strategy, the residual at the
trial point is compared with the current residual, and the cases
of possible violation of the linear decrease test on later
iterations are treated by means of backup safeguards. Thus, some
iterates computed by active-set steps can be eventually
disregarded, and the overall efficiency of the algorithm certainly
depends on the actual amount of these useless computations.
Alternatively, in the second globalization strategy, the residual
at the trial point is compared with record (rather than current)
residual, and the record is updated only when the linear decrease
test turns satisfied, or when the step of the outer-phase method
turns to decrease the residual with respect to the record. This
strategy allows to avoid using the backup safeguards.
Preliminary numerical results demonstrate that our approaches are
rather promissing and competitive with respect to the existing
alternatives.