3,060 research outputs found
On barrier and modified barrier multigrid methods for 3d topology optimization
One of the challenges encountered in optimization of mechanical structures,
in particular in what is known as topology optimization, is the size of the
problems, which can easily involve millions of variables. A basic example is
the minimum compliance formulation of the variable thickness sheet (VTS)
problem, which is equivalent to a convex problem. We propose to solve the VTS
problem by the Penalty-Barrier Multiplier (PBM) method, introduced by R.\
Polyak and later studied by Ben-Tal and Zibulevsky and others. The most
computationally expensive part of the algorithm is the solution of linear
systems arising from the Newton method used to minimize a generalized augmented
Lagrangian. We use a special structure of the Hessian of this Lagrangian to
reduce the size of the linear system and to convert it to a form suitable for a
standard multigrid method. This converted system is solved approximately by a
multigrid preconditioned MINRES method. The proposed PBM algorithm is compared
with the optimality criteria (OC) method and an interior point (IP) method,
both using a similar iterative solver setup. We apply all three methods to
different loading scenarios. In our experiments, the PBM method clearly
outperforms the other methods in terms of computation time required to achieve
a certain degree of accuracy
A two-phase gradient method for quadratic programming problems with a single linear constraint and bounds on the variables
We propose a gradient-based method for quadratic programming problems with a
single linear constraint and bounds on the variables. Inspired by the GPCG
algorithm for bound-constrained convex quadratic programming [J.J. Mor\'e and
G. Toraldo, SIAM J. Optim. 1, 1991], our approach alternates between two phases
until convergence: an identification phase, which performs gradient projection
iterations until either a candidate active set is identified or no reasonable
progress is made, and an unconstrained minimization phase, which reduces the
objective function in a suitable space defined by the identification phase, by
applying either the conjugate gradient method or a recently proposed spectral
gradient method. However, the algorithm differs from GPCG not only because it
deals with a more general class of problems, but mainly for the way it stops
the minimization phase. This is based on a comparison between a measure of
optimality in the reduced space and a measure of bindingness of the variables
that are on the bounds, defined by extending the concept of proportioning,
which was proposed by some authors for box-constrained problems. If the
objective function is bounded, the algorithm converges to a stationary point
thanks to a suitable application of the gradient projection method in the
identification phase. For strictly convex problems, the algorithm converges to
the optimal solution in a finite number of steps even in case of degeneracy.
Extensive numerical experiments show the effectiveness of the proposed
approach.Comment: 30 pages, 17 figure
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