9,895 research outputs found
The parallel surrogate constraint approach to the linear feasibility problem
The linear feasibility problem arises in several areas of applied mathematics and medical science, in several forms of image reconstruction problems. The surrogate constraint algorithm of Yang and Murty for the linear feasibility problem is implemented and analyzed. The sequential approach considers projections one at a time. In the parallel approach, several projections are made simultaneously and their convex combination is taken to be used at the next iteration. The sequential method is compared with the parallel method for varied numbers of processors. Two improvement schemes for the parallel method are proposed and tested. © Springer-Verlag Berlin Heidelberg 1996
Accelerating two projection methods via perturbations with application to Intensity-Modulated Radiation Therapy
Constrained convex optimization problems arise naturally in many real-world
applications. One strategy to solve them in an approximate way is to translate
them into a sequence of convex feasibility problems via the recently developed
level set scheme and then solve each feasibility problem using projection
methods. However, if the problem is ill-conditioned, projection methods often
show zigzagging behavior and therefore converge slowly.
To address this issue, we exploit the bounded perturbation resilience of the
projection methods and introduce two new perturbations which avoid zigzagging
behavior. The first perturbation is in the spirit of -step methods and uses
gradient information from previous iterates. The second uses the approach of
surrogate constraint methods combined with relaxed, averaged projections.
We apply two different projection methods in the unperturbed version, as well
as the two perturbed versions, to linear feasibility problems along with
nonlinear optimization problems arising from intensity-modulated radiation
therapy (IMRT) treatment planning. We demonstrate that for all the considered
problems the perturbations can significantly accelerate the convergence of the
projection methods and hence the overall procedure of the level set scheme. For
the IMRT optimization problems the perturbed projection methods found an
approximate solution up to 4 times faster than the unperturbed methods while at
the same time achieving objective function values which were 0.5 to 5.1% lower.Comment: Accepted for publication in Applied Mathematics & Optimizatio
Ghost Penalties in Nonconvex Constrained Optimization: Diminishing Stepsizes and Iteration Complexity
We consider nonconvex constrained optimization problems and propose a new
approach to the convergence analysis based on penalty functions. We make use of
classical penalty functions in an unconventional way, in that penalty functions
only enter in the theoretical analysis of convergence while the algorithm
itself is penalty-free. Based on this idea, we are able to establish several
new results, including the first general analysis for diminishing stepsize
methods in nonconvex, constrained optimization, showing convergence to
generalized stationary points, and a complexity study for SQP-type algorithms.Comment: To appear on Mathematics of Operations Researc
State-of-the-art in aerodynamic shape optimisation methods
Aerodynamic optimisation has become an indispensable component for any aerodynamic design over the past 60 years, with applications to aircraft, cars, trains, bridges, wind turbines, internal pipe flows, and cavities, among others, and is thus relevant in many facets of technology. With advancements in computational power, automated design optimisation procedures have become more competent, however, there is an ambiguity and bias throughout the literature with regards to relative performance of optimisation architectures and employed algorithms. This paper provides a well-balanced critical review of the dominant optimisation approaches that have been integrated with aerodynamic theory for the purpose of shape optimisation. A total of 229 papers, published in more than 120 journals and conference proceedings, have been classified into 6 different optimisation algorithm approaches. The material cited includes some of the most well-established authors and publications in the field of aerodynamic optimisation. This paper aims to eliminate bias toward certain algorithms by analysing the limitations, drawbacks, and the benefits of the most utilised optimisation approaches. This review provides comprehensive but straightforward insight for non-specialists and reference detailing the current state for specialist practitioners
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