320 research outputs found
Bounded perturbation resilience of extragradient-type methods and their applications
In this paper we study the bounded perturbation resilience of the
extragradient and the subgradient extragradient methods for solving variational
inequality (VI) problem in real Hilbert spaces. This is an important property
of algorithms which guarantees the convergence of the scheme under summable
errors, meaning that an inexact version of the methods can also be considered.
Moreover, once an algorithm is proved to be bounded perturbation resilience,
superiorizion can be used, and this allows flexibility in choosing the bounded
perturbations in order to obtain a superior solution, as well explained in the
paper. We also discuss some inertial extragradient methods. Under mild and
standard assumptions of monotonicity and Lipschitz continuity of the VI's
associated mapping, convergence of the perturbed extragradient and subgradient
extragradient methods is proved. In addition we show that the perturbed
algorithms converges at the rate of . Numerical illustrations are given
to demonstrate the performances of the algorithms.Comment: Accepted for publication in The Journal of Inequalities and
Applications. arXiv admin note: text overlap with arXiv:1711.01936 and text
overlap with arXiv:1507.07302 by other author
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
PDEs with Compressed Solutions
Sparsity plays a central role in recent developments in signal processing,
linear algebra, statistics, optimization, and other fields. In these
developments, sparsity is promoted through the addition of an norm (or
related quantity) as a constraint or penalty in a variational principle. We
apply this approach to partial differential equations that come from a
variational quantity, either by minimization (to obtain an elliptic PDE) or by
gradient flow (to obtain a parabolic PDE). Also, we show that some PDEs can be
rewritten in an form, such as the divisible sandpile problem and
signum-Gordon. Addition of an term in the variational principle leads to
a modified PDE where a subgradient term appears. It is known that modified PDEs
of this form will often have solutions with compact support, which corresponds
to the discrete solution being sparse. We show that this is advantageous
numerically through the use of efficient algorithms for solving based
problems.Comment: 21 pages, 15 figure
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