2,497 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
Formulation, existence, and computation of boundedly rational dynamic user equilibrium with fixed or endogenous user tolerance
This paper analyzes dynamic user equilibrium (DUE) that incorporates the notion of boundedly rational (BR) user behavior in the selection of departure times and routes. Intrinsically, the boundedly rational dynamic user equilibrium (BR-DUE) model we present assumes that travelers do not always seek the least costly route-and-departure-time choice. Rather, their perception of travel cost is affected by an indifference band describing travelersâ tolerance of the difference between their experienced travel costs and the minimum travel cost. An extension of the BR-DUE problem is the so-called variable tolerance dynamic user equilibrium (VT-BR-DUE) wherein endogenously determined tolerances may depend not only on paths, but also on the established path departure rates. This paper presents a unified approach for modeling both BR-DUE and VT-BR-DUE, which makes significant contributions to the model formulation, analysis of existence, solution characterization, and numerical computation of such problems. The VT-BR-DUE problem, together with the BR-DUE problem as a special case, is formulated as a variational inequality. We provide a very general existence result for VT-BR-DUE and BR-DUE that relies on assumptions weaker than those required for normal DUE models. Moreover, a characterization of the solution set is provided based on rigorous topological analysis. Finally, three computational algorithms with convergence results are proposed based on the VI and DVI formulations. Numerical studies are conducted to assess the proposed algorithms in terms of solution quality, convergence, and computational efficiency
First order algorithms in variational image processing
Variational methods in imaging are nowadays developing towards a quite
universal and flexible tool, allowing for highly successful approaches on tasks
like denoising, deblurring, inpainting, segmentation, super-resolution,
disparity, and optical flow estimation. The overall structure of such
approaches is of the form ; where the functional is a data fidelity term also
depending on some input data and measuring the deviation of from such
and is a regularization functional. Moreover is a (often linear)
forward operator modeling the dependence of data on an underlying image, and
is a positive regularization parameter. While is often
smooth and (strictly) convex, the current practice almost exclusively uses
nonsmooth regularization functionals. The majority of successful techniques is
using nonsmooth and convex functionals like the total variation and
generalizations thereof or -norms of coefficients arising from scalar
products with some frame system. The efficient solution of such variational
problems in imaging demands for appropriate algorithms. Taking into account the
specific structure as a sum of two very different terms to be minimized,
splitting algorithms are a quite canonical choice. Consequently this field has
revived the interest in techniques like operator splittings or augmented
Lagrangians. Here we shall provide an overview of methods currently developed
and recent results as well as some computational studies providing a comparison
of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure
A General Self-Adaptive Relaxed-PPA Method for Convex Programming with Linear Constraints
We present an efficient
method for solving linearly constrained convex programming. Our algorithmic framework employs an implementable proximal step by a slight relaxation to the subproblem of proximal point algorithm (PPA). In particular, the stepsize choice condition of our algorithm is weaker than some elegant PPA-type methods. This condition is flexible and effective. Self-adaptive strategies are proposed to improve the convergence in practice. We theoretically show under mild conditions that our method converges in a global sense. Finally, we discuss applications and perform numerical experiments which confirm the efficiency of the proposed method. Comparisons of our method with some state-of-the-art algorithms are also provided
Iterative Methods for the Elasticity Imaging Inverse Problem
Cancers of the soft tissue reign among the deadliest diseases throughout the world and effective treatments for such cancers rely on early and accurate detection of tumors within the interior of the body. One such diagnostic tool, known as elasticity imaging or elastography, uses measurements of tissue displacement to reconstruct the variable elasticity between healthy and unhealthy tissue inside the body. This gives rise to a challenging parameter identification inverse problem, that of identifying the LamĂ© parameter ÎŒ in a system of partial differential equations in linear elasticity. Due to the near incompressibility of human tissue, however, common techniques for solving the direct and inverse problems are rendered ineffective due to a phenomenon known as the âlocking effectâ. Alternative methods, such as mixed finite element methods, must be applied to overcome this complication. Using these methods, this work reposes the problem as a generalized saddle point problem along with a presentation of several optimization formulations, including the modified output least squares (MOLS), energy output least squares (EOLS), and equation error (EE) frameworks, for solving the elasticity imaging inverse problem. Subsequently, numerous iterative optimization methods, including gradient, extragradient, and proximal point methods, are explored and applied to solve the related optimization problem. Implementations of all of the iterative techniques under consideration are applied to all of the developed optimization frameworks using a representative numerical example in elasticity imaging. A thorough analysis and comparison of the methods is subsequently presented
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