An inertial modified algorithm for solving variational inequalities

2020 ◽  
Vol 54 (1) ◽  
pp. 163-178 ◽  
Author(s):  
Dang Van Hieu ◽  
Pham Kim Quy
Keyword(s):  
Hilbert Spaces ◽  
Lipschitz Constant ◽  
Inertial Effects ◽  
Convergence Properties ◽  
Pseudomonotone Operators ◽  
Lipschitz Continuous ◽  
Strong Pseudomonotonicity ◽  
Computational Performance ◽  
Speed Up ◽  
The Cost

The paper deals with an inertial-like algorithm for solving a class of variational inequality problems involving Lipschitz continuous and strongly pseudomonotone operators in Hilbert spaces. The presented algorithm can be considered a combination of the modified subgradient extragradient-like algorithm and inertial effects. This is intended to speed up the convergence properties of the algorithm. The main feature of the new algorithm is that it is done without the prior knowledge of the Lipschitz constant and the modulus of strong pseudomonotonicity of the cost operator. Several experiments are performed to illustrate the convergence and computational performance of the new algorithm in comparison with others having similar features. The numerical results have confirmed that the proposed algorithm has a competitive advantage over the existing methods.

scholarly journals Contributions to the Normalized Correlation and the Mean Squares Metric

2006 ◽  
Author(s):  
Marius Staring
Keyword(s):  
Source Code ◽  
Computation Time ◽  
Convergence Properties ◽  
Random Subset ◽  
Speed Up ◽  
The Mean ◽  
The Cost

This document describes contributions on the NormalizedCorrelationImageToImageMetric and the MeanSquaresImageToImageMetric of the Insight Toolkit ITK . For the first metric a two time speed-up can be achieved by rewriting the code to loop only once over the fixed image. This is instead of the two times that is used in the current ITK code. The reduction in computation time comes at the cost of an additional storage of a parameters array. For both metric we have implemented the option to use only a random subset of the fixed image voxels for calculating the metric value and its derivatives. This reduces the computation time (substantially), while convergence properties are maintained. This paper is accompanied with the source code.

scholarly journals Projected-Reflected Subgradient-Extragradient Method and Its Real-World Applications

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 489
Author(s):  
Aviv Gibali ◽  
Olaniyi S. Iyiola ◽  
Lanre Akinyemi ◽  
Yekini Shehu
Keyword(s):  
Extragradient Method ◽  
A Priori ◽  
Lipschitz Constant ◽  
Monotone Mapping ◽  
Subgradient Extragradient Method ◽  
Lipschitz Continuous ◽  
Real World Applications ◽  
Potential Applicability ◽  
Tomography Reconstruction ◽  
The Cost

Our main focus in this work is the classical variational inequality problem with Lipschitz continuous and pseudo-monotone mapping in real Hilbert spaces. An adaptive reflected subgradient-extragradient method is presented along with its weak convergence analysis. The novelty of the proposed method lies in the fact that only one projection onto the feasible set in each iteration is required, and there is no need to know/approximate the Lipschitz constant of the cost function a priori. To illustrate and emphasize the potential applicability of the new scheme, several numerical experiments and comparisons in tomography reconstruction, Nash–Cournot oligopolistic equilibrium, and more are presented.

scholarly journals Two Bregman Projection Methods for Solving Variational Inequality Problems in Hilbert Spaces with Applications to Signal Processing

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2007
Author(s):  
Lateef Olakunle Jolaoso ◽  
Maggie Aphane ◽  
Safeer Hussain Khan
Keyword(s):  
Signal Processing ◽  
Variational Inequality ◽  
Hilbert Spaces ◽  
Convex Functions ◽  
Lipschitz Constant ◽  
Variational Inequality Problems ◽  
Bregman Distance ◽  
Bregman Projection ◽  
Prior Estimate ◽  
The Cost

Studying Bregman distance iterative methods for solving optimization problems has become an important and very interesting topic because of the numerous applications of the Bregman distance techniques. These applications are based on the type of convex functions associated with the Bregman distance. In this paper, two different extragraident methods were proposed for studying pseudomonotone variational inequality problems using Bregman distance in real Hilbert spaces. The first algorithm uses a fixed stepsize which depends on a prior estimate of the Lipschitz constant of the cost operator. The second algorithm uses a self-adaptive stepsize which does not require prior estimate of the Lipschitz constant of the cost operator. Some convergence results were proved for approximating the solutions of pseudomonotone variational inequality problem under standard assumptions. Moreso, some numerical experiments were also given to illustrate the performance of the proposed algorithms using different convex functions such as the Shannon entropy and the Burg entropy. In addition, an application of the result to a signal processing problem is also presented.

scholarly journals An Efficient Parallel Extragradient Method for Systems of Variational Inequalities Involving Fixed Points of Demicontractive Mappings

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1915
Author(s):  
Lateef Olakunle Jolaoso ◽  
Maggie Aphane
Keyword(s):  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Line Search ◽  
Convex Combination ◽  
Extragradient Method ◽  
Convergence Result ◽  
Armijo Line Search ◽  
Systems Of Variational Inequalities ◽  
Demicontractive Mappings ◽  
The Cost

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method.

Sparse Incremental Delta-Bar-Delta for System Identification

2014 ◽  
Vol 665 ◽  
pp. 643-646
Author(s):  
Ying Liu ◽  
Yan Ye ◽  
Chun Guang Li
Keyword(s):  
System Identification ◽  
Cost Function ◽  
Learning Algorithm ◽  
Learning System ◽  
The Other ◽  
Sparse System ◽  
Speed Up ◽  
Sparse System Identification ◽  
The Cost ◽  
Zero Attractor

Metalearning algorithm learns the base learning algorithm, targeted for improving the performance of the learning system. The incremental delta-bar-delta (IDBD) algorithm is such a metalearning algorithm. On the other hand, sparse algorithms are gaining popularity due to their good performance and wide applications. In this paper, we propose a sparse IDBD algorithm by taking the sparsity of the systems into account. Thenorm penalty is contained in the cost function of the standard IDBD, which is equivalent to adding a zero attractor in the iterations, thus can speed up convergence if the system of interest is indeed sparse. Simulations demonstrate that the proposed algorithm is superior to the competing algorithms in sparse system identification.

scholarly journals Fine-Tuning by Curriculum Learning for Non-Autoregressive Neural Machine Translation

2020 ◽  
Vol 34 (05) ◽  
pp. 7839-7846
Author(s):  
Junliang Guo ◽  
Xu Tan ◽  
Linli Xu ◽  
Tao Qin ◽  
Enhong Chen ◽  
...  
Keyword(s):  
Machine Translation ◽  
Fine Tuning ◽  
Inference Process ◽  
Neural Machine Translation ◽  
Training Strategy ◽  
Speed Up ◽  
Good Improvement ◽  
Tuning Process ◽  
Translation Accuracy ◽  
The Cost

Non-autoregressive translation (NAT) models remove the dependence on previous target tokens and generate all target tokens in parallel, resulting in significant inference speedup but at the cost of inferior translation accuracy compared to autoregressive translation (AT) models. Considering that AT models have higher accuracy and are easier to train than NAT models, and both of them share the same model configurations, a natural idea to improve the accuracy of NAT models is to transfer a well-trained AT model to an NAT model through fine-tuning. However, since AT and NAT models differ greatly in training strategy, straightforward fine-tuning does not work well. In this work, we introduce curriculum learning into fine-tuning for NAT. Specifically, we design a curriculum in the fine-tuning process to progressively switch the training from autoregressive generation to non-autoregressive generation. Experiments on four benchmark translation datasets show that the proposed method achieves good improvement (more than 1 BLEU score) over previous NAT baselines in terms of translation accuracy, and greatly speed up (more than 10 times) the inference process over AT baselines.

scholarly journals AUTOMATIC MRF-BASED REGISTRATION OF HIGH RESOLUTION SATELLITE VIDEO DATA

2016 ◽  
Vol III-1 ◽  
pp. 121-128 ◽  
Author(s):  
C. Platias ◽  
M. Vakalopoulou ◽  
K. Karantzalos
Keyword(s):  
High Resolution ◽  
Markov Random Fields ◽  
Video Data ◽  
Registration Accuracy ◽  
Registration Method ◽  
Computational Performance ◽  
Markov Random ◽  
Video Frames ◽  
Reference Map ◽  
The Cost

In this paper we propose a deformable registration framework for high resolution satellite video data able to automatically and accurately co-register satellite video frames and/or register them to a reference map/image. The proposed approach performs non-rigid registration, formulates a Markov Random Fields (MRF) model, while efficient linear programming is employed for reaching the lowest potential of the cost function. The developed approach has been applied and validated on satellite video sequences from Skybox Imaging and compared with a rigid, descriptor-based registration method. Regarding the computational performance, both the MRF-based and the descriptor-based methods were quite efficient, with the first one converging in some minutes and the second in some seconds. Regarding the registration accuracy the proposed MRF-based method significantly outperformed the descriptor-based one in all the performing experiments.

scholarly journals An Efficient Probability Estimation Decision Tree Postprocessing Method for Mining Optimal Profitable Knowledge for Enterprises with Multi-Class Customers

2019 ◽  
Vol 22 (64) ◽  
pp. 63-84
Author(s):  
JanapatyI Naga Muneiah ◽  
Ch D V SubbaRao
Keyword(s):  
Decision Tree ◽  
Probability Estimation ◽  
Automatic Extraction ◽  
Business Applications ◽  
Computational Performance ◽  
Novel Approach ◽  
Benchmark Datasets ◽  
Phone Service ◽  
The Cost ◽  
Service Data

Enterprises often classify their customers based on the degree of profitability in decreasing order like C1, C2, ..., Cn. Generally, customers representing class Cn are zero profitable since they migrate to the competitor. They are called as attritors (or churners) and are the prime reason for the huge losses of the enterprises. Nevertheless, customers of other intermediary classes are reluctant and offer an insignificant amount of profits in different degrees and lead to uncertainty. Various data mining models like decision trees, etc., which are built using the customers’ profiles, are limited to classifying the customers as attritors or non-attritors only and not providing profitable actionable knowledge. In this paper, we present an efficient algorithm for the automatic extraction of profit-maximizing knowledge for business applications with multi-class customers by postprocessing the probability estimation decision tree (PET). When the PET predicts a customer as belonging  to any of the lesser profitable classes, then, our algorithm suggests the cost-sensitive actions to change her/him to a maximum possible higher profitable status. In the proposed novel approach, the PET is represented in the compressed form as a Bit patterns matrix and the postprocessing task is performed on the bit patterns by applying the bitwise AND operations. The computational performance of the proposed method is strong due to the employment of effective data structures. Substantial experiments conducted on UCI datasets, real Mobile phone service data and other benchmark datasets demonstrate that the proposed method remarkably outperforms the state-of-the-art methods.

The Multidirectional Mean Value Theorem in Banach Spaces

1997 ◽  
Vol 40 (1) ◽  
pp. 88-102 ◽  
Author(s):  
M. L. Radulescu ◽  
F. H. Clarke
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Locally Lipschitz Functions ◽  
Bump Function ◽  
Lipschitz Continuous ◽  
Lower Semi ◽  
Locally Lipschitz

AbstractRecently, F. H. Clarke and Y. Ledyaev established a multidirectional mean value theorem applicable to lower semi-continuous functions on Hilbert spaces, a result which turns out to be useful in many applications. We develop a variant of the result applicable to locally Lipschitz functions on certain Banach spaces, namely those that admit a C1-Lipschitz continuous bump function.

Piecewise continuous solutions of nonlinear pseudoparabolic equations in two space dimensions*

1992 ◽  
Vol 121 (3-4) ◽  
pp. 203-217 ◽  
Author(s):  
Dao-Qing Dai ◽  
Wei Lin
Keyword(s):  
Lipschitz Constant ◽  
Complex Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Lipschitz Continuous ◽  
Two Space Dimensions ◽  
Image Position ◽  
Initial Boundary Value ◽  
Pseudoparabolic Equations ◽  
Nonlinear Pseudoparabolic Equation

SynopsisAn initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variablesThe complex functionHis measurable on ℂ ×I × ℂ5, withIbeing an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.

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