{"title":"A New Prediction–Correction Primal–Dual Hybrid Gradient Algorithm for Solving Convex Minimization Problems with Linear Constraints","authors":"","doi":"10.1007/s10851-024-01173-2","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>The primal–dual hybrid gradient (PDHG) algorithm has been applied for solving linearly constrained convex problems. However, it was shown that without some additional assumptions, convergence may fail. In this work, we propose a new competitive prediction–correction primal–dual hybrid gradient algorithm to solve this kind of problem. Under some conditions, we prove the global convergence for the proposed algorithm with the rate of <em>O</em>(1/<em>T</em>) in a nonergodic sense, and also in the ergodic sense, in terms of the objective function value gap and the constraint violation. Comparative performance analysis of our method with other related methods on some matrix completion and wavelet-based image inpainting test problems shows the outperformance of our approach, in terms of iteration number and CPU time.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Imaging and Vision","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10851-024-01173-2","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE","Score":null,"Total":0}
引用次数: 0
Abstract
The primal–dual hybrid gradient (PDHG) algorithm has been applied for solving linearly constrained convex problems. However, it was shown that without some additional assumptions, convergence may fail. In this work, we propose a new competitive prediction–correction primal–dual hybrid gradient algorithm to solve this kind of problem. Under some conditions, we prove the global convergence for the proposed algorithm with the rate of O(1/T) in a nonergodic sense, and also in the ergodic sense, in terms of the objective function value gap and the constraint violation. Comparative performance analysis of our method with other related methods on some matrix completion and wavelet-based image inpainting test problems shows the outperformance of our approach, in terms of iteration number and CPU time.
摘要 原始-双重混合梯度（PDHG）算法已被用于求解线性约束凸问题。然而，研究表明，如果没有一些额外的假设，收敛可能会失败。在这项研究中，我们提出了一种新的竞争性预测-修正原始-双重混合梯度算法来解决这类问题。在某些条件下，我们证明了所提算法的全局收敛性，在非遍历意义上收敛率为 O(1/T)，而且在遍历意义上，在目标函数值差距和约束违反方面也是如此。我们的方法与其他相关方法在一些矩阵补全和基于小波的图像绘制测试问题上的性能对比分析表明，我们的方法在迭代次数和 CPU 时间上都优于其他方法。
期刊介绍：
The Journal of Mathematical Imaging and Vision is a technical journal publishing important new developments in mathematical imaging. The journal publishes research articles, invited papers, and expository articles.
Current developments in new image processing hardware, the advent of multisensor data fusion, and rapid advances in vision research have led to an explosive growth in the interdisciplinary field of imaging science. This growth has resulted in the development of highly sophisticated mathematical models and theories. The journal emphasizes the role of mathematics as a rigorous basis for imaging science. This provides a sound alternative to present journals in this area. Contributions are judged on the basis of mathematical content. Articles may be physically speculative but need to be mathematically sound. Emphasis is placed on innovative or established mathematical techniques applied to vision and imaging problems in a novel way, as well as new developments and problems in mathematics arising from these applications.
The scope of the journal includes:
computational models of vision; imaging algebra and mathematical morphology
mathematical methods in reconstruction, compactification, and coding
filter theory
probabilistic, statistical, geometric, topological, and fractal techniques and models in imaging science
inverse optics
wave theory.
Specific application areas of interest include, but are not limited to:
all aspects of image formation and representation
medical, biological, industrial, geophysical, astronomical and military imaging
image analysis and image understanding
parallel and distributed computing
computer vision architecture design.
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