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primal dual algorithm for solving the nonsmooth Twin SVM

ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE 2024年 128卷

作者： Lyaqini, S. Hadri, A. Ellahyani, A. Nachaoui, M. Hassan First Univ Settat LAMSAD Lab Settat Morocco Ibn Zohr Univ LAB SIV Agadir Morocco Sultan Moulay Slimane Univ EMI Beni Mellal Morocco

In this paper, we propose an improved version of Twin SVM using a non-smooth optimization method. Twin SVM generally consists in determining two non-parallel planes by alternately solving two constrained optimization models. Solving this problem using the classical Lagrangian method has many limitations, notably: its only limited to handle Gaussian noise, generally exaggerates the influence of outliers and cannot handle unbalanced data, this due to the differentiability of the model. To circumvent these issues, we transform two-constraint optimization models using the penalty method into an unconstrained non-smooth optimization one. The non-smoothness nature of the problem has many advantages, but it requires special treatment, which is why we use the primal dual method to solve it, since it is the most appropriate and it is robust in terms of stability, convergence and speed (Lyaqini, Nachaoui and Hadri, 2022). To demonstrate the effectiveness of the proposed approach, several experiments were carried out on numerous UCI benchmarks, medical image and HandPD datasets. These experiments demonstrated the effectiveness and applicability of the proposed approach, with satisfactory results compared to the state of the art.

关键词： Twin SVM primal dual algorithm Hinge loss

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A fully stochastic primal-dual algorithm

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OPTIMIZATION LETTERS 2021年第2期15卷 701-710页

作者： Bianchi, Pascal Hachem, Walid Salim, Adil IP Paris Telecom Paris IP Paris F-75013 Paris France Univ Gustave Eiffel LIGM CNRS F-77454 Marne La Vallee France KAUST Visual Comp Ctr Thuwal 23955 Saudi Arabia

A new stochastic primal-dual algorithm for solving a composite optimization problem is proposed. It is assumed that all the functions / operators that enter the optimization problem are given as statistical expectations. These expectations are unknown but revealed across time through i.i.d realizations. The proposed algorithm is proven to converge to a saddle point of the Lagrangian function. In the framework of the monotone operator theory, the convergence proof relies on recent results on the stochastic Forward Backward algorithm involving random monotone operators. An example of convex optimization under stochastic linear constraints is considered.

关键词： Stochastic optimization primal dual algorithm

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primal dual based algorithm for degree-balanced spanning tree problem

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APPLIED MATHEMATICS AND COMPUTATION 2018年 316卷 167-173页

作者： Ran, Yingli Chen, Zhihao Tang, Shaojie Zhang, Zhao Xinjiang Univ Coll Math & Syst Sci Urumqi 830046 Xinjiang Peoples R China Zhejiang Normal Univ Coll Math Phys & Informat Engn Jinhua 321004 Zhejiang Peoples R China Univ Texas Dallas Naveen Jindal Sch Management Dallas TX 75080 USA

This paper studies approximation algorithm for the degree-balanced spanning tree (DBST) problem. Given a graph G = (V, E), the goal is to find a spanning tree T such that Sigma v is an element of V deg(T)(v)(2) is minimized, where deg T (v) denotes the degree of node v in tree T. The idea of taking squares on node degrees is to manifest the role of nodes with large degree, and thus minimizing the sum will result in a comparatively balanced degree distribution. This is a non-linear objective function. We prove that DBST is NP-hard, and then develop a primal-dual based algorithm with a guaranteed performance ratio. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Degree-balanced spanning tree Nonlinear objective function primal dual algorithm

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Online Ranking with Constraints: A primal-dual algorithm and Applications to Web Traffic-Shaping 17

Online Ranking with Constraints: A Primal-Dual Algorithm and...

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23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD)

作者： Shah, Parikshit Soni, Akshay Chevalier, Troy Yahoo Res New York NY 10036 USA

ISBN: (纸本)9781450348874

We study the online constrained ranking problem motivated by an application to web-traffic shaping: an online stream of sessions arrive in which, within each session, we are asked to rank items. The challenge involves optimizing the ranking in each session so that local vs. global objectives are controlled: within each session one wishes to maximize a reward (local) while satisfying certain constraints over the entire set of sessions (global). A typical application of this setup is that of page optimization in a web portal. We wish to rank items so that not only is user engagement maximized in each session, but also other business constraints (such as the number of views/clicks delivered to various publishing partners) are satisfied. We describe an online algorithm for performing this optimization. A novel element of our approach is the use of linear programming duality and connections to the celebrated Hungarian algorithm. This framework enables us to determine a set of shadow prices for each traffic-shaping constraint that can then be used directly in the final ranking function to assign near-optimal rankings. The (dual) linear program can be solved off-line periodically to determine the prices. At serving time these prices are used as weights to compute weighted rank-scores for the items, and the simplicity of the approach facilitates scalability to web applications. We provide rigorous theoretical guarantees for the performance of our online algorithm and validate our approach using numerical experiments on real web-traffic data from a prominent internet portal.

关键词： online ranking traffic shaping constrained ranking primal dual algorithm

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Impact of angular sampling interval on image reconstruction from limited-angular-range data

Impact of angular sampling interval on image reconstruction ...

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Conference on Medical Imaging - Physics of Medical Imaging

作者： Zhang, Zheng Chen, Buxin Xia, Dan Sidky, Emil Y. Pan, Xiaochuan Univ Chicago Dept Radiol Chicago IL 60637 USA Univ Chicago Dept Radiat & Cellular Oncol Chicago IL 60637 USA

ISBN: (数字)9781510649385

ISBN: (纸本)9781510649385;9781510649378

In computed tomography (CT) imaging, recent developments in reconstruction algorithm and scan configuration design have provided useful tools for image reconstruction from data collected over a limited-angular range (LAR). In this work, we aim to investigate the impact of angular sampling interval on the accuracy of reconstruction from LAR data. In specific, we employ a two-orthogonal-arc scan configuration, and collect data from a numerical chest phantom over an LAR with various angular intervals. We then investigate image reconstruction by using the directional-total-variation (DTV) algorithm and evaluate reconstructions qualitatively and quantitatively. Results show that increased angular sampling interval results in degraded image quality. Results of the simulation study also indicate an appropriate interval for sufficient reconstruction accuracy under specific imaging conditions, which may provide insights for the upper-bound performance of reconstructions in practical use.

关键词： Optimization-based reconstruction directional total variation primal dual algorithm two orthogonal arcs

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Maximum shortest path interdiction problem by upgrading edges on trees under weighted l1 norm

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JOURNAL OF GLOBAL OPTIMIZATION 2021年第4期79卷 959-987页

作者： Zhang, Qiao Guan, Xiucui Pardalos, Panos M. Southeast Univ Sch Math Nanjing 210096 Peoples R China Univ Florida Dept Ind & Syst Engn Ctr Appl Optimizat Gainesville FL 32611 USA Higher Sch Econ LATNA Moscow Russia

Network interdiction problems by deleting critical edges have wide applicatio ns. However, in some practical applications, the goal of deleting edges is difficult to achieve. We consider the maximum shortest path interdiction problem by upgrading edges on trees (MSPIT) under unit/weighted l(1) norm. We aim to maximize the the length of the shortest path from the root to all the leaves by increasing the weights of some edges such that the upgrade cost under unit/weighted l(1) norm is upper-bounded by a given value. We construct their mathematical models and prove some properties. We propose a revised algorithm for the problem (MSPIT) under unit l(1) norm with time complexity O(n), where n is the number of vertices in the tree. We put forward a primal dual algorithm in O(n(2)) time to solve the problem (MSPIT) under weighted l(1) norm, in which a minimum cost cut is found in each iteration. We also solve the problem to minimize the cost to upgrade edges such that the length of the shortest path is lower bounded by a value and present an O(n(2)) algorithm. Finally, we perform some numerical experiments to compare the results obtained by these algorithms.

关键词： Network interdiction problem Upgrading critical edges Shortest path Weighted l(1) norm primal dual algorithm Minimum cost cut

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Accurate reconstruction of cross-section images from limited-angular-range data in human-limb imaging

Accurate reconstruction of cross-section images from limited...

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Medical Imaging Conference - Physics of Medical Imaging

作者： Zhang, Zheng Chen, Buxin Xia, Dan Sidky, Emil Y. Pan, Xiaochuan Univ Chicago Dept Radiol Chicago IL 60637 USA Univ Chicago Dept Radiat & Cellular Oncol Chicago IL 60637 USA

ISBN: (纸本)9781510640207

In this work, we investigate and develop a method for cross-section image reconstruction from data collected over limited-angular ranges in the context of human-limb imaging. We first design a convex optimization program with constraints on directional image total-variations (TVs), and then tailor a convex primal-dual algorithm, which is referred to as the directional TV (DTV) algorithm, for solving this program. By using the proposed DTV algorithm, we investigate image reconstructions in studies with data collected from numerical thigh phantoms over a limited-angular range of 60 degrees. The results of the numerical studies demonstrate that the method proposed can yield, from limited-angular-range data, cross-section images with significantly reduced artifacts that are observed otherwise in images obtained with existing algorithms.

关键词： Optimization-based reconstruction directional total variation primal dual algorithm limb CT

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Investigation of image reconstruction for digital breast tomosynthesis imaging by use of directional TV constraint

Investigation of image reconstruction for digital breast tom...

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Medical Imaging Conference - Physics of Medical Imaging

作者： Zhang, Zheng Chen, Buxin Xia, Dan Sidky, Emil Y. Pan, Xiaochuan Univ Chicago Dept Radiol Chicago IL 60637 USA Univ Chicago Dept Radiat & Cellular Oncol Chicago IL 60637 USA

ISBN: (纸本)9781510640207

In digital breast tomosynthesis (DBT), in-plane images are of clinical utility, whereas images within transverse planes contain significant artifacts simply because the existing algorithms are not designed for reconstructing accurately images within transverse planes from extremely limited-angular-range data. In this work, we investigate and develop a convex primal-dual (CPD) algorithm that incorporates directional total-variation (DTV) constraints for yielding breast images within transverse planes with substantially reduced artifacts when images are reconstructed from DBT data. We have performed numerical studies to demonstrate that the algorithm proposed has the potential to yield breast images with substantially reduced artifacts within transverse planes observed in images obtained otherwise with existing algorithms.

关键词： digital breast tomosynthesis (DBT) optimization-based reconstruction directional total variation primal dual algorithm

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A Dynamical Approach to Two-Block Separable Convex Optimization Problems with Linear Constraints

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2020年第1期42卷 1-38页

作者： Bitterlich, Sandy Csetnek, Erno Robert Wanka, Gert Tech Univ Chemnitz Fac Math Chemnitz Germany Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria

The aim of this manuscript is to approach by means of first order differential equations/inclusions convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. Each block of the objective contains a further smooth convex function. We investigate the dynamical system proposed and prove that its trajectories converge weakly to a saddle point of the Lagrangian of the convex optimization problem. The dynamical system provides through time discretization the alternating minimization algorithm AMA and also its proximal variant recently introduced in the literature.

关键词： Convex optimization duality dynamical system Lyapunov analysis Lagrangian primal dual algorithm Proximal AMA saddle points structured convex minimization subdifferential

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Variational Approach for Restoring Blurred Images with Cauchy Noise

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SIAM JOURNAL ON IMAGING SCIENCES 2015年第3期8卷 1894-1922页

作者： Sciacchitano, Federica Dong, Yiqiu Zeng, Tieyong Tech Univ Denmark Dept Appl Math & Comp Sci DK-2800 Lyngby Denmark Hong Kong Baptist Univ Dept Math Kowloon Tong Hong Kong Peoples R China

The restoration of images degraded by blurring and noise is one of the most important tasks in image processing. In this paper, based on the total variation (TV) we propose a new variational method for recovering images degraded by Cauchy noise and blurring. In order to obtain a strictly convex model, we add a quadratic penalty term, which guarantees the uniqueness of the solution. Due to the convexity of our model, the primal dual algorithm is employed to solve the minimization problem. Experimental results show the effectiveness of the proposed method for simultaneously deblurring and denoising images corrupted by Cauchy noise. Comparison with other existing and well-known methods is provided as well.

关键词： Cauchy noise image deblurring image denoising primal dual algorithm total variation regularization variational model

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