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Continuous-Time Distributed proximal gradient algorithms for Nonsmooth Resource Allocation Over General Digraphs

IEEE TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING 2021年第2期8卷 1733-1744页

作者： Zhu, Yanan Wen, Guanghui Yu, Wenwu Yu, Xinghuo Nanjing Univ Informat Sci & Technol Sch Automat Nanjing 210044 Peoples R China Southeast Univ Sch Math Jiangsu Prov Key Lab Networked Collect Intelligen Nanjing 210096 Peoples R China RMIT Univ Sch Engn Melbourne Vic 3001 Australia

This paper studies a nonsmooth resource allocation problem with network resource constraints and local set constraints, where the interaction graphs among agents are generally strongly connected digraphs. First, we design a centralized continuous-time proximal gradient algorithm, where each agent uses the global Lagrangian multipliers and the global values of constraint functions. For the case that the agents' private information could not be leaked and the global Lagrangian multipliers are not available, the agents are endowed with some additional variables to estimate those global information via consensus protocols. Then, we construct a class of continuous-time distributed proximal gradient algorithms by using a two-time scale mechanism to integrate the proposed proximal gradient algorithm and consensus protocols. By adopting Lyapunov stability theory and convex optimization theory, we prove that the decision variables asymptotically converge to the optimal solution of the nonsmooth resource allocation problem. Finally, numerical simulations are applied to illustrate the effectiveness of the proposed algorithms.

关键词： Nonsmooth resource allocation proximal gradient algorithm consensus protocol two-time scale mechanism

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proximal gradient/Semismooth Newton Methods for Projection onto a Polyhedron via the Duality-Gap-Active-Set Strategy

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JOURNAL OF SCIENTIFIC COMPUTING 2023年第1期97卷 1-34页

作者： Wang, Yunlong Shen, Chungen Zhang, Lei-Hong Yang, Wei Hong Shanghai Jiao Tong Univ Antai Coll Econ & Management Shanghai 200030 Peoples R China Univ Shanghai Sci & Technol Coll Sci Shanghai 200093 Peoples R China Soochow Univ Sch Math Sci Suzhou 215006 Peoples R China Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China

The polyhedral projection problem arises in various applications. To efficiently solve the dual problem, one of the crucial issues is to safely identify zero-elements as well as the signs of nonzero elements at the optimal solution. In this paper, relying on its nonsmooth dual problem and active set techniques, we first propose a Duality-Gap-Active-Set strategy (DGASS) to effectively identify the indices of zero-elements and the signs of nonzero entries of the optimal solution. Serving as an efficient acceleration strategy, DGASS can be embedded into certain iterative methods. In particular, by applying DGASS to both the proximal gradient algorithm (PGA) and the proximal semismooth Newton algorithm (PSNA), we propose the method of PGA-DGASS and PSNA-DGASS, respectively. Global convergence and local quadratic convergence rate are discussed. We report on numerical results over both synthetic and real data sets to demonstrate the high efficiency of the two DGASS-accelerated methods.

关键词： Polyhedral projection problem Active set Duality gap proximal gradient algorithm proximal semismooth Newton method

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On convergence of iterative thresholding algorithms to approximate sparse solution for composite nonconvex optimization

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MATHEMATICAL PROGRAMMING 2025年第1-2期211卷 181-206页

作者： Hu, Yaohua Hu, Xinlin Yang, Xiaoqi Shenzhen Univ Sch Math Sci Shenzhen 518060 Peoples R China Shenzhen Univ WeBank Inst Fintech Shenzhen Audencia Business Sch Shenzhen 518060 Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China

This paper aims to find an approximate true sparse solution of an underdetermined linear system. For this purpose, we propose two types of iterative thresholding algorithms with the continuation technique and the truncation technique respectively. We introduce a notion of limited shrinkage thresholding operator and apply it, together with the restricted isometry property, to show that the proposed algorithms converge to an approximate true sparse solution within a tolerance relevant to the noise level and the limited shrinkage magnitude. Applying the obtained results to nonconvex regularization problems with SCAD, MCP and lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _p$$\end{document} penalty (0 <= p <= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le p \le 1$$\end{document}) and utilizing the recovery bound theory, we establish the convergence of their proximal gradient algorithms to an approximate global solution of nonconvex regularization problems. The established results include the existing convergence theory for l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document} or l0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _0$$\end{document} regularization problems for finding a true sparse solution as special c

关键词： Nonconvex sparse optimization Iterative thresholding algorithm proximal gradient algorithm Sparse solution Global solution

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Inertial self-adaptive algorithms for solving non-smooth convex optimization problems

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NUMERICAL algorithmS 2025年第1期98卷 133-163页

作者： Chen, Xin Duan, Peichao Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

In this paper, for two different forms of non-smooth convex optimization problems, we investigate the self-adaptive algorithms with inertia acceleration. Firstly, we propose a self-adaptive proximal gradient algorithm with an inertial step. Under reasonable parameters, the strong convergence theorem is established. Secondly, we propose a self-adaptive split proximal algorithm with inertial acceleration. We prove that our algorithm converges strongly under suitable conditions. Notably, both inertial algorithms are extended to multi-step inertial version to accelerate the convergence of the algorithms. Finally, numerical results illustrate the performances of our algorithms.

关键词： Convex optimization Self-adaptive algorithm Inertial acceleration proximal gradient algorithm Split proximal algorithm

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Joint sparse optimization: lower-order regularization method and application in cell fate conversion

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INVERSE PROBLEMS 2024年第9期40卷 095003-095003页

作者： Hu, Yaohua Hu, Xinlin Kwok Wai Yu, Carisa Qin, Jing Shenzhen Univ Sch Math Sci Shenzhen 518060 Peoples R China Guangdong Univ Technol Sch Math & Stat Guangzhou 510520 Peoples R China Hang Seng Univ Hong Kong Dept Math Stat & Insurance Shatin Hong Kong Peoples R China Sun Yat sen Univ Sch Pharmaceut Sci Shenzhen Shenzhen Campus Shenzhen 518107 Guangdong Peoples R China

Multiple measurement signals are commonly collected in practical applications, and joint sparse optimization adopts the synchronous effect within multiple measurement signals to improve model analysis and sparse recovery capability. In this paper, we investigate the joint sparse optimization problem via l(p,q) regularization (0 <= q <= 1 <= p) in three aspects: theory, algorithm and application. In the theoretical aspect, we introduce a weak notion of joint restricted Frobenius norm condition associated with the l(p,q) regularization, and apply it to establish an oracle property and a recovery bound for the l(p,q) regularization of joint sparse optimization problem. In the algorithmic aspect, we apply the well-known proximal gradient algorithm to solve the l(p,q) regularization problems, provide analytical formulas for proximal subproblems of certain specific l(p,q) regularizations, and establish the global convergence and linear convergence rate of the proximal gradient algorithm under some mild conditions. More importantly, we propose two types of proximal gradient algorithms with the truncation technique and the continuation technique, respectively, and establish their convergence to the ground true joint sparse solution within a tolerance relevant to the noise level and the recovery bound under the assumption of restricted isometry property. In the aspect of application, we develop a novel method, based on joint sparse optimization with lower-order regularization and proximal gradient algorithm, to infer the master transcription factors for cell fate conversion, which is a powerful tool in developmental biology and regenerative medicine. Numerical results indicate that the novel method facilitates fast identification of master transcription factors, give raise to the possibility of higher successful conversion rate and in the hope of reducing biological experimental cost.

关键词： joint sparse optimization lower-order regularization consistency theory proximal gradient algorithm global convergence cell fate conversion

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A varying coefficient model with matrix valued covariates

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JOURNAL OF NONPARAMETRIC STATISTICS 2024年第3期36卷 673-705页

作者： Zhang, Hong-Fan Southwest Jiaotong Univ Dept Stat Chengdu Peoples R China

Modern data are often collected in a matrix form. In this paper, we consider modelling the varying coefficient regression with matrix valued covariate X and scalar index variable U. The proposed model simultaneously makes principal component analysis for both the row and column dimensions of the matrix objects, maintaining the matrix structure while achieving substantial dimension reduction. We develop an iterative estimation method for the involved principal parameters and nonparametric functions. Under regularity conditions, the asymptotic distributions of the estimators are derived. In addition, by incorporating the estimation with the adaptive group Lasso and the group SCAD penalties, variables of X in entire rows or columns are selected. The proximal gradient algorithm is further utilised to solve the regularised optimisation problems. The asymptotic properties of the penalised estimators are also studied. Our model and estimation methods are demonstrated by simulated experiments. Real applications to the primary biliary cirrhosis (PBC) data reveal that the effects of the blood measurements to the survival time vary with levels of age.

关键词： Varying coefficient model matrix covariates principal component analysis variable selection proximal gradient algorithm

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A two-phase rank-based algorithm for low-rank matrix completion

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OPTIMIZATION LETTERS 2023年第7期17卷 1679-1695页

作者： Araujo, Tacildo de S. Goncalves, Douglas S. Torezzan, Cristiano Univ Estadual Campinas Dept Appl Math IMECC Campinas Brazil Univ Fed Santa Catarina Dept Math Florianopolis SC Brazil Univ Estadual Campinas Sch Appl Sci Limeira Brazil

Matrix completion aims to recover an unknown low-rank matrix from a small subset of its entries. In many applications, the rank of the unknown target matrix is known in advance and this information can be useful in the completion process. In this paper, first, we revisit a recently proposed rank-based heuristic for "known-rank " matrix completion and establish a condition under which the generated sequence is quasi-Fejer convergent to the solution set. Then, by including an acceleration mechanism similar to Nesterov's acceleration, we obtain a new heuristic. Even though the convergence of this new heuristic cannot be granted in general, it turns out that it can be very useful as a warm-start phase (phase one), providing a suitable estimate for the regularization parameter and a good starting point to an accelerated proximal gradient algorithm (phase two) aimed to solve a nuclear-norm regularized problem. Numerical experiments with both synthetic and real data show that the resulting two-phase rank-based algorithm can recover low-rank matrices, with relatively high precision, faster than other well-established matrix completion algorithms.

关键词： Matrix completion proximal gradient algorithm Soft-thresholding

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Incremental proximal gradient scheme with penalization for constrained composite convex optimization problems

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OPTIMIZATION 2021年第5-6期70卷 1307-1336页

作者： Petrot, Narin Nimana, Nimit Naresuan Univ Fac Sci Dept Math Phitsanulok Thailand Naresuan Univ Fac Sci Ctr Excellence Nonlinear Anal & Optimizat Phitsanulok Thailand Khon Kaen Univ Fac Sci Dept Math Khon Kaen Thailand

We consider the problem of minimizing a finite sum of convex functions subject to the set of minimizers of a convex differentiable function. In order to solve the problem, an algorithm combining the incremental proximal gradient method with smooth penalization technique is proposed. We show the convergence of the generated sequence of iterates to an optimal solution of the optimization problems, provided that a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. Finally, the functionality of the method is illustrated by some numerical experiments addressing image inpainting problems and generalized Heron problems with least squares constraints.

关键词： Convex optimization Fenchel conjugate incremental proximal method penalization proximal gradient algorithm

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Convergence rates of accelerated proximal gradient algorithms under independent noise

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NUMERICAL algorithmS 2019年第2期81卷 631-654页

作者： Sun, Tao Barrio, Roberto Jiang, Hao Cheng, Lizhi Natl Univ Def Technol Dept Math Changsha 410073 Hunan Peoples R China Univ Zaragoza Dept Matemat Aplicada IUMA Grp Pedro Cerbuna 12 E-50009 Zaragoza Spain Univ Zaragoza Dept Matemat Aplicada CoDy Grp Pedro Cerbuna 12 E-50009 Zaragoza Spain Natl Univ Def Technol Coll Comp Changsha 410073 Hunan Peoples R China Natl Univ Def Technol Coll Sci Changsha 410073 Hunan Peoples R China Natl Univ Def Technol State Key Lab High Performance Computat Changsha 410073 Hunan Peoples R China

We consider an accelerated proximal gradient algorithm for the composite optimization with independent errors (errors little related with historical information) for solving linear inverse problems. We present a new inexact version of FISTA algorithm considering deterministic and stochastic noises. We prove some convergence rates of the algorithm and we connect it with the current existing catalyst framework for many algorithms in machine learning. We show that a catalyst can be regarded as a special case of the FISTA algorithm where the smooth part of the function vanishes. Our framework gives a more generic formulation that provides convergence results for the deterministic and stochastic noise cases and also to the catalyst framework. Some of our results provide simpler alternative analysis of some existing results in literature, but they also extend the results to more generic situations.

关键词： proximal gradient algorithm Inexact algorithm Independent noise Convergence rates Acceleration

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Robust Multivariate Lasso Regression with Covariance Estimation

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2023年第3期32卷 961-973页

作者： Chang, Le Welsh, A. H. Australian Natl Univ Res Sch Finance Actuarial Studies & Stat Canberra ACT Australia

Multivariate regression with covariance estimation (MRCE) is a method that performs sparse estimation of multivariate regression coefficients, while taking account the covariance structure of the response variables. MRCE uses a penalized likelihood approach to simultaneously estimate the regression coefficients and the inverse covariance matrix so that prediction accuracy can be significantly improved. However, traditional likelihood-based methods such as MRCE can produce very misleading results in the presence of outliers. In this work, we propose an extension of MRCE, namely, a robust multivariate lasso regression with covariance estimation (RMLC) to handle potential outliers within the data. By using Huber's loss or Tukey's biweight loss, RMLC can be resistant to outliers in the responses or in both the responses and the covariates. A novel optimization algorithm that incorporates a 2-fold accelerated proximal gradient (APG) algorithm is developed to solve RMLC efficiently. We also demonstrate that our proposed RMLC enjoys the oracle property. Our simulation study shows that RMLC produces very reliable results for both the regression coefficients and the correlation structure of the responses, even if the data are contaminated. A real analysis on hyperspectral data further demonstrates the utility of RMLC. Supplementary materials for this article are available online.

关键词： Graphical lasso Lasso Multivariate regression Oracal property proximal gradient algorithm Robust estimation

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