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EXACT augmented lagrangian function FOR NONLINEAR PROGRAMMING PROBLEMS WITH INEQUALITY CONSTRAINTS

Applied Mathematics and Mechanics(English Edition) 2005年第12期26卷 1649-1656页

作者：杜学武张连生尚有林李铭明 Department of Mathematics Shanghai University

An exact augmented lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, f^om the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented lagrangian function presented.

关键词： local minimizer global minimizer nonlinear programming exact penalty function augmented lagrangian function

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Non-intrusive Load Monitoring Based on Graph Total Variation for Residential Appliances

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Journal of Modern Power Systems and Clean Energy 2024年第3期12卷 947-957页

作者： Xiaoyang Ma Diwen Zheng Xiaoyong Deng Ying Wang Dawei Deng Wei Li the College of Electrical Engineering Sichuan UniversityChengdu 610065China State Grid Chongqing Electric Power Company Chongqing 400014China State Grid Sichuan Electric Power Company Chengdu 610095China State Grid Hubei Electric Power Company Limited Research Institute Wuhan 430077China

Non-intrusive load monitoring is a technique for monitoring the operating conditions of electrical appliances by collecting the aggregated electrical information at the household power *** several studies on the mining of unique load characteristics,few studies have extensively considered the high computational burden and sample *** on lowfrequency sampling data,a non-intrusive load monitoring algorithm utilizing the graph total variation(GTV)is proposed in this *** algorithm can effectively depict the load state without the need for prior ***,the combined Kmeans clustering algorithm and graph signals are used to build concise and accurate graph structures as load *** GTV representing the internal structure of the graph signal is introduced as the optimization model and solved using the augmented lagrangian iterative *** introduction of the difference operator reduces the computing cost and addresses the inaccurate reconstruction of the graph *** low-frequency sampling data,the algorithm only requires a little prior data and no training,thereby reducing the computing *** conducted using the reference energy disaggregation dataset and almanac of minutely power dataset demonstrated the stable superiority of the algorithm and its low computational burden.

关键词： Non-intrusive load monitoring graph total variation augmented lagrangian function smart grid

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On Iteration Complexity of a First-Order Primal-Dual Method for Nonlinear Convex Cone Programming

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Journal of the Operations Research Society of China 2022年第1期10卷 53-87页

作者： Lei Zhao Dao-Li Zhu School of Naval Architecture Ocean and Civil EngineeringShanghai Jiao Tong UniversityShanghai 200240China Antai College of Economics and Management and Sino-US Global Logistics Institute Shanghai Jiao Tong UniversityShanghai 200230China School of Data Science The Chinese University of Hong KongShenzhen 518172GuangdongChina

Nonlinear convex cone programming(NCCP)models have found many practical *** this paper,we introduce a flexible first-order primal-dual algorithm,called the variant auxiliary problem principle(VAPP),for solving NCCP problems when the objective function and constraints are convex but may be *** each iteration,VAPP generates a nonlinear approximation of the primal augmented lagrangian *** approximation incorporates both linearization and a distance-like proximal term,and then the iterations of VAPP are shown to possess a decomposition property for *** by recent applications in big data analytics,there has been a growing interest in the convergence rate analysis of algorithms with parallel computing capabilities for large scale optimization *** establish O(1/t)convergence rate towards primal optimality,feasibility and dual *** adaptively setting parameters at different iterations,we show an O(1/t2)rate for the strongly convex ***,we discuss some issues in the implementation of VAPP.

关键词： Nonlinear convex cone programming First-order method Primal-dual method augmented lagrangian function

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Sparse Estimation of High-Dimensional Inverse Covariance Matrices with Explicit Eigenvalue Constraints

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Journal of the Operations Research Society of China 2021年第3期9卷 543-568页

作者： Yun-Hai Xiao Pei-Li Li Sha Lu Institute of Applied Mathematics College of Mathematics and StatisticsHenan UniversityKaifeng 475000HenanChina School of Statistics Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOEEast China Normal UniversityShanghai 200062China Department of Mathematics Nanning Normal UniversityNanning 530299GuangxiChina

Firstly,this paper proposes a generalized log-determinant optimization model with the purpose of estimating the high-dimensional sparse inverse covariance *** the normality assumption,the zero components in the inverse covariance matrices represent the conditional independence between pairs of variables given all the other *** generalized model considered in this study,because of the setting of the eigenvalue bounded constraints,covers a large number of existing estimators as special ***,rather than directly tracking the challenging optimization problem,this paper uses a couple of alternating direction methods of multipliers(ADMM)to solve its dual model where 5 separable structures are *** first implemented algorithm is based on a single Gauss–Seidel iteration,but it does not necessarily converge *** contrast,the second algorithm employs the symmetric Gauss–Seidel(sGS)based ADMM which is equivalent to the 2-block iterative scheme from the latest sGS decomposition ***,we do numerical simulations using the synthetic data and the real data set which show that both algorithms are very effective in estimating high-dimensional sparse inverse covariance matrix.

关键词： Non-smooth convex minimization Inverse covariance matrix Maximum likelihood estimation augmented lagrangian function Symmetric Gauss–Seidel iteration

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AN augmented lagrangian TRUST REGION METHOD WITH A BI-OBJECT STRATEGY

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Journal of Computational Mathematics 2018年第3期36卷 331-350页

作者： Caixia Kou Zhongwen Chen Yuhong Dai Haifei Han School of Sciences Beijing University of Posts and Telecommunications Beijing 100876 China School of Mathematical Sciences Soochow University Suzhou 215006 China LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China School of Mathematical Sciences Soochow University Suzhou 215006 China

An augmented lagrangian trust region method with a bi=object strategy is proposed for solving nonlinear equality constrained optimization, which falls in between penalty-type methods and penalty-free ones. At each iteration, a trial step is computed by minimizing a quadratic approximation model to the augmented lagrangian function within a trust region. The model is a standard trust region subproblem for unconstrained optimization and hence can efficiently be solved by many existing methods. To choose the penalty parameter, an auxiliary trust region subproblem is introduced related to the constraint violation. It turns out that the penalty parameter need not be monotonically increasing and will not tend to infinity. A bi-object strategy, which is related to the objective function and the measure of constraint violation, is utilized to decide whether the trial step will be accepted or not. Global convergence of the method is established under mild assumptions. Numerical experiments are made, which illustrate the efficiency of the algorithm on various difficult situations.

关键词： Nonlinear constrained optimization augmented lagrangian function Bi-object strategy Global convergence.

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augmented lagrangian Alternating Direction Method for Tensor RPCA

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Journal of Mathematical Research with Applications 2017年第3期37卷 367-378页

作者： Ruru HAO Zhixun SU Department of Mathematics Dalian University of Technology

Tensor robust principal component analysis(TRPCA) problem aims to separate a low-rank tensor and a sparse tensor from their sum. This problem has recently attracted considerable research attention due to its wide range of potential applications in computer vision and pattern recognition. In this paper, we propose a new model to deal with the TRPCA problem by an alternation minimization algorithm along with two adaptive rankadjusting strategies. For the underlying low-rank tensor, we simultaneously perform low-rank matrix factorizations to its all-mode matricizations; while for the underlying sparse tensor,a soft-threshold shrinkage scheme is applied. Our method can be used to deal with the separation between either an exact or an approximate low-rank tensor and a sparse one. We established the subsequence convergence of our algorithm in the sense that any limit point of the iterates satisfies the KKT conditions. When the iteration stops, the output will be modified by applying a high-order SVD approach to achieve an exactly low-rank final result as the accurate rank has been calculated. The numerical experiments demonstrate that our method could achieve better results than the compared methods.

关键词： tensor RPCA alternating direction method augmented lagrangian function high-order SVD

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