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A global two-stage algorithm for non-convex penalized high-dimensional linear regression problems

COMPUTATIONAL STATISTICS 2023年第2期38卷 871-898页

作者： Li, Peili Liu, Min Yu, Zhou East China Normal Univ KLATASDS MOE Sch Stat Shanghai 200062 Peoples R China Wuhan Univ Sch Math & Stat Wuhan 430072 Peoples R China

By the asymptotic oracle property, non-convex penalties represented by minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD) have attracted much attentions in high-dimensional data analysis, and have been widely used in signal processing, image restoration, matrix estimation, etc. However, in view of their non-convex and non-smooth characteristics, they are computationally challenging. Almost all existing algorithms converge locally, and the proper selection of initial values is crucial. Therefore, in actual operation, they often combine a warm-starting technique to meet the rigid requirement that the initial value must be sufficiently close to the optimal solution of the corresponding problem. In this paper, based on the DC (difference of convex functions) property of MCP and SCAD penalties, we aim to design a global two-stage algorithm for the high-dimensional least squares linear regression problems. A key idea for making the proposed algorithm to be efficient is to use the primal dual active set with continuation (PDASC) method to solve the corresponding sub-problems. Theoretically, we not only prove the global convergence of the proposed algorithm, but also verify that the generated iterative sequence converges to a d-stationary point. In terms of computational performance, the abundant research of simulation and real data show that the algorithm in this paper is superior to the latest SSN method and the classic coordinate descent (CD) algorithm for solving non-convex penalized high-dimensional linear regression problems.

关键词： High-dimensional linear regression Global convergence Two-stage algorithm Primal dual active set with continuation algorithm difference of convex functions

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Completion of Traffic Matrix by Tensor Nuclear Norm Minus Frobenius Norm Minimization and Time Slicing

Completion of Traffic Matrix by Tensor Nuclear Norm Minus Fr...

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IEEE/IFIP Network Operations and Management Symposium (NOMS)

作者： Miyata, Takamichi Chiba Inst Technol 2-17-1 Tsudanuma Narashino Chiba Japan

ISBN: (纸本)9798350327939;9798350327946

The goal of traffic matrix completion is to estimate the elements of the network traffic matrix that could not be observed. Conventional methods can be divided into mathematical optimization-based methods and machine learning-based methods, with the former suffering from low prediction performance and the latter depending on the dataset and missing patterns used for training. We propose an optimization-based traffic matrix completion method with high accuracy that uses the nuclear norm minus Frobenius norm, which has recently been shown to be effective in image denoising. The proposed method combines the construction of a tensor by dividing the traffic matrix by a predetermined time interval and the introducing of regularization by a newly defined tensor nuclear norm minus Frobenius norm. Experimental results show that the proposed method achieves traffic matrix completion comparable to or better than state-of-the-art neural network-based methods, without the need for prior training.

关键词： Traffic matrices tensor completion non-convex optimization difference of convex functions nuclear norm minus Frobenius norm

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The Boosted DC Algorithm for Linearly Constrained DC Programming

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SET-VALUED AND VARIATIONAL ANALYSIS 2022年第4期30卷 1265-1289页

作者： Aragon-Artacho, F. J. Campoy, R. Vuong, P. T. Univ Alicante Dept Math Alicante 03690 Spain Univ Valencia Dept Stat & Operat Res Valencia Spain Univ Southampton Math Sci Sch Southampton England

The Boosted difference of convex functions Algorithm (BDCA) has been recently introduced to accelerate the performance of the classical difference of convex functions Algorithm (DCA). This acceleration is achieved thanks to an extrapolation step from the point computed by DCA via a line search procedure. In this work, we propose an extension of BDCA that can be applied to difference of convex functions programs with linear constraints, and prove that every cluster point of the sequence generated by this algorithm is a Karush-Kuhn-Tucker point of the problem if the feasible set has a Slater point. When the objective function is quadratic, we prove that any sequence generated by the algorithm is bounded and R-linearly (geometrically) convergent. Finally, we present some numerical experiments where we compare the performance of DCA and BDCA on some challenging problems: to test the copositivity of a given matrix, to solve one-norm and infinity-norm trust-region subproblems, and to solve piecewise quadratic problems with box constraints. Our numerical results demonstrate that this new extension of BDCA outperforms DCA.

关键词： difference of convex functions Boosted difference of convex functions algorithm Global convergence Constrained DC program Copositivity problem Trust region subproblem

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Optimization of Workload Balancing and Power Allocation for Wireless Distributed Computing

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2022年第9期21卷 7682-7695页

作者： Sun, Chen Gao, Xiqi Ding, Zhi Southeast Univ Natl Mobile Commun Res Lab Nanjing 210096 Peoples R China Purple Mt Labs Nanjing 211100 Peoples R China Univ Calif Davis Dept Elect & Comp Engn Davis CA 95616 USA

Distributed computing systems, such as Hadoop, have been widely studied and used for executing and analyzing large data. In this paper, we investigate an emerging resource allocation problem for wireless distributed computing systems consisting of multifunctional nodes in charge of both numerical computation and wireless communication with master nodes. We focus on a computation power consumption model based on CMOS devices and a communication power consumption model involving multiple antenna transceivers against mutual interference. We present a joint optimization problem for workload scheduling and power allocation for achieving maximum computational speed under total power constraint. We simplify the joint optimization into two sub-problems. For workload scheduling as an integer programming sub-problem, we relax the integer constraint and establish the equivalence between relaxed and original problems. For the power allocation sub-problem, we maximize a difference of convex functions by utilizing the concave-convex procedure. We prove our proposed algorithm to converge to a stationary point of the original program. Simulation results confirm the efficiency and near-optimal performance of our proposed algorithms.

关键词： Resource management Wireless communication Task analysis Power demand Processor scheduling Optimization Distributed computing difference of convex functions power allocation wireless distributed computing workload scheduling Hadoop

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A DC approach for minimax fractional optimization programs with ratios of convex functions

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OPTIMIZATION METHODS & SOFTWARE 2022年第2期37卷 639-657页

作者： Ghazi, A. Roubi, A. Univ Hassan 1 Lab MISI Fac Sci & Tech Settat Morocco

This paper deals with minimax fractional programs whose objective functions are the maximum of finite ratios of convex functions, with arbitrary convex constraints set. For such problems, Dinkelbach-type algorithms fail to work since the parametric subproblems may be nonconvex, whereas the latter need a global optimal solution of these subproblems. We give necessary optimality conditions for such problems, by means of convex analysis tools. We then propose a method, based on solving approximately a sequence of parametric convex problems, which acts as dc (difference of convex functions) algorithm, if the parameter is positive and as Dinkelbach algorithm if not. We show that every cluster point of the sequence of optimal solutions of these subproblems satisfies necessary optimality conditions of KKT criticality type, that are also of Clarke stationarity type. Finally we end with some numerical tests to illustrate the behaviour of the algorithm.

关键词： Fractional programming quotient of convex functions difference of convex functions optimality conditions Dinkelbach algorithms

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Solving a continuous multifacility location problem by DC algorithms

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OPTIMIZATION METHODS & SOFTWARE 2022年第1期37卷 338-360页

作者： Bajaj, Anuj Mordukhovich, Boris S. Nam, Nguyen Mau Tran, Tuyen Wayne State Univ Dept Math Detroit MI 48202 USA Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97207 USA

The paper presents a new approach to solve multifacility location problems, which is based on mixed integer programming and algorithms for minimizing differences of convex (DC) functions. The main challenges for solving the multifacility location problems under consideration come from their intrinsic discrete, nonconvex, and nondifferentiable nature. We provide a reformulation of these problems as those of continuous optimization and then develop a new DC type algorithm for their solutions involving Nesterov's smoothing. The proposed algorithm is computationally implemented via MATLAB numerical tests on both artificial and real data sets.

关键词： Mixed integer programming multifacility location difference of convex functions Nesterov's smoothing the DCA

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COPOSITIVITY MEETS D. C. OPTIMIZATION

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JOURNAL OF DYNAMICS AND GAMES 2022年第1期9卷 27-32页

作者： Moudafi, Abdellatif Mainge, Paul-Emile Aix Marseille Univ CNRS UMR 7020 AMUUTLNLab Informat & Syst LIS Marseille France Univ Antilles F-97233 Martinique France

Based on a work by M. Dur and J.-B. Hiriart-Urruty[3], we consider the problem of whether a symmetric matrix is copositive formulated as a difference of convex functions problem. The convex nondifferentiable function in this d.c. decomposition being proximable, we then apply a proximal-gradient method to approximate the related stationary points. Whereas, in [3], the DCA algorithm was used.

关键词： Copositive matrices difference of convex functions proximal-gradient algorithm DC optimization critical point

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Representations of Continuous Piecewise Affine functions

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VESTNIK ST PETERSBURG UNIVERSITY-MATHEMATICS 2022年第1期55卷 39-47页

作者： Malozemov, V. N. Tamasyan, G. Sh. St Petersburg State Univ St Petersburg 199034 Russia Russian Acad Sci Inst Problems Mech Engn St Petersburg 199178 Russia

Continuous piecewise affine functions are widely used in computational mathematics. In the one-dimensional case, such functions are called broken lines. The paper analyzes the analytical representations of broken lines both in the forms accepted in the theory of polynomial splines and in the form of the difference of the maxima of two finite families of affine functions. We establish a correlation between these representations.

关键词： piecewise affine function broken line analytical representations of broken lines difference of convex functions

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Comparative study of non-convex penalties and related algorithms in compressed sensing

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DIGITAL SIGNAL PROCESSING 2023年第1期135卷

作者： Xu, Fanding Duan, Junbo Liu, Wenyu Xi An Jiao Tong Univ Sch Life Sci & Technol Xian 710049 Peoples R China Xi An Jiao Tong Univ Key Lab Biomed Informat Engn Minist Educ Xian 710049 Peoples R China

Non-convex penalties are widely used in sparse representation models and fields such as compressed sensing, because they can efficiently recover signals with a high sparsity level compared with the well-known B1-norm. However, few classic penalties have been selected in most applications without exploring the performance of other non-convex penalties. Therefore, the objective of this study is to provide a comparative and comprehensive study on non-convex penalties. We first collected nine representative non-convex penalties and then applied two optimization frameworks: the difference of convex function algorithm (DCA) and iterative shrinkage/thresholding algorithm (ISTA) to solve the optimization problems associated with the nine non-convex penalties. The performances of the nine penalties and two algorithms were systematically compared and analyzed experimentally. We hope that these results that concern non-convex penalty selection in sparse representation models can be used as a general reference for research.(c) 2023 Elsevier Inc. All rights reserved.

关键词： Compressed sensing Non -convex penalty Sparse representation difference of convex functions Iterative thresholding

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A Generalized Non-convex Method for Robust Tensor Completion

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JOURNAL OF SCIENTIFIC COMPUTING 2023年第3期96卷 91-91页

作者： Zhang, Zhechen Liu, Sanyang Lin, Zhiping Xidian Univ Sch Math & Stat Xian 710126 Peoples R China Nanyang Technol Univ Sch Elect & Elect Engn Singapore 639798 Singapore

In this paper, we concentrate on the robust tensor completion (RTC) problem, which aims to recover a low-rank tensor from partial observations corrupted by sparse noise. Most existing methods for RTC utilize the tensor nuclear norm (TNN) to evaluate the tensor rank. However, the TNN often yields biased solutions due to its loose approximation for the tensor rank. To solve this problem, we derive a unified non-convex surrogate that better approximates the tensor rank. Our surrogate is composed of several non-convex penalty functions. Further, we propose a generalized non-convex model, which minimizes a weighted combination of the unified non-convex surrogate and the t1-norm data fidelity term. To solve the proposed model, we devise a simple but efficient algorithm called the proximal alternating difference of convex functions (PADCF) algorithm. Moreover, we prove the sequence generated by the PADCF algorithm converges to the critical point under some mild conditions. Numerical experiments are provided for illustrations and comparisons.

关键词： Robust tensor completion Transformed tensor nuclear norm Low-tubal-rank Non-convex difference of convex functions

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