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Asymptotic Performance of Sparse Signal Detection Using convex programming Method

Chinese Journal of Aeronautics 2012年第3期25卷 396-405页

作者： LEI Chuan ZHANG Jun School of Electronics and Information Engineering Beihang UniversityBeijing 100191China National Key Laboratory of CNS/ATM Beijing 100191China

The detection of sparse signals against background noise is considered. Detecting signals of such kind is difficult since only a small portion of the signal carries information. Prior knowledge is usually assumed to ease detection. In this paper, we consider the general unknown and arbitrary sparse signal detection problem when no prior knowledge is available. Under a Ney- man-Pearson hypothesis-testing framework, a new detection scheme is proposed by combining a generalized likelihood ratio test （GLRT）-Iike test statistic and convex programming methods which directly exploit sparsity in an underdetermined system of linear equations. We characterize large sample behavior of the proposed method by analyzing its asymptotic performance. Specifically, we give the condition for the Chernoff-consistent detection which shows that the proposed method is very sensitive to the norm energy of the sparse signals. Both the false alam rate and the miss rate tend to zero at vanishing signal-to-noise ratio （SNR）, as long as the signal energy grows at least logarithmically with the problem dimension. Next we give a large deviation analysis to characterize the error exponent for the Neyman-Pearson detection. We derive the oracle error exponent assuming signal knowledge. Then we explicitly derive the error exponent of the proposed scheme and compare it with the oracle exponent. We complement our study with numerical experiments, showing that the proposed method performs in the vicinity of the likelihood ratio test （LRT） method in the finite sample scenario and the error probability degrades exponentially with the number of observations.

关键词： signal detection convex programming asymptotic analysis signal reconstruction sparse signals

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NEWTON METHOD FOR SOLVING A CLASS OF SMOOTH convex programming

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Applied Mathematics and Mechanics(English Edition) 2005年第11期26卷 1491-1498页

作者：姚奕荣张连生韩伯顺 DAI Shi-qiang Department of Mathematics Shanghai University Shanghai 200444 P. R. China 不详

An algorithm for solving a class of smooth convex programming is given. Using smooth exact multiplier penalty function, a smooth convex programming is minimized to a minimizing strongly convex function on the compact ... 详细信息

关键词： convex programming Newton method KKT multiplier

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Relaxed inertial proximal Peaceman-Rachford splitting method for separable convex programming

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Frontiers of Mathematics in China 2018年第3期13卷 555-578页

作者： Yongguang HE Huiyun LI Xinwei LIU Institute of Mathematics Hebei University of Technology Tianjin 300401 China School of Control Science and Engineering Hebei University of Technology Tianjin 300401 China

The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman- Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman- Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.

关键词： convex programming inertial proximal Peaceman-Rachford splitting method relaxation factor global convergence

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PARALLEL MULTIPLICATIVE ITERATIVE METHODS FOR convex programming

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Acta Mathematica Scientia 1997年第2期17卷 205-210页

作者：陈忠费浦生 WUHAN UNIV DEPT MATH WUHAN 430072 PEOPLES R CHINA

In this paper, we present two parallel multiplicative algorithms for convex programming. If the objective function has compact level sets and has a locally Lipschitz continuous gradient, we discuss convergence of the ... 详细信息

关键词： parallel algorithm convex programming

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Block-Wise ADMM with a Relaxation Factor for Multiple-Block convex programming

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Journal of the Operations Research Society of China 2018年第4期6卷 485-505页

作者： Bing-Sheng He Ming-Hua Xu Xiao-Ming Yuan Department of Mathematics South University of Science and TechnologyShenzhen 518055GuangdongChina Department of Mathematics Nanjing UniversityNanjing 210093China School of Mathematics and Physics Changzhou UniversityChangzhou 213164JiangsuChina Department of Mathematics The University of Hong KongHong KongChina

It has been shown that the alternating direction method of multipliers(ADMM)is not necessarily convergent when it is directly extended to a multiple-block linearly constrained convex minimization model with an objective function that is in the sum of more than two functions without coupled ***,we pro-posed the block-wise ADMM,which was obtained by regrouping the variables and functions of such a model as two blocks and then applying the original ADMM in *** note is a further study on this topic with the purpose of showing that a well-known relaxation factor proposed by Fortin and Glowinski for iteratively updat-ing the Lagrangian multiplier of the originalADMMcan also be used in the block-wise *** thus propose the block-wise ADMM with Fortin and Glowinski’s relax-ation factor for the multiple-block convex minimization *** the block-wise ADMM,we also suggest further decomposing the resulting subproblems and regular-izing them by proximal terms to ensure the *** the block-wise ADMM with Fortin and Glowinski's relaxation factor,its convergence and worst-case conver-gence rate measured by the iteration complexity in the ergodic sense are derived.

关键词： convex programming Operator splitting methods Alternating direction method of multipliers Fortin and Glowinski’s relaxation factor

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The Second-Order Differential Equation System with the Feedback Controls for Solving convex programming

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Open Journal of Applied Sciences 2022年第6期12卷 977-989页

作者： Xingxu Chen Li Wang Juhe Sun Yanhong Yuan School of Science Shenyang Aerospace University Shenyang China College of Economics and Management Taiyuan University of Technology Taiyuan China

In this paper, we establish the second-order differential equation system with the feedback controls for solving the problem of convex programming. Using Lagrange function and projection operator, the equivalent operator equations for the convex programming problems under the certain conditions are obtained. Then a second-order differential equation system with the feedback controls is constructed on the basis of operator equation. We prove that any accumulation point of the trajectory of the second-order differential equation system with the feedback controls is a solution to the convex programming problem. In the end, two examples using this differential equation system are solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the feedback controls for solving the convex programming problem.

关键词： convex programming Lagrange Function Projection Operator Second-Order Differential Equation

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Incorrect Results for E-convex Functions and E-convex programming

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Journal of Mathematical Research and Exposition 2003年第3期23卷 461-466页

作者：简金宝广西大学数学与信息科学系广西南宁530004

A class of functions and a sort of nonlinear programming called respectively E-convex functions and E-convex programming were presented and studied recently by Youness in [1], In this paper, we point out the most resu... 详细信息

关键词： convex sets convex functions convex programming generalized converx-ity.

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Research of Urban Green Space Distribution Based on convex programming Model

Research of Urban Green Space Distribution Based on Convex P...

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第26届中国控制与决策会议

作者： Yue Xiao-ning Wu Meng-xiang Ding Yue College of Normal Shenyang University College of Information Engineering Shenyang University

The product diffusion theory and the optimization theory are introduced to research the quantification of the livable index satisfaction for green coverage of urban and performed the quantitative analysis on the rationality of distribution. Using the statistical curve regression theory and mathematical statistics software SPSS, the utility function of livable index satisfaction for green coverage of urban is analyzed with using the curve of Logistic regression. In which urban residents are regarded as the subject, a convex programming model for green coverage ratio of urban is established, and the existence condition for the optimal solution of convex programming is proposed. In the meantime, the optimum distributions of green coverage are discussed. Some references for the theory and practice of livable urban are provided by the results of the study.

关键词： Green Coverage Livable City Utility Function convex programming Logistic Curve

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A Trust Region Algorithm for a Class of Nonsmooth convex programming

A Trust Region Algorithm for a Class of Nonsmooth Convex Pro...

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2001年全国数学规划及运筹研讨会

作者： Yang Yinmin(Anhui Mechanical and Electrical College, Wuhu 241000 China)

Author give a trust region algorithm for a class of nonsmooth conuex pvogremming.

关键词： nonsmooth convex programming trust region algorithm global con-vergence.

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一类凸规划的多项式预估校正内点法

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高等学校计算数学学报 2005年第3期27卷 5-13页

作者：余谦黄崇超王先甲武汉大学数学与统计学院武汉大学系统工程研究所武汉大学系统工程研究所武汉 430072

1引言1990年由Mehrotra对线性规划问题提出了一个称为预估校正的方法,并在1992年给出了其数值算法.1993年Mizuno,Todd和***.给出了改进的预估校正内点法,使得一个预估步后只跟一个校正步.1994年***给出了不可行预估校正内点法,使得可以... 详细信息

1引言1990年由Mehrotra对线性规划问题提出了一个称为预估校正的方法,并在1992年给出了其数值算法.1993年Mizuno,Todd和***.给出了改进的预估校正内点法,使得一个预估步后只跟一个校正步.1994年***给出了不可行预估校正内点法,使得可以从一个不可行的初始点开始算法的迭代,并证明了其为二次收敛.实际上预估校正内点法等价于应用简化一步牛顿法与一步牛顿法的合成方法求解该

关键词： convex programming predictor-corrector interior-point algorithm.

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