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| 1 | Theory of Compressive Sensing via l1-Minimization:a Non-RIP Analysis and Extensions显示文摘Compressive sensing(CS)is an emerging methodology in computational signal processing that has recently attracted intensive research activities.At present,the basic CS theory includes recoverability and stability:the former quantifies the central fact that a sparse signal of length n can be exactly recovered from far fewer than n measurements via l1-minimization or other recovery techniques,while the latter specifies the stability of a recovery technique in the presence of measurement errors and inexact sparsity.So far,most analyses in CS rely heavily on the Restricted Isometry Property(RIP)for matrices.In this paper,we present an alternative,non-RIP analysis for CS via l1-minimization.Our purpose is three-fold:(a)to introduce an elementary and RIP-free treatment of the basic CS theory;(b)to extend the current recoverability and stability results so that prior knowledge can be utilized to enhance recovery via l1-minimization;and(c)to substantiate a property called uniform recoverability of l1-minimization;that is,for almost all random measurement matrices recoverability is asymptotically identical.With the aid of two classic results,the non-RIP approach enables us to quickly derive from scratch all basic results for the extended theory. | Yin Zhang | 2013 | Journal of the Operations Research Society of China2013，1，1： | 9 |
| 2 | Alternating Direction Method of Multipliers for Sparse Principal Component Analysis显示文摘We consider a convex relaxation of sparse principal component analysisproposed by d' Aspremont et al. (SIAM Rev. 49:434 448, 2007). This convex relax-ation is a nonsmooth semidefinite programming problem in which the ξ1 norm of thedesired matrix is imposed in either the objective function or the constraint to improvethe sparsity of the resulting matrix. The sparse principal component is obtained by arank- one decomposition of the resulting sparse matrix. We propose an alternating di-rection method based on a variable-splitting technique and an augmented I agrangianframework for solving this nonsmooth semidefinite programming problem. In con-trast to the first-order method proposed in d' Aspremont et al. (SIAM Rev. 49:434448, 2007), which solves approximately the dual problem of the original semidefiniteprogramming problem, our method deals with the primal problem directly and solvesit exactly, which guarantees that the resulting matrix is a sparse matrix. A globalconvergence result is established for the proposed method. Numerical results on bothsynthetic problems and the real applications from classification of text data and senatevoting data are reported to demonstrate the efficacy of our method. | Shiqian Ma | 2013 | Journal of the Operations Research Society of China2013，1，2： | 5 |
| 3 | First-Order Algorithms for Convex Optimization with Nonseparable Objective and Coupled Constraints显示文摘In this paper,we consider a block-structured convex optimization model,where in the objective the block variables are nonseparable and they are further linearly coupled in the constraint.For the 2-block case,we propose a number of first-order algorithms to solve this model.First,the alternating direction method of multipliers(ADMM)is extended,assuming that it is easy to optimize the augmented Lagrangian function with one block of variables at each time while fixing the other block.We prove that O(1/t)iteration complexity bound holds under suitable conditions,where t is the number of iterations.If the subroutines of the ADMM cannot be implemented,then we propose new alternative algorithms to be called alternating proximal gradient method of multipliers,alternating gradient projection method of multipliers,and the hybrids thereof.Under suitable conditions,the O(1/t)iteration complexity bound is shown to hold for all the newly proposed algorithms.Finally,we extend the analysis for the ADMM to the general multi-block case. | Xiang Gao Shu-Zhong Zhang | 2017 | Journal of the Operations Research Society of China2017，5，2： | 4 |
| 4 | Pricing Decisions in Dual-Channel Closed-Loop Supply Chain Under Retailer’s Risk Aversion and Fairness Concerns显示文摘This paper studies the price decisions in a dual-channel closed-loop supply chain with a risk-averse retailer and a risk-neutral manufacturer by modeling and analyzing three cases:(1)the retailer does not have fairness concerns;(2)the retailer has fairness concerns and the manufacturer considers it;and(3)the retailer has fairness concerns and the manufacturer does not consider it.The effects of risk aversion and fairness concerns on the pricing decisions,profits and demand are examined in differing scenarios. | Chun-Fa Li Xue-Qing Guo Dong-Lei Du | 2021 | Journal of the Operations Research Society of China2021，9，3： | 4 |
| 5 | On the Linear Convergence of a Proximal Gradient Method for a Class of Nonsmooth Convex Minimization Problems显示文摘We consider a class of nonsmooth convex optimization problems where the objective function is the composition of a strongly convex differentiable function with a linear mapping,regularized by the sum of both l1-norm and l2-norm of the optimization variables.This class of problems arise naturally from applications in sparse group Lasso,which is a popular technique for variable selection.An effective approach to solve such problems is by the Proximal Gradient Method(PGM).In this paper we prove a local error bound around the optimal solution set for this problem and use it to establish the linear convergence of the PGM method without assuming strong convexity of the overall objective function. | Haibin Zhang Jiaojiao Jiang Zhi-Quan Luo | 2013 | Journal of the Operations Research Society of China2013，1，2： | 4 |
| 6 | A Subspace Version of the Powell–Yuan Trust-Region Algorithm for Equality Constrained Optimization显示文摘This paper studied subspace properties of the Celis–Dennis–Tapia(CDT)subproblem that arises in some trust-region algorithms for equality constrained optimization.The analysis is an extension of that presented by Wang and Yuan(Numer.Math.104:241–269,2006)for the standard trust-region subproblem.Under suitable conditions,it is shown that the trial step obtained from the CDT subproblem is in the subspace spanned by all the gradient vectors of the objective function and of the constraints computed until the current iteration.Based on this observation,a subspace version of the Powell–Yuan trust-region algorithm is proposed for equality constrained optimization problems where the number of constraints is much lower than the number of variables. The convergence analysis is given and numerical results arealso reported. | Geovani Nunes Grapiglia Jinyun Yuan Ya-xiang Yuan | 2013 | Journal of the Operations Research Society of China2013，1，4： | 3 |
| 7 | Optimality Conditions of Approximate Solutions for Nonsmooth Semi-infinite Programming Problems显示文摘In this paper,we study optimality conditions of approximate solutions for nonsmooth semi-infinite programming problems.Three new classes of functions,namelyε-pseudoconvex functions of type I and type II andε-quasiconvex functions are introduced,respectively.By utilizing these new concepts,sufficient optimality conditions of approximate solutions for the nonsmooth semi-infinite programming problem are established.Some examples are also presented.The results obtained in this paper improve the corresponding results of Son et al.(J Optim Theory Appl 141:389–409,2009). | Xian-Jun Long Yi-Bin Xiao Nan-Jing Huang | 2018 | Journal of the Operations Research Society of China2018，6，2： | 3 |
| 8 | A Brief Introduction to Manifold Optimization显示文摘Manifold optimization is ubiquitous in computational and appliedmathematics,statistics,engineering,machine learning,physics,chemistry,etc.One of the main challenges usually is the non-convexity of the manifold constraints.By utilizing the geometry of manifold,a large class of constrained optimization problems can be viewed as unconstrained optimization problems on manifold.From this perspective,intrinsic structures,optimality conditions and numerical algorithms for manifold optimization are investigated.Some recent progress on the theoretical results of manifold optimization is also presented. | Jiang Hu Xin Liu Zai-Wen Wen Ya-Xiang Yuan | 2020 | Journal of the Operations Research Society of China2020，8，2： | 3 |
| 9 | Analysis of Stationary Queue Length Distribution for Geo/T-IPH/1 Queue显示文摘In this paper we study a Geo/T-IPH/1 queue model,where T-IPH denotes the discrete time phase type distribution defined on a birth-and-death process with countably many states.The queue model can be described by a quasi-birth-anddeath(QBD)process with countably phases.Using the operator-geometric solution method,we first give the expression of the operator and the joint stationary distribution.Then we obtain the probability generating function(PGF)for stationary queue length distribution and sojourn time distribution,respectively. | Hongbo Zhang Zhenting Hou Dinghua Shi | 2013 | Journal of the Operations Research Society of China2013，1，3： | 3 |
| 10 | A New Analysis on the Barzilai-Borwein Gradient Method显示文摘Due to its simplicity and efficiency,the Barzilai and Borwein(BB)gradi-ent method has received various attentions in different fields.This paper presents a new analysis of the BB method for two-dimensional strictly convex quadratic func-tions.The analysis begins with the assumption that the gradient norms at the first two iterations are fixed.We show that there is a superlinear convergence step in at most three consecutive steps.Meanwhile,we provide a better convergence relation for the BB method.The influence of the starting point and the condition number to the convergence rate is comprehensively addressed. | Yu-Hong Dai | 2013 | Journal of the Operations Research Society of China2013，1，2： | 3 |
| 11 | New Bounds for RIC in Compressed Sensing显示文摘This paper gives new bounds for restricted isometry constant(RIC)in compressed sensing.LetΦbe an m×n real matrix and k be a positive integer with k≤n.The main results of this paper show that if the restricted isometry constant ofΦsat-isfiesδ8ak<1 andδk+ak<3/2−1+√(4a+3)^(2)−8/8aforα>3/8,then k-sparse solution can be recovered exactly via l1 minimization in the noiseless case.In particular,whenα=1,1.5,2 and3,we haveδ2k<0.5746 andδ8k<1,orδ2.5k<0.7046 andδ12k<1,orδ3k<0.7731 andδ16k<1 orδ4k<0.8445 andδ24k<1. | Shenglong Zhou Lingchen Kong Naihua Xiu | 2013 | Journal of the Operations Research Society of China2013，1，2： | 3 |
| 12 | Optimal Control of Nonlinear Switched Systems:Computational Methods and Applications显示文摘A switched system is a dynamic system that operates by switching between different subsystems or modes.Such systems exhibit both continuous and discrete characteristics—a dual nature that makes designing effective control policies a challenging task.The purpose of this paper is to review some of the latest computational techniques for generating optimal control laws for switched systems with nonlinear dynamics and continuous inequality constraints.We discuss computational strategies for optimizing both the times at which a switched system switches from one mode to another(the so-called switching times)and the sequence in which a switched system operates its various possible modes(the so-called switching sequence).These strategies involve novel combinations of the control parameterization method,the timescaling transformation,and bilevel programming and binary relaxation techniques.We conclude the paper by discussing a number of switched system optimal control models arising in practical applications. | Qun Lin Ryan Loxton Kok Lay Teo | 2013 | Journal of the Operations Research Society of China2013，1，3： | 2 |
| 13 | An Approximation Bound Analysis for Lasserre’s Relaxation in Multivariate Polynomial Optimization显示文摘Suppose f,g1,…,gm are multivariate polynomials in x∈R^(n)and their degrees are at most 2d.Consider the problem:Minimize f(x)subject to g1(x)≥0,…,gm(x)≥0.Let f_(min)(resp.,f_(max))be the minimum(resp.,maximum)of f on the feasible set S,and f_(sos)be the lower bound of_(fmin)given by Lasserre’s relaxation of order d.This paper studies its approximation bound.Under a suitable condition on g1,…,gm,we prove that(f_(max)−f_(sos))≤Q(f_(max)−f_(min))with Q a constant depending only on g1,…,gm but not on f.In particular,if S is the unit ball,Q=O(n^(d));if S is the hypercube,Q=O(n^(2d));if S is the boolean set,Q=O(n^(d)). | Jiawang Nie | 2013 | Journal of the Operations Research Society of China2013，1，3： | 2 |
| 14 | The Time-Scaling Transformation Technique for Optimal Control Problems with Time-Varying Time-Delay Switched Systems显示文摘In this paper,we consider a class of optimal control problems where the dynamical systems are time-delay switched systems with the delay being a function of time.By applying the control parameterization method,the control heights and switching times become decision variables that need to be optimized.It is well-known that,for this type problem,the variable switching times cannot be optimized directly.To work around this problem,we introduce a time-scaling transformation technique so that the original system is transformed an equivalent system,which is defined on a new time horizon with fixed switching times.Based on the relationship between the original time scale and the new time scale,we derive the gradients of the objective and constraint functions with respect to the control heights and durations.Then,the new problem can be solved by gradient-based optimization approach.To demonstrate the effectiveness of the time-scaling transformation technique,two example problems are solved. | Ning Zhang Chang-Jun Yu Fu-Sheng Xie | 2020 | Journal of the Operations Research Society of China2020，8，4： | 2 |
| 15 | A Note on“Prospect Theory and the Newsvendor Problem”显示文摘Based on the newsvendor setting,many behavioral models are proposed to predict the biases of decision makers in inventory management.Recently,Nagarajan and Shechter(Manag Sci 60:1057–1062,2014)claimed that prospect theory cannot explain the consistent empirical findings.However,it is noticed that their model is a special case of the general prospect theorymodel.In this note,we showthat the general prospect theory model may be powerful in predicting the preferences of decision makers in inventory management. | Xiao-Bo Zhao Wei Geng | 2015 | Journal of the Operations Research Society of China2015，3，1： | 2 |
| 16 | A Survey on Some Recent Developments of Alternating Direction Method of Multipliers显示文摘Recently, alternating direction method of multipliers (ADMM) attracts much attentions from various fields and there are many variant versions tailored for differentmodels. Moreover, its theoretical studies such as rate of convergence and extensionsto nonconvex problems also achieve much progress. In this paper, we give a surveyon some recent developments of ADMM and its variants. | De-Ren Han | 2022 | Journal of the Operations Research Society of China2022，10，1： | 2 |
| 17 | Optimization for Deep Learning:An Overview显示文摘Optimization is a critical component in deep learning.We think optimization for neural networks is an interesting topic for theoretical research due to various reasons.First,its tractability despite non-convexity is an intriguing question and may greatly expand our understanding of tractable problems.Second,classical optimization theory is far from enough to explain many phenomena.Therefore,we would like to understand the challenges and opportunities from a theoretical perspective and review the existing research in this field.First,we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum and then discuss practical solutions including careful initialization,normalization methods and skip connections.Second,we review generic optimization methods used in training neural networks,such as stochastic gradient descent and adaptive gradient methods,and existing theoretical results.Third,we review existing research on the global issues of neural network training,including results on global landscape,mode connectivity,lottery ticket hypothesis and neural tangent kernel. | Ruo-Yu Sun | 2020 | Journal of the Operations Research Society of China2020，8，2： | 2 |
| 18 | On the l_(1)-Norm Invariant Convex k-Sparse Decomposition of Signals显示文摘Inspired by an interesting idea of Cai and Zhang,we formulate and prove the convex k-sparse decomposition of vectors that is invariant with respect to the l_(1) norm.This result fits well in discussing compressed sensing problems under the Restricted Isometry property,but we believe it also has independent interest.As an application,a simple derivation of the RIP recovery conditionδk+θk,k<1 is presented. | Guangwu Xu Zhiqiang Xu | 2013 | Journal of the Operations Research Society of China2013，1，4： | 2 |
| 19 | Conditional Quadratic Semidefinite Programming:Examples and Methods显示文摘The conditional quadratic semidefinite programming(cQSDP)refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace,and the objectives are quadratic.The chief purpose of this paper is to focus on two primal examples of cQSDP:the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem.For the latter problem,we review some classical contributions and establish certain links among them.Moreover,we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy.We also include an application in calibrating the correlation matrix in Libor market models.We hope this work will stimulate new research in cQSDP. | Hou-Duo Qi | 2014 | Journal of the Operations Research Society of China2014，2，2： | 2 |
| 20 | Equivalence and Strong Equivalence Between the Sparsest and Least l1-Norm Nonnegative Solutions of Linear Systems and Their Applications显示文摘Many practical problems can be formulated as l0-minimization problems with nonnegativity constraints,which seek the sparsest nonnegative solutions to underdetermined linear systems.Recent study indicates that l1-minimization is efficient for solving l0-minimization problems.From a mathematical point of view,however,the understanding of the relationship between l0-and l1-minimization remains incomplete.In this paper,we further address several theoretical questions associated with these two problems.We prove that the fundamental strict complementarity theorem of linear programming can yield a necessary and sufficient condition for a linear system to admit a unique least l1-norm nonnegative solution.This condition leads naturally to the so-called range space property(RSP)and the “full-column-rank”property,which altogether provide a new and broad understanding of the equivalence and the strong equivalence between l0-and l1-minimization.Motivated by these results,we introduce the concept of “RSP of order K”that turns out to be a full characterization of uniform recovery of all K-sparse nonnegative vectors.This concept also enables us to develop a nonuniform recovery theory for sparse nonnegative vectors via the so-called weak range space property. | Yun-Bin Zhao | 2014 | Journal of the Operations Research Society of China2014，2，2： | 2 |
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