Geometric interpretation of several classical iterative methods for linear system of equations and diverse relaxation parameter of the SOR method

Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterative methods and provide a reference for further study and design. Finally, a new iterative method is designed named as the diverse relaxation parameter of the SOR method which, in particular, demonstrates the geometric characteristics. Many examples prove that the method is quite effective.

LINEARIZATION RELAXATION PROCEDURE FOR A KIND OF N-DIMENSIONAL NONLINEAR OPTIMIZATION PROBLEM

This paper is concerned with a class of convex multivariable nonlinear program problems. By virtue of linearization philosophy, a linearization problem (LP) is constructed and theoretical equivalence between (LP) and the original problem established. Based on relaxation techniques an algorithm for solving (LP) is proposed, which is efficient from a computational viewpoint, since at each iteration the only program that needs to solve is a standard linear program. Furthermore, the optimality criterion is derived. The convergence analysis conducted in this paper indicates that the algorithm guarantees finite ε convergence.

Continuous Iteratively Reweighted Least Squares Algorithm for Solving Linear Models by Convex Relaxation

In this paper, we present continuous iteratively reweighted least squares algorithm (CIRLS) for solving the linear models problem by convex relaxation, and prove the convergence of this algorithm. Under some conditions, we give an error bound for the algorithm. In addition, the numerical result shows the efficiency of the algorithm.

Measuring method of water surface wave relaxation rate based on linear array CCD

Relaxation rate is a very crucial parameter in physics.For the water surface wave,its relaxation rate is directly relevantto the response time of disturbed spectrum returning back to its quasi-steady state.It is difficult to be calculated directly asa function of different oceanographic and meteorological parameters.Previous researches were mainly based on experimentalmeasurements or parameterization.In this paper,a method based on the liner array charge-coupled device(CCD)is proposed tomeasure the relaxation rate of the water surface wave.Compared with the traditional methods?it can obtain the information ofsurface wave and current synchronously,and works well under a multi wind-wave environment.Wind wav^tank experimentswere carried out based on this method.The good consistency between the results calculated by this method and the traditionalrelaxation rate models shows the validity of the proposed method.This method can be further used to study the modulation theoryof surface waves by currents.

STABLE NONLINEAR RELAXATIONS IN A MULTIPHASIC Al-Zn ALLOY

The experiments reveal the characteristics of stable damping in a multiphasic Al-Zn eutectoid alloy:（1）Thewhole damping(Q-1)has the same dependence on measured frequency（f）,i.e.Q-1ocf-n,where n is a parameterand is independent of temperature.（2）In a low-temperature（low-T）and low-strain-amplitude（low-A）region,Q-1=（B/）exp（-nH/kT）,where B is a parameter,H the atomic diffusion activation energy,k Boltzmann′sconstant,and T the absolute temperature.n,Ho（=nH）and H are all independent of A.The damping comesfrom an anelastic motion of the phase-interface.（3）In an intermediate region including a low-Tand a high-A,a middle-T and middle A and a high-T and low-A regions,the equation Q-1=（C/fn）exp（nH/kT）stillholds,but the damping has a normal amplitude effect C,n,and H all vary with A;the damping results from anonlinear relaxation of phase-interface.（4）In a high-T and high-A region,there is no longer a linear relationship between InQ-1 and T-1,whereas the relation Q-1f-n is still satisfied,where n increases as A increases,andthe damping has a normal amplitude effect but one which is weaker than that in the case（3）.The damping maybe attributed to another kind of nonlinear relaxation between phase-interfaces.

A CLASS OF GENERALIZED MULTISPLITTING RELAXATION METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

Abstract In this paper,a class of generalized parallel matrix multisplitting relaxation methods for solving linear complementarity problems on the high speed multiprocessor systems is set up.This class of methods not only includes all the existing relaxation methods for the linear complementarity problems,but also yields a lot of novel ones in the sense of multisplitting.We establish the convergence theories of this class of generalized parallel multisplitting relaxation methods under the condition that the system matrix is an H matrix with positive diagonal elements.

GENERALIZED MATRIX MULTISPLITTING RELAXATION METHODS AND THEIR CONVERGENCE

In this paper, we set up a general framework of parallel matrix mullisplitting relaxation methods for solving large scale system of linear equations. We investigate the convergence properties of this framework and give several sufficient conditions ensuring it to converge as well as diverge. At last, we conclude a necessary and sufficient condition for the convergence of this framework when the coefficient matrix is an L-matrix.

ASYNCHRONOUS RELAXED ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS OF EQUATIONS

In this paper, the asynchronous versions of classical iterative methods for solving linear systems of equations are considered. Sufficient conditions for convergence of asynchronous relaxed processes are given for H-matrix by which nor only the requirements of [3] on coefficient matrix are lowered, but also a larger region of convergence than that in [3] is obtained.

A Relaxed Greedy Block Kaczmarz Method for Solving Large Consistent Linear Systems

Many problems in science and engineering require solving large consistent linear systems. This paper presents a relaxed greedy block Kaczmarz method (RGBK) and an accelerated greedy block Kaczmarz method (AGBK) for solving large-size consistent linear systems. The RGBK algorithm extends the greedy block Kaczmarz algorithm (GBK) presented by Niu and Zheng in [1] by introducing a relaxation parameter to the iteration formulation of GBK, and the AGBK algorithm uses different iterative update rules to minimize the running time. The convergence of the RGBK is proved and a method to determine an optimal parameter is provided. Several examples are presented to show the effectiveness of the proposed methods for overdetermined and underdetermined consistent linear systems with dense and sparse coefficient matrix.

A Lagrange Relaxation Based Approach to Solve a Discrete-Continous Bi-Level Model

In this work we propose a solution method based on Lagrange relaxation for discrete-continuous bi-level problems, with binary variables in the leading problem, considering the optimistic approach in bi-level programming. For the application of the method, the two-level problem is reformulated using the Karush-Kuhn-Tucker conditions. The resulting model is linearized taking advantage of the structure of the leading problem. Using a Lagrange relaxation algorithm, it is possible to find a global solution efficiently. The algorithm was tested to show how it performs.

面向高速移动环境的二级信号检测算法

正交时间序列复用(OTSM)可以以更低的复杂度实现类似正交时频空间(OTFS)调制的传输性能,为未来需要低复杂度收发器的高速移动性通信系统提供一种有前景的解决方法。针对现有的基于时域的高斯-赛德尔(GS)迭代均衡效率不高的问题,提出二级信号检测算法。首先在时域进行低复杂度线性最小均方误差(LMMSE)检测,其次采用连续超松弛(SOR)迭代算法进一步消除残余符号干扰。为进一步提高收敛效率和检测性能,对SOR算法进行线性优化得到改进SOR(ISOR)算法。仿真实验结果表明,与SOR算法相比,ISOR算法在增加较低复杂度前提下可以提升检测性能并加快算法收敛。与GS迭代算法相比,ISOR算法采用16QAM调制且误码率为10-4时有1.61 dB的增益。

潮汐干线交通信号协调控制模型

为缓解城市布局等问题导致的潮汐交通拥堵现象,提出了一种潮汐交通状态下的干线信号协调控制策略.依据交通供需关系及交通量进出平衡,构建了以干线吞吐量最大化为优化目标的混合整数线性规划模型.基于冲击波理论,构建了以过饱和方向车均延误最小化为优化目标的二次规划模型.通过引入松弛变量实现可变带宽绿波的分段优化,构建了以非饱和方向带宽最大化为目标的混合整数线性规划模型.采用三阶段优化方法求解模型,获得潮汐干线的最优控制方案.算例结果表明,本文提出的模型能优化干线吞吐量,控制过饱和方向的延误,并为非饱和方向提供可变带宽绿波,从而满足潮汐交通需求.

基于线性化潮流模型的配电网电压无功控制策略

针对现有基于二阶锥松弛的电压无功控制模型无法以电压波动为目标且求解速度较慢等问题,文中提出了一种基于线性化潮流模型的配电网电压无功控制策略。基于支路潮流方程建立了包含有载调压分接头、电容器组和光伏逆变器等调控手段的电压无功控制模型,并将其离散变量松弛为连续变量,以形成二阶锥规划。进一步以二阶锥规划所得结果为基准点构造线性化潮流模型,并以电压波动最小为目标,将连续变量还原为离散变量。最后基于改进的IEEE-33节点测试系统进行了仿真,仿真结果表明所提出的线性化潮流模型具有较高的精度和更快的求解速度,能够适用于配电网实时电压无功控制。

基于自适应步长的McCormick包络方法的配电网能源市场交易策略

针对配电网能源市场交易中双线性项非凸且求解精度差等问题,提出基于自适应步长的McCormick包络方法;该方法利用McCormick包络方法松弛双线性项,实现双线性项的线性化处理,采用边界收紧方法迭代收紧双线性项中变量的边界,并基于迭代梯度结果更新步长,在已知可行解基础上,迭代添加原目标函数的上界约束,进一步提升上层模型松弛后的求解质量;将该方法应用于双层能源市场交易模型,对零售商与微电网交易电价引导下的多微电网能源交易运营进行仿真。结果表明,与定步长的McCormick包络方法相比,基于自适应步长的McCormick包络方法实现了配电网侧含双线性项双层模型的有效求解,显著提升交易电价平均值和边界值场景下含双线性项双层模型的求解精度。

一类非凸约束二次规划问题的分支定界算法

针对一类非凸约束二次规划问题,提出一种新的参数化线性松弛分支定界算法,主要利用线性松弛技术求得原问题的全局最优值下界,以及区域删除规则缩减不可行区域,证明了算法的收敛性,最后通过数值实验表明算法的收敛速度加快,且该算法有效可行。

Two-Level Linear Relaxation Method for Generalized Linear Fractional Programming

This paper presents an efficient algorithm for globally solving a generalized linear fractional programming problem.For establishing this algorithm,we firstly construct a two-level linear relaxation method,and by utilizing the method,we can convert the initial generalized linear fractional programming problem and its subproblems into a series of linear programming relaxation problems.Based on the branch-and-bound framework and linear programming relaxation problems,a branch-and-bound algorithm is presented for globally solving the generalized linear fractional programming problem,and the computational complexity of the algorithm is given.Finally,numerical experimental results demonstrate the feasibility and efficiency of the proposed algorithm.

On designing memory state feedback controller for linear systems with interval time-varying delay

This paper is concerned with the design of a memory state feedback controller for linear systems with interval time-varying delays.The time delay is assumed to be a time-varying continuous function belonging to a given interval,which means that the lower and upper bounds of time-varying delay are available.First,a less conservative delay-range-dependent stability criteria is proposed by using a new interval fraction method.In the process of controller synthesis,the history information of system is considered in the controller design by introducing the lower delay state.Moreover,the usual memoryless state feedback controller for the underlying systems could be considered as a special case of the memory case.Finally,two numerical examples are given to show the effectiveness of the proposed method.

Global Optimization of Nonlinear Blend-Scheduling Problems

The scheduling of gasoline-blending operations is an important problem in the oil refining industry. Thisproblem not only exhibits the combinatorial nature that is intrinsic to scheduling problems, but alsonon-convex nonlinear behavior, due to the blending of various materials with different quality properties.In this work, a global optimization algorithm is proposed to solve a previously published continuous-timemixed-integer nonlinear scheduling model for gasoline blending. The model includes blend recipe optimi-zation, the distribution problem, and several important operational features and constraints. The algorithmemploys piecewise McCormick relaxation （PMCR） and normalized multiparametric disaggregation tech-nique （NMDT） to compute estimates of the global optimum. These techniques partition the domain of oneof the variables in a bilinear term and generate convex relaxations for each partition. By increasing the num-ber of partitions and reducing the domain of the variables, the algorithm is able to refine the estimates ofthe global solution. The algorithm is compared to two commercial global solvers and two heuristic methodsby solving four examples from the literature. Results show that the proposed global optimization algorithmperforms on par with commercial solvers but is not as fast as heuristic approaches.

Iterative Methods for Parametric Linear Systems with Linear Functions

This paper mainly proposes a new C-XSC （C- for eXtended Scientific Computing） software for the symmetric single step method and relaxation method for computing an enclosure for the solution set and compares the methods with others＇ and then makes some modifications and finally, examples illustrating the applicability of the proposed methods are given.

IRS辅助认知无线携能通信网络的发射功率最小化算法

智能反射平面(IRS)和认知无线携能通信技术被视为是提高能量效率和频谱利用率的潜在关键技术。文中研究了基于非线性能量采集模型的IRS辅助认知无线携能通信网络,其中次用户发射机同时给多个次用户接收机发送信息和能量,每个次用户接收机采用功率分割方式实现信息解码与能量采集,目的是通过联合优化次用户发射机的波束成形矢量、次用户接收机的功率分割系数以及IRS相移使次用户发射机的发射功率最小化。为了保证次用户发射机的信息与能量传输效率并限制次用户发射机对主用户接收机的同频干扰,考虑次用户接收机具有最小信干噪比约束、最小能量采集约束与功率分割系数约束,次用户发射机对主用户接收机有最大干扰功率值约束,以及IRS具有反射相移约束。所构建的问题属于非凸的二次约束二次规划问题,并且优化变量之间高度耦合,难以求解。文中提出一种基于半正定松弛法和连续秩一约束松弛法的交替优化算法进行高效求解。为了降低复杂度,进一步提出一种基于IRS分组的低复杂度优化算法。仿真结果表明,与几种基准算法相比,所提算法能够有效降低次用户发射机的发射功率。