A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

Distributed Alternating Direction Method of Multipliers for Multi-Objective Optimization

In this paper, a distributed algorithm is proposed to solve a kind of multi-objective optimization problem based on the alternating direction method of multipliers. Compared with the centralized algorithms, this algorithm does not need a central node. Therefore, it has the characteristics of low communication burden and high privacy. In addition, numerical experiments are provided to validate the effectiveness of the proposed algorithm.

An improved optimal elemental method for updating finite element models

The optimal matrix method and optimal elemental method used to update finite element models may not provide accurate results.This situation occurs when the test modal model is incomplete,as is often the case in practice.An improved optimal elemental method is presented that defines a new objective function,and as a byproduct,circumvents the need for mass normalized modal shapes,which are also not readily available in practice.To solve the group of nonlinear equations created by the improved optimal method,the Lagrange multiplier method and Matlab function fmincon are employed.To deal with actual complex structures, the float-encoding genetic algorithm(FGA)is introduced to enhance the capability of the improved method.Two examples,a 7- degree of freedom(DOF)mass-spring system and a 53-DOF planar frame,respectively,are updated using the improved method. The example results demonstrate the advantages of the improved method over existing optimal methods,and show that the genetic algorithm is an effective way to update the models used for actual complex structures.

Modified Augmented Lagrange Multiplier Methods for Large-Scale Chemical Process Optimization

Chemical process optimization can be described as large-scale nonlinear constrained minimization. The modified augmented Lagrange multiplier methods (MALMM) for large-scale nonlinear constrained minimization are studied in this paper. The Lagrange function contains the penalty terms on equality and inequality constraints and the methods can be applied to solve a series of bound constrained sub-problems instead of a series of unconstrained sub-problems. The steps of the methods are examined in full detail. Numerical experiments are made for a variety of problems, from small to very large-scale, which show the stability and effectiveness of the methods in large-scale problems.

Nested Alternating Direction Method of Multipliers to Low-Rank and Sparse-Column Matrices Recovery

The task of dividing corrupted-data into their respective subspaces can be well illustrated,both theoretically and numerically,by recovering low-rank and sparse-column components of a given matrix.Generally,it can be characterized as a matrix and a 2,1-norm involved convex minimization problem.However,solving the resulting problem is full of challenges due to the non-smoothness of the objective function.One of the earliest solvers is an 3-block alternating direction method of multipliers(ADMM)which updates each variable in a Gauss-Seidel manner.In this paper,we present three variants of ADMM for the 3-block separable minimization problem.More preciously,whenever one variable is derived,the resulting problems can be regarded as a convex minimization with 2 blocks,and can be solved immediately using the standard ADMM.If the inner iteration loops only once,the iterative scheme reduces to the ADMM with updates in a Gauss-Seidel manner.If the solution from the inner iteration is assumed to be exact,the convergence can be deduced easily in the literature.The performance comparisons with a couple of recently designed solvers illustrate that the proposed methods are effective and competitive.

Two-stage ADMM-based distributed optimal reactive power control method for wind farms considering wake effects

Since the connection of small-scale wind farms to distribution networks,power grid voltage stability has been reduced with increasing wind penetration in recent years,owing to the variable reactive power consumption of wind generators.In this study,a two-stage reactive power optimization method based on the alternating direction method of multipliers(ADMM)algorithm is proposed for achieving optimal reactive power dispatch in wind farm-integrated distribution systems.Unlike existing optimal reactive power control methods,the proposed method enables distributed reactive power flow optimization with a two-stage optimization structure.Furthermore,under the partition concept,the consensus protocol is not needed to solve the optimization problems.In this method,the influence of the wake effect of each wind turbine is also considered in the control design.Simulation results for a mid-voltage distribution system based on MATLAB verified the effectiveness of the proposed method.

Distributed Lagrange Multiplier/Fictitious Domain Finite Element Method for a Transient Stokes Interface Problem with Jump Coefficients

The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface,where the arbitrary Lagrangian-Eulerian(ALE)technique is employed to deal with the moving and immersed subdomain.Stability and optimal convergence properties are obtained for both schemes.Numerical experiments are carried out for different scenarios of jump coefficients,and all theoretical results are validated.

A Parameter-Free Approach to Determine the Lagrange Multiplier in the Level Set Method by Using the BESO

A parameter-free approach is proposed to determine the Lagrange multiplier for the constraint of material volume in the level set method.It is inspired by the procedure of determining the threshold of sensitivity number in the BESO method.It first computes the difference between the volume of current design and the upper bound of volume.Then,the Lagrange multiplier is regarded as the threshold of sensitivity number to remove the redundant material.Numerical examples proved that this approach is effective to constrain the volume.More importantly,there is no parameter in the proposed approach,which makes it convenient to use.In addition,the convergence is stable,and there is no big oscillation.

Application of Linearized Alternating Direction Multiplier Method in Dictionary Learning

The Alternating Direction Multiplier Method (ADMM) is widely used in various fields, and different variables are customized in the literature for different application scenarios [1] [2] [3] [4]. Among them, the linearized alternating direction multiplier method (LADMM) has received extensive attention because of its effectiveness and ease of implementation. This paper mainly discusses the application of ADMM in dictionary learning (non-convex problem). Many numerical experiments show that to achieve higher convergence accuracy, the convergence speed of ADMM is slower, especially near the optimal solution. Therefore, we introduce the linearized alternating direction multiplier method (LADMM) to accelerate the convergence speed of ADMM. Specifically, the problem is solved by linearizing the quadratic term of the subproblem, and the convergence of the algorithm is proved. Finally, there is a brief summary of the full text.

Lagrangian Relaxation Method for Multiobjective Optimization Methods: Solution Approaches

This paper introduces the Lagrangian relaxation method to solve multiobjective optimization problems. It is often required to use the appropriate technique to determine the Lagrangian multipliers in the relaxation method that leads to finding the optimal solution to the problem. Our analysis aims to find a suitable technique to generate Lagrangian multipliers, and later these multipliers are used in the relaxation method to solve Multiobjective optimization problems. We propose a search-based technique to generate Lagrange multipliers. In our paper, we choose a suitable and well-known scalarization method that transforms the original multiobjective into a scalar objective optimization problem. Later, we solve this scalar objective problem using Lagrangian relaxation techniques. We use Brute force techniques to sort optimum solutions. Finally, we analyze the results, and efficient methods are recommended.

Optimal Designs Technique for Locating the Optimum of a Second Order Response Function

A more efficient method of locating the optimum of a second order response function was of interest in this work. In order to do this, the principles of optimal designs of experiment is invoked and used for this purpose. At the end, it was discovered that the noticeable pitfall in response surface methodology (RSM) was circumvented by this method as the step length was obtained by taking the derivative of the response function rather than doing so by intuition or trial and error as is the case in RSM. A numerical illustration shows that this method is suitable for obtaining the desired optimizer in just one move which compares favourably with other known methods such as Newton-Raphson method which requires more than one iteration to reach the optimizer.

A Bregman-style Partially Symmetric Alternating Direction Method of Multipliers for Nonconvex Multi-block Optimization

The alternating direction method of multipliers(ADMM)is one of the most successful and powerful methods for separable minimization optimization.Based on the idea of symmetric ADMM in two-block optimization,we add an updating formula for the Lagrange multiplier without restricting its position for multiblock one.Then,combining with the Bregman distance,in this work,a Bregman-style partially symmetric ADMM is presented for nonconvex multi-block optimization with linear constraints,and the Lagrange multiplier is updated twice with different relaxation factors in the iteration scheme.Under the suitable conditions,the global convergence,strong convergence and convergence rate of the presented method are analyzed and obtained.Finally,some preliminary numerical results are reported to support the correctness of the theoretical assertions,and these show that the presented method is numerically effective.

A Symmetric Linearized Alternating Direction Method of Multipliers for a Class of Stochastic Optimization Problems

Alternating direction method of multipliers(ADMM)receives much attention in the recent years due to various demands from machine learning and big data related optimization.In 2013,Ouyang et al.extend the ADMM to the stochastic setting for solving some stochastic optimization problems,inspired by the structural risk minimization principle.In this paper,we consider a stochastic variant of symmetric ADMM,named symmetric stochastic linearized ADMM(SSL-ADMM).In particular,using the framework of variational inequality,we analyze the convergence properties of SSL-ADMM.Moreover,we show that,with high probability,SSL-ADMM has O((ln N)·N^(-1/2))constraint violation bound and objective error bound for convex problems,and has O((ln N)^(2)·N^(-1))constraint violation bound and objective error bound for strongly convex problems,where N is the iteration number.Symmetric ADMM can improve the algorithmic performance compared to classical ADMM,numerical experiments for statistical machine learning show that such an improvement is also present in the stochastic setting.

CONVERGENCE PROPERTIES OF IMPROVED SECANT METHODS WITH TRUST REGION MULTIPLIER

The secant methods discussed by Fontecilla (in 1988) are considerably revised through employing a trust region multiplier strategy and introducing a nondifferentiable merit function. In this paper the secant methods are also improved by adding a dogleg typed movement which allows to overcome a phenomena similar to the Maratos effect. Furthermore, these algorithms are analyzed and global convergence theorems as well as local superlinear convergence rate are proved.

考虑异步通信的配电-气网分布式有功-无功协同优化

配电-气网(EGDN)中新能源大量并网和燃气运输过程中管网压力变化均可能导致配电网的电压波动增大。文中提出一种考虑异步通信的EGDN有功-无功协同优化方法。首先,研究EGDN中各元件的有功-无功特性;其次,根据EGDN能量运输过程中无功需求特性,建立EGDN无功供需模型;然后,考虑EGDN无功平衡和网络运行约束,以最小化电网电压波动、网损和EGDN购电/气成本为目标,建立EGDN无功电压优化控制模型;最后,为保障EGDN中各能源主体信息的安全性与私密性,考虑配电网与配气网信息采集传输情况存在差异,可能存在通信延迟和信息丢包情况,采用异步交替方向乘子法对模型进行求解。通过对改进的IEEE 33节点系统和7节点配气网构成的EGDN系统进行仿真,结果表明,所提方法可有效缓解EGDN电压波动问题,减小网损并降低系统总运行成本。

求解不可分离非凸非光滑问题的线性惯性ADMM算法

针对目标函数中包含耦合函数H(x,y)的非凸非光滑极小化问题,提出了一种线性惯性交替乘子方向法(Linear Inertial Alternating Direction Method of Multipliers,LIADMM)。为了方便子问题的求解,对目标函数中的耦合函数H(x,y)进行线性化处理,并在x-子问题中引入惯性效应。在适当的假设条件下,建立了算法的全局收敛性;同时引入满足Kurdyka-Lojasiewicz不等式的辅助函数,验证了算法的强收敛性。通过两个数值实验表明,引入惯性效应的算法比没有惯性效应的算法收敛性能更好。

考虑碳-绿证耦合机制的电-气-热综合能源系统分布式鲁棒低碳经济调度

为了保护综合能源系统中不同运营主体的信息隐私,权衡系统运行低碳性和经济性,解决天然气网络鲁棒模型难以求解的问题,提出了考虑碳-绿证耦合机制的电-气-热综合能源系统分布式鲁棒低碳经济调度方法。考虑天然气网络和热力网络的动态特性、碳-绿证耦合机制,构建综合能源系统动态低碳经济调度模型;为了保护各网络运营商的数据隐私,根据能量耦合关系对综合能源系统解耦,建立电-气-热网络的分布式协同优化模型;在此基础上,考虑风电出力和多能负荷的不确定性,提出基于一致性交替方向乘子法的分布式鲁棒优化框架,并利用二阶锥对偶理论与交替优化方法实现含二阶锥约束和二元变量的鲁棒子问题求解。以修改的IEEE 39节点电力网络、比利时20节点天然气网络和15节点热力网络为算例进行仿真分析,验证所提方法能够在源荷不确定性条件下实现综合能源系统各网络分散自治运行,同时兼顾系统运行的低碳性和经济性。

一种基于矩阵填充的稀疏阵波达方向估计技术

为了提高稀疏阵列波达方向(DOA)估计的性能,该文将低秩矩阵重构理论应用到DOA估计中,提出了一种改进的矩阵填充模型及其优化求解方法。该方法利用Sigmoid函数实现核范数约束并建立最小化模型;然后基于粒子群算法改进增广拉格朗日乘子法,对模型实现低秩优化求解;最后利用多信号分类(MUSIC)算法实现DOA估计。仿真结果表明,该方法能有效实现稀疏阵重构,DOA估计的性能优良,且能够适用于相关信源。

Application of the Alternating Direction Method of Multipliers to Control Constrained Parabolic Optimal Control Problems and Beyond

Control constrained parabolic optimal control problems are generally challenging,from either theoretical analysis or algorithmic design perspectives.Conceptually,the well-known alternating direction method of multipliers(ADMM)can be directly applied to such problems.An attractive advantage of this direct ADMM application is that the control constraints can be untied from the parabolic optimal control problem and thus can be treated individually in the iterations.At each iteration of the ADMM,the main computation is for solving an unconstrained parabolic optimal control subproblem.Because of its inevitably high dimensionality after space-time discretization,the parabolicoptimal control subproblem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative algorithm is required.It then becomes important to find an easily implementable and efficient inexactness criterion to perform the internal iterations,and to prove the overall convergence rigorously for the resulting two-layer nested iterative algorithm.To implement the ADMM efficiently,we propose an inexactness criterion that is independent of the mesh size of the involved discretization,and that can be performed automatically with no need to set empirically perceived constant accuracy a priori.The inexactness criterion turns out to allow us to solve the resulting parabolic optimal control subproblems to medium or even low accuracy and thus save computation significantly,yet convergence of the overall two-layer nested iterative algorithm can be still guaranteed rigorously.Efficiency of this ADMM implementation is promisingly validated by some numerical results.Our methodology can also be extended to a range of optimal control problems modeled by other linear PDEs such as elliptic equations,hyperbolic equations,convection-diffusion equations,and fractional parabolic equations.

基于集群动态划分的配电网无功电压自律-协同控制

针对高渗透率分布式光伏接入配电网导致的电压越限问题,以及传统集中式控制面临的通信延时、计算量大与控制设备过多等问题,提出一种基于集群动态划分的配电网无功电压自律-协同控制方法。首先,建立日前集中式无功电压优化模型,安排离散无功补偿设备投退计划,日内调度以节点电压偏差最小为目标建立日内无功电压滚动优化模型,结合同步型交替方向乘子法实现各集群间的分布式协调控制;然后,通过群内本地电压实时控制抑制群内节点电压的波动;最终,以改进的IEEE 33节点系统及某地实际60节点配电网为例,验证了该方法的正确性与有效性。