Higher-order approximations for testing neglected nonlinearity

We illustrate the need to use higher-order (specifically sixth-order) expansions in order to properly determine the asymptotic distribution of a standard artificial neural network test for neglected nonlinearity. The test statistic is a quasi-likelihood ratio (QLR) statistic designed to test whether the mean square prediction error improves by including an additional hidden unit with an activation function violating the no-zero condition in Cho, Ishida, and White (2011). This statistic is also shown to be asymptotically equivalent under the null to the Lagrange multiplier (LM) statistic of Luukkonen, Saikkonen, and Ter?svirta (1988) and Ter?svirta (1994). In addition, we compare the power properties of our QLR test to one satisfying the no-zero condition and find that the latter is not consistent for detecting a DGP with neglected nonlinearity violating an analogous no-zero condition, whereas our QLR test is consistent.     

Consistent treatment of boundaries with mortar contact formulations using dual Lagrange multipliers

Computational modelling of contact problems raises two basic questions: Which method should be used to enforce the contact conditions and how should this method be discretised? The most popular enforcement methods are the Lagrange multiplier method, the penalty method and combinations of these two. A frequently used discretisation method is the so called node-to-segment approach. However, this approach might lead to problems like jumps in contact forces, loss of convergence or failure to pass the patch test. Thus in the last few years, several segment-to-segment contact algorithms based on the mortar method were proposed.Combination of a mortar discretisation with a penalty based enforcement of the contact conditions leads to unphysical penetrations. On the other hand, a Lagrange multiplier mortar method requires additional unknowns. Hence, condensation of the Lagrange multipliers is desirable to preserve the initial size of the system of equations. This can be achieved by interpolating the Lagrange multipliers with so-called dual shape functions.Discretising two contacting bodies leads to opposed contact surface representations of finite element edges, called slave and master elements, respectively. In current versions of dual Lagrange multiplier mortar formulations an inconsistency at the boundary appears when only a part of a slave element (instead of the entire element) belongs to the contact area. We present a modified definition of the dual shape functions in such slave elements. The basic idea is to construct dual shape functions that fulfill the so-called biorthogonality condition within the contact area. This leads to consistent mortar matrices also in the boundary region. To avoid ill-conditioning of the stiffness matrix, the modified mortar matrices are weighted with appropriate weighting factors. In doing so, the corresponding modified Lagrange multiplier nodal values are of the same order as the unmodified ones. Various examples demonstrate the performance of the modified mortar contact algorithm.     

基于氧气A吸收带高光谱反射信息的气溶胶廓线反演算法及模拟反演验证

为反演对流层气溶胶消光系数廓线,提出了一种新的、基于氧气A光谱吸收带高光谱反射信息的反演方法。这种方法在经典的线性约束反演算法的基础上引入了两个与气溶胶柱光学厚度相关的订正因子,通过实现目标廓线和反演廓线的光学柱厚度差异最小,进而动态选择拉格朗日系数。为验证新方法的有效性和精度,借助了CALIPSO (Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations)气溶胶廓线产品和高光谱辐射传输模拟,模拟反演海洋下垫面晴空和沙尘两种情景下气溶胶消光系数廓线,结果表明新的方法能够比较准确地获得对流层内气溶胶廓线的数值,尤其在多次散射比较强的沙尘条件下,相对误差能够控制在51%以下;在晴空时,相对误差能够控制在20%以下。如果加入订正信息,反演精度能够进一步提高。     

Testing for jumps in the EGARCH process

This paper considers testing for jumps in the exponential GARCH (EGARCH) models with Gaussian and Student-t innovations. The Wald and log likelihood ratio tests contain a nuisance parameter unidentified under the null hypothesis of no jumps, and hence are unavailable for this problem, because jump probability and variance of jumps in the test statistic cannot be estimated under the null hypothesis of no jumps. It is shown that the nuisance parameter is cancelled out in the Lagrange multiplier (LM) test statistic, and hence that the test is nuisance parameter-free. The one-sided test is also proposed using the nonnegative constraint on jump variance. The actual size and power of the tests are examined in a Monte Carlo experiment. The test is applied to daily returns of S&P 500 as an illustrative example.     

Testing for jumps in the stochastic volatility models

This paper proposes the Lagrange multiplier (LM) test, or the score test, for jumps in the stochastic volatility (SV) model in the cases where the innovation term follows the normal and Student t-distributions. The tested null hypothesis is that the jump density has zero variance, which is expressed by Dirac’s delta function. It is shown that the unknown jump probability, which is an unidentified parameter under the null hypothesis, is cancelled out in the LM test statistic, and hence this test is free from the estimation problem of unidentified parameters, which is known as the Davies problem [R.B. Davies, Hypothesis testing when a nuisance parameter is present only under the alternative, Biometrika 64 (1977) 247–254]. Monte Carlo experiments show that the null distribution of the LM test statistic can be approximated by the normal distribution with sufficient accuracy.     

Alternative approaches to implementing Lagrange multiplier tests for serial correlation in dynamic regression models

An approximate F-form of the Lagrange multiplier (LM) test for serial correlation in dynamic regression models is compared with three bootstrap tests. In one bootstrap procedure, residuals from restricted estimation under the null hypothesis are resampled. The other two bootstrap tests use residuals from unrestricted estimation under an alternative hypothesis. A fixed autocorrelation alternative is assumed in one of the two unrestricted bootstrap tests and the other is based upon a Pitman-type sequence of local alternatives. Monte Carlo experiments are used to estimate rejection probabilities under the null hypothesis and in the presence of serial correlation.     

Testing correct model specification using extreme learning machines

Testing the correct model specification hypothesis for artificial neural network (ANN) models of the conditional mean is not standard. The traditional Wald, Lagrange multiplier, and quasi-likelihood ratio statistics weakly converge to functions of Gaussian processes, rather than to convenient chi-squared distributions. Also, their large-sample null distributions are problem dependent, limiting applicability. We overcome this challenge by applying functional regression methods of Cho et al. [8] to extreme learning machines (ELM). The Wald ELM (WELM) test statistic proposed here is easy to compute and has a large-sample standard chi-squared distribution under the null hypothesis of correct specification. We provide associated theory for time-series data and affirm our theory with some Monte Carlo experiments.     

参数不确定线性系统混合H2/H∞状态反馈控制

对一类含有范数有界参数不确定线性系统的混合H2/H∞状态反馈控制问题进行了 研究.给出了系统一个输出在满足给定的H∞干扰衰减约束条件下,其另一个输出的H2性能 指标所满足的上界,并利用Lagrange乘子法导出了使得该界达到最小的"最优的"状态反馈 控制器.结果仅需求解一个含有两个尺度参数的修正代数Riccati方程.而且数值结果表明 该方法是非常有效的.     

改进Lagrange乘子法及收敛性分析

将与不等式约束相关的乘子重新定义为原乘子的正定函数,则Karush-Kuhn-Tucker必要条件中关于不等式约束乘子的非负约束可以去掉,并能构造出直接处理不等式约束的Lagrange乘子法.分析了算法的收敛性,利用LaSalle不变集原理揭示其稳定机制,并讨论如何减弱收敛条件和扩大收敛域.     

Modified augmented Lagrangian coordination and alternating direction method of multipliers with parallelization in non-hierarchical analytical target cascading

Analytical Target Cascading (ATC) is a decomposition-based optimization methodology that partitions a system into subsystems and then coordinates targets and responses among subsystems. Augmented Lagrangian with Alternating Direction method of multipliers (AL-AD), one of efficient ATC coordination methods, has been widely used in both hierarchical and non-hierarchical ATC and theoretically guarantees convergence under the assumption that all subsystem problems are convex and continuous. One of the main advantages of distributed coordination which consists of several non-hierarchical subproblems is that it can solve subsystem problems in parallel and thus reduce computational time. Therefore, previous studies have proposed an augmented Lagrangian coordination strategy for parallelization by eliminating interactions among subproblems. The parallelization is achieved by introducing a master problem and support variables or by approximating a quadratic penalty function to make subproblems separable. However, conventional AL-AD does not guarantee convergence in the case of parallel solving. Our study shows that, in parallel solving using targets and responses of the current iteration, conventional AL-AD causes mismatch of information in updating the Lagrange multiplier. Therefore, the Lagrange multiplier may not reach the optimal point, and as a result, increasing penalty weight causes numerical difficulty in the augmented Lagrangian coordination approach. To solve this problem, we propose a modified AL-AD with parallelization in non-hierarchical ATC. The proposed algorithm uses the subgradient method with adaptive step size in updating the Lagrange multiplier and also maintains penalty weight at an appropriate level not to cause oscillation. Without approximation or introduction of an artificial master problem, the modified AL-AD with parallelization can achieve similar accuracy and convergence with much less computational cost compared with conventional AL-AD with sequential solving.     

约束随机线性二次最优控制的研究

本文研究线性终端状态约束下不定随机线性二次最优控制问题.首先利用Lagrange Multiplier 定理得到了存在最优线性状态反馈解的必要条件, 而在加强的条件下也得到了最优控制存在的充分条件. 从某种意义上讲, 以往关于无约束随机线性二次最优控制的一些结果可以看成本文主要定理的推论.     

基于Hopfield 神经网络的污水处理过程优化控制

针对前置反硝化污水处理过程的优化控制问题,提出一种基于拉格朗日乘子法的Hofield神经网络优化方法.构造了污水处理过程约束优化问题的数学表达式,通过Hopfield神经网络优化计算生化池第5分区溶解氧浓度和第2分区硝态氮浓度的设定值,并采用PID控制器实现底层的跟踪控制.基于国际标准的Benchmark基准仿真平台进行仿真实验,结果表明污水处理系统在出水关键水质达标的基础上,能够显著降低能耗.     

串并联复式系统可靠性最优分配

本文讨论独立故障及混合故障串并联复式系统中,在给定可靠性条件下寻求总成本最小 的最优分配问题.运用拉格朗日乘子法,得到了计算分配向量的公式,并给出了两个计算实 例.     

A robust forward weighted Lagrange multiplier test for conditional heteroscedasticity

Statistical tests routinely adopted for detecting nonlinear components in time series rely on the auxiliary regression of ARMA lagged residuals, and the Lagrange multiplier test to detect ARCH components is an example. The size distortion of such test suggests adopting a weighted test, where the weights are computed through a forward search algorithm. Simulations show that the forward weighted robust test is preferable to the classical Lagrange test and to existing robust tests, which are based on backward weighted regression or on estimated autocorrelation function. The forward weighted robust test is applied to daily financial and quarterly macroeconomic time series, showing its usefulness in detecting ARCH effects, even when outliers are present.     

基于中间变量的乘子法

为了更有效地用乘子法求解结构优化问题,将倒变量和混合变量这两种中间变量引入乘子法,提出了基于中间变量的乘子法。在这种新方法中,基于增广拉格朗日函数的无约束子问题的构造和求解,以及算法的迭代和收敛,都是对中间变量进行的。只有目标函数与约束函数的函数值及梯度值的求取,是对原变量进行的。用两个具有较高非线性的工程算例对新方法进行测试,测试结果表明新方法具有良好的收敛性,且比一般的乘子法的收敛速度更快。     

Two-Way中继系统中基于最小化总功率的功率分配策略*

在保证系统QoS前提下,为了降低Two-Way中继系统的总功率开销,提出了一种基于最小化总功率的优化功率分配策略。首先对Two-Way中继系统中断概率的闭合表达式进行了推导;在系统中断概率等于某一恒定值的条件下,建立了最小化系统总功率的优化模型。通过拉格朗日乘数法对优化模型进行了求解。仿真结果证明了此策略能降低系统总功率开销。     

一种自适应全变分图像去噪算法

针对传统全变分去噪算法存在需要已知噪声方差和阶梯效应的问题,提出一种自适应全变分图像去噪算法。该算法通过修改传统算法中的逼近项,将经模糊处理后的图像代替原含噪声图像,使得在算法的计算过程中不需要已知图像的噪声方差,也减少逼近项因图像噪声所产生的误差影响。并且拉格朗日因子不再是一个全局变量,而是由图像局部信息决定其数值大小,同时用近似的边缘信息对拉格朗日因子进行加权,从而使图像的演化过程可以由统一的演化公式来表示。最后从实验和数据分析的结果可以看出,该算法要明显优于传统的全变分图像去噪算法。     

A Lagrangian relaxation technique for certain inventory models

This paper examines the n-item deterministic inventory model with one linear capacity constraint. An algorithm to compute the optimal Lagrange multiplier of the capacity constraint is presented. The results of a computational analysis conducted to test the performance of the algorithm are also included.     

一种求解约束优化问题的混合算法

提出一种基于修改增广Lagrange函数和PSO的混合算法用于求解约束优化问题。将约束优化问题转化为界约束优化问题,混合算法由两层迭代结构组成,在内层迭代中,利用改进PSO算法求解界约束优化问题得到下一个迭代点。外层迭代主要修正Lagrange乘子和罚参数,检查收敛准则是否满足,重构下次迭代的界约束优化子问题,检查收敛准则是否满足。数值实验结果表明该混合算法的有效性。     

Applicability of the Lagrange multiplier method to the optimization of quadratic systems

Conditions under which the Lagrange multiplier method (LMM) can be applied to optimization of the class of quadratic systems are considered. It is shown that quadratic systems can be optimised by LMM under rather general, and to some extent constructive, assumptions. The applicability of the results is illustrated with examples.