Iterative optimization algorithms for discrete-continuous processes

We propose a series of approximate and iterative optimizational methods for discrete-continuous processes based on Krotov’s sufficient optimality conditions. Iterations are constructed by localizing the global optimality conditions and Krotov’s minimax scheme with various approximations. The very concept of a discrete-continuous system and the corresponding optimality conditions and algorithms represent a convenient formalism to study a wide class of complex systems and processes, in particular, magistral solutions of singular problems that are significantly non-uniform in structure.     

Iterative algorithms for the linear complementarity problem

Direct complementary pivot algorithms for the linear complementarity problem with P-matrices are known to have exponential computational complexity. The analog of Gauss-Seidel and SOR iteration for linear complementarity problems with P-matrices has not been extensively developed. This paper extends some work of van Bokhoven to a class of nonsymmetric P-matrices, and develops and compares several new iterative algorithms for the linear complementarity problem. Numerical results for several hundred test problems are presented. Such indirect iterative algorithms may prove useful for large sparse complementarity problems.     

Iterative ADP learning algorithms for discrete-time multi-player games

Adaptive dynamic programming (ADP) is an important branch of reinforcement learning to solve various optimal control issues. Most practical nonlinear systems are controlled by more than one controller. Each controller is a player, and to make a tradeoff between cooperation and conflict of these players can be viewed as a game. Multi-player games are divided into two main categories: zero-sum game and non-zero-sum game. To obtain the optimal control policy for each player, one needs to solve Hamilton–Jacobi–Isaacs equations for zero-sum games and a set of coupled Hamilton–Jacobi equations for non-zero-sum games. Unfortunately, these equations are generally difficult or even impossible to be solved analytically. To overcome this bottleneck, two ADP methods, including a modified gradient-descent-based online algorithm and a novel iterative offline learning approach, are proposed in this paper. Furthermore, to implement the proposed methods, we employ single-network structure, which obviously reduces computation burden compared with traditional multiple-network architecture. Simulation results demonstrate the effectiveness of our schemes.     

Iterative algorithms for discrete periodic Riccati matrix equations

In this paper, two iterative algorithms are constructed to obtain the positive definite solutions of the discrete periodic algebraic Riccati matrix equations. In these two algorithms, the estimation of the unknown matrices are updated by using the available estimation information at the current iteration step. The convergence properties of the proposed algorithms are also given. Finally, numerical examples are employed to illustrate the effectiveness of the proposed algorithms.     

Iterative feature construction for improving inductive learning algorithms

Inductive learning algorithms, in general, perform well on data that have been pre-processed to reduce complexity. By themselves they are not particularly effective in reducing data complexity while learning difficult concepts. Feature construction has been shown to reduce complexity of space spanned by input data. In this paper, we present an iterative algorithm for enhancing the performance of any inductive learning process through the use of feature construction as a pre-processing step. We apply the procedure on three learning methods, namely genetic algorithms, C4.5 and lazy learner, and show improvement in performance.     

Iterative weighted least squares algorithms for neural networks classifiers

This paper discusses learning algorithms of layered neural networks from the standpoint of maximum likelihood estimation. At first we discuss learning algorithms for the most simple network with only one neuron. It is shown that Fisher information of the network, namely minus expected values of Hessian matrix, is given by a weighted covariance matrix of input vectors. A learning algorithm is presented on the basis of Fisher's scoring method which makes use of Fisher information instead of Hessian matrix in Newton's method. The algorithm can be interpreted as iterations of weighted least squares method. Then these results are extended to the layered network with one hidden layer. Fisher information for the layered network is given by a weighted covariance matrix of inputs of the network and outputs of hidden units. Since Newton's method for maximization problems has the difficulty when minus Hessian matrix is not positive definite, we propose a learning algorithm which makes use of Fisher information matrix, which is non-negative, instead of Hessian matrix. Moreover, to reduce the computation of full Fisher information matrix, we propose another algorithm which uses only block diagonal elements of Fisher information. The algorithm is reduced to an iterative weighted least squares algorithm in which each unit estimates its own weights by a weighted least squares method. It is experimentally shown that the proposed algorithms converge with fewer iterations than error back-propagation (BP) algorithm.     

Iterative identification algorithms for input nonlinear output error autoregressive systems

This paper focuses on the parameter estimation problems of input nonlinear output error autoregressive systems. Based on the key variables separation technique and the auxiliary model identification idea, the output of the system is expressed as a linear combination of all the system parameters, the unknown inner variables in the information vector are replaced with the outputs of the auxiliary model and a gradient based and a least squares based iterative identification algorithms are derived. Simulation example is provided to illustrate the effectiveness of the proposed algorithms.     

Iterative RELIEF for feature weighting: algorithms, theories, and applications

RELIEF is considered one of the most successful algorithms for assessing the quality of features. In this paper, we propose a set of new feature weighting algorithms that perform significantly better than RELIEF, without introducing a large increase in computational complexity. Our work starts from a mathematical interpretation of the seemingly heuristic RELIEF algorithm as an online method solving a convex optimization problem with a margin-based objective function. This interpretation explains the success of RELIEF in real application and enables us to identify and address its following weaknesses. RELIEF makes an implicit assumption that the nearest neighbors found in the original feature space are the ones in the weighted space and RELIEF lacks a mechanism to deal with outlier data. We propose an iterative RELIEF (I-RELIEF) algorithm to alleviate the deficiencies of RELIEF by exploring the framework of the expectation-maximization algorithm. We extend I-RELIEF to multiclass settings by using a new multiclass margin definition. To reduce computational costs, an online learning algorithm is also developed. Convergence analysis of the proposed algorithms is presented. The results of large-scale experiments on the UCI and microarray data sets are reported, which demonstrate the effectiveness of the proposed algorithms, and verify the presented theoretical results     

Data mining for state space orthogonalization in adaptive dynamic programming

Dynamic programming (DP) is a mathematical programming approach for optimizing a system that changes over time and is a common approach for developing intelligent systems. Expert systems that are intelligent must be able to adapt dynamically over time. An optimal DP policy identifies the optimal decision dependent on the current state of the system. Hence, the decisions controlling the system can intelligently adapt to changing system states. Although DP has existed since Bellman introduced it in 1957, exact DP policies are only possible for problems with low dimension or under very limiting restrictions. Fortunately, advances in computational power have given rise to approximate DP (ADP). However, most ADP algorithms are still computationally-intractable for high-dimensional problems. This paper specifically considers continuous-state DP problems in which the state variables are multicollinear. The issue of multicollinearity is currently ignored in the ADP literature, but in the statistics community it is well known that high multicollinearity leads to unstable (high variance) parameter estimates in statistical modeling. While not all real world DP applications involve high multicollinearity, it is not uncommon for real cases to involve observed state variables that are correlated, such as the air quality ozone pollution application studied in this research. Correlation is a common occurrence in observed data, including sources in meteorology, energy, finance, manufacturing, health care, etc.ADP algorithms for continuous-state DP achieve an approximate solution through discretization of the state space and model approximations. Typical state space discretizations involve full-dimensional grids or random sampling. The former option requires exponential growth in the number of state points as the state space dimension grows, while the latter option is typically inefficient and requires an intractable number of state points. The exception is computationally-tractable ADP methods based on a design and analysis of computer experiments (DACE) approach. However, the DACE approach utilizes ideal experimental designs that are (nearly) orthogonal, and a multicollinear state space will not be appropriately represented by such ideal experimental designs. While one could directly build approximations over the multicollinear state space, the issue of unstable model approximations remains unaddressed. Our approach for handling multicollinearity employs data mining methods for two purposes: (1) to reduce the dimensionality of a DP problem and (2) to orthogonalize a multicollinear DP state space and enable the use of a computationally-efficient DACE-based ADP approach. Our results demonstrate the risk of ignoring high multicollinearity, quantified by high variance inflation factors representing model instability. Our comparisons using an air quality ozone pollution case study provide guidance on combining feature selection and feature extraction to guarantee orthogonality while achieving over 95% dimension reduction and good model accuracy.     

Subspace identification by data orthogonalization and model decoupling

It has been observed that identification of state-space models with inputs may lead to unreliable results in certain experimental conditions even when the input signal excites well within the bandwidth of the system. This may be due to ill-conditioning of the identification problem, which occurs when the state space and the future input space are nearly parallel.We have in particular shown in the companion papers (Automatica 40(4) (2004) 575; Automatica 40(4) (2004) 677) that, under these circumstances, subspace methods operating on input-output data may be ill-conditioned, quite independently of the particular algorithm which is used. In this paper, we indicate that the cause of ill-conditioning can sometimes be cured by using orthogonalized data and by recasting the model into a certain natural block-decoupled form consisting of a “deterministic” and a “stochastic” subsystem. The natural subspace algorithm for the identification of the deterministic subsystem is then a weighted version of the PI-MOESP method of Verhaegen and Dewilde (Int. J. Control 56 (1993) 1187-1211). The analysis shows that, under certain conditions, methods based on the block-decoupled parametrization and orthogonal decomposition of the input-output data, perform better than traditional joint-model-based methods in the circumstance of nearly parallel regressors.     

A neurofuzzy network knowledge extraction and extended Gram-Schmidt algorithm for model subspace decomposition

This paper introduces a new neurofuzzy model construction and parameter estimation algorithm from observed finite data sets, based on a Takagi-Sugeno (T-S) inference mechanism and a new extended Gram-Schmidt orthogonal decomposition algorithm, for the modeling of a priori unknown dynamical systems in the form of a set of fuzzy rules. The paper introduces a one to one mapping between a fuzzy rule-base and a model matrix feature subspace. Hence, rule-based knowledge can be extracted to enhance model transparency. Model transparency is explored by the derivation of an equivalence between an A-optimality experimental design criterion of the weighting matrix and the average model output sensitivity to the fuzzy rule. The A-optimality experimental design criterion of the weighting matrices of fuzzy rules is used to construct an initial model rule-base. An extended Gram-Schmidt algorithm is then developed to estimate the parameter vector for each rule. This new algorithm decomposes the model rule-bases via an orthogonal subspace decomposition approach, so as to enhance model transparency with the capability of interpreting the derived rule-base energy level.     

基于修正Gram-Schmidt算法预白的水下目标检测方法

将基于最小二乘问题的修正Gram—Schmidt算法运用于白化滤波器,提出一种基于修正Gram—Schmidt算法的白化滤波器。推导了该白化滤波器的权值表达式,分析了其白化性能和滤波器阶数的选择应注意的问题。提出了基于修正Gram—Schmidt算法预白的水下目标回波检测方法。蒙特卡洛仿真显示该方法的检测性能优于传统的基于AR模型预白的检测方法,尤其是在低信混比或（和）低多普勒的情况下。     

多项式方程的迭代方法

基于韦达定理,给出了求解高次代数方程迭代方法,可同时迭代出所有实解。对其收敛性作了初步讨论。给出了实例以及MATLAB源程序.     

多视三角化的迭代算法

多视三角化是在给定测量点对应和摄像机投影矩阵的情况下,求解相应的空间点的过程.由于测量点存在测量误差,所以只能求解在某种准则下的最优空间点.文中提出一种新的优化准则:在空间平面矩阵最小奇异值为0的约束下最小化估计点到测量点的L2-范数距离.在此基础上,采用该准则约束的Sampson近似得到一种简单的迭代求解方法;通过空间平面矩阵最小奇异值单调递减的条件和共轭梯度方法得到另一种收敛性更好的迭代算法.实验结果表明,这2种迭代算法不仅迭代次数及运算时间明显少于黄金标准算法,而且能得到基本相同的计算精度.     

Iterative method for fuzzy equations

In this paper, we propose numerical solution for solving a system of fuzzy nonlinear equations based on Fixed point method. The convergence theorem is proved in detail. In this method the algorithm is illustrated by solving several numerical examples.     

Joint-space orthogonalization and passivity for physical interpretations of dextrous robot motions under geometric constraints

A principle of ‘joint-space orthogonalization’ is proposed as an extended notion of hybrid (force and position) control for robot manipulators under geometric constraints. The principle realizes the hybrid control in a strict sense by letting position feedback signals be orthogonal in joint space to the contact force vector whose components exert at corresponding joints. This orthogonalization is executed via a projection matrix computed in real-time from a Jacobian matrix of the constraint equation in joint coordinates. To show the important role of the principle in control of robot manipulators, two basic set-point control problems are analysed. One is a hybrid PID control problem for robot manipulators under geometric endpoint constraint and another is a coordinated control problem of two arms. It is shown that passivity properties of residual dynamics of robots follow from the introduction of a quasi-natural potential and the joint-space orthogonalization. Various stability problems of PID-type feedback control schemes without compensating for the gravity force and with or without use of a force sensor are discussed from passivity properties of robot dynamics with the aid of the hyper-stability theory.     

How Iterative are Iterative Algebras?

Iterative algebras are defined by the property that every guarded system of recursive equations has a unique solution. We prove that they have a much stronger property: every system of recursive equations has a unique strict solution. And we characterize those systems that have a unique solution in every iterative algebra.