Regularization and general methods in the theory of functional equations

Summary The aim of this paper is to give a survey of two fields of the theory of non-iterated functional equations with two variables. One is the application of new, general methods of functional analysis, harmonic analysis and other topics to get a unified treatment of several kinds of equations. The other includes general regularity results for non-iterated functional equations with two variables.     

Error analysis of variable degree mixed methods for elliptic problems via hybridization

A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new variable degree Raviart-Thomas method, but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates. The error analysis is achieved by using a variational characterization of the Lagrange multipliers wherein the other unknowns do not appear. This approach can be applied to other hybridized mixed methods as well.

     

Multivariate Normal Slice Sampling

By introducing auxiliary variables, the traditional Markov chain Monte Carlo method can be improved in certain cases by implementing a “slice sampler.” In the current literature, this sampling technique is used to sample from multivariate distributions with both single and multiple auxiliary variables. When the latter is employed, it generally updates one component at a time.

In this article, we propose two variations of a new multivariate normal slice sampling method that uses multiple auxiliary variables to perform multivariate updating. These methods are flexible enough to allow for truncation to a rectangular region and/or exclusion of any n-dimensional hyper-quadrant. We present results of our methods and existing state-of-the-art slice samplers by comparing efficiency and accuracy. We find that we can generate approximately iid samples at a rate that is more efficient than other methods that update all dimensions at once. Supplemental materials are available online.     

The splitting of variables and constraints in the formulation of integer programming models

The splitting of variables in an integer programming model into the sum of other variables can allow the constraints to be disaggregated, leading to a more constrained (tighter) linear programming relaxation. Well known examples of such reformulations are quoted from the literature. They can be viewed as instances of some general methods of performing such reformulations, namely disjunctive formulations, partial network reformulations and a method based on the introduction of auxiliary variables.     

Coefficient reduction for inequalities in 0–1 variables

For a given inequality with 0–1 variables, there are many other “equivalent” inequalities with exactly the same 0–1 feasible solutions. The set of all equivalent inequalities is characterized, and methods to construct the equivalent inequality with smallest coefficients are described.     

Interior point algorithms for linear programming with inequality constraints

Interior methods for linear programming were designed mainly for problems formulated with equality constraints and non-negative variables. The formulation with inequality constraints has shown to be very convenient for practical implementations, and the translation of methods designed for one formulation into the other is not trivial. This paper relates the geometric features of both representations, shows how to transport data and procedures between them and shows how cones and conical projections can be associated with inequality constraints.     

不同差补方法的比较

本文针对缺失数据提出几种差补方法 ,通过模拟实验 ,考察文些方法的适用性及优缺点。结果表明 ,控制变量的恰当引入有利于提高估算效果。从与真值的拟合角度看 ,均值差补法有优势 ;而从保持样本分布的角度看 ,含有随机过程的差补法效果显著。在使用差补后的“完整”数据集时始终保持客观谨慎的态度是非常重要的。     

Solving dual problems using a coevolutionary optimization algorithm

In solving certain optimization problems, the corresponding Lagrangian dual problem is often solved simply because in these problems the dual problem is easier to solve than the original primal problem. Another reason for their solution is the implication of the weak duality theorem which suggests that under certain conditions the optimal dual function value is smaller than or equal to the optimal primal objective value. The dual problem is a special case of a bilevel programming problem involving Lagrange multipliers as upper-level variables and decision variables as lower-level variables. Another interesting aspect of dual problems is that both lower and upper-level optimization problems involve only box constraints and no other equality of inequality constraints. In this paper, we propose a coevolutionary dual optimization (CEDO) algorithm for co-evolving two populations—one involving Lagrange multipliers and other involving decision variables—to find the dual solution. On 11 test problems taken from the optimization literature, we demonstrate the efficacy of CEDO algorithm by comparing it with a couple of nested smooth and nonsmooth algorithms and a couple of previously suggested coevolutionary algorithms. The performance of CEDO algorithm is also compared with two classical methods involving nonsmooth (bundle) optimization methods. As a by-product, we analyze the test problems to find their associated duality gap and classify them into three categories having zero, finite or infinite duality gaps. The development of a coevolutionary approach, revealing the presence or absence of duality gap in a number of commonly-used test problems, and efficacy of the proposed coevolutionary algorithm compared to usual nested smooth and nonsmooth algorithms and other existing coevolutionary approaches remain as the hallmark of the current study.     

Deconvolution and Regularization with Toeplitz Matrices

By deconvolution we mean the solution of a linear first-kind integral equation with a convolution-type kernel, i.e., a kernel that depends only on the difference between the two independent variables. Deconvolution problems are special cases of linear first-kind Fredholm integral equations, whose treatment requires the use of regularization methods. The corresponding computational problem takes the form of structured matrix problem with a Toeplitz or block Toeplitz coefficient matrix. The aim of this paper is to present a tutorial survey of numerical algorithms for the practical treatment of these discretized deconvolution problems, with emphasis on methods that take the special structure of the matrix into account. Wherever possible, analogies to classical DFT-based deconvolution problems are drawn. Among other things, we present direct methods for regularization with Toeplitz matrices, and we show how Toeplitz matrix–vector products are computed by means of FFT, being useful in iterative methods. We also introduce the Kronecker product and show how it is used in the discretization and solution of 2-D deconvolution problems whose variables separate.     

Extending Mosaic Displays: Marginal,Conditional, and Partial Views of Categorical Data

Abstract

This article first illustrates the use of mosaic displays for the analysis of multiway contingency tables. We then introduce several extensions of mosaic displays designed to integrate graphical methods for categorical data with those used for quantitative data. The scatterplot matrix shows all pairwise (bivariate marginal) views of a set of variables in a coherent display. One analog for categorical data is a matrix of mosaic displays showing some aspect of the bivariate relation between all pairs of variables. The simplest case shows the bivariate marginal relation for each pair of variables. Another case shows the conditional relation between each pair, with all other variables partialled out. For quantitative data this represents (a) a visualization of the conditional independence relations studied by graphical models, and (b) a generalization of partial residual plots. The conditioning plot, or coplot shows a collection of partial views of several quantitative variables, conditioned by the values of one or more other variables. A direct analog of the coplot for categorical data is an array of mosaic plots of the dependence among two or more variables, stratified by the values of one or more given variables. Each such panel then shows the partial associations among the foreground variables; the collection of such plots shows how these associations change as the given variables vary.     

Estimating VAR models for the term structure of interest rates

In this paper we follow the work of Evans and Marshall and propose new approaches for modelling the joint development of macro variables and the returns of government bond yields of several maturities. The models are estimated and compared with other forecasting schemes previously proposed in the literature, especially those relying on univariate, VAR and error correction methods. The models are then used to judge the hypothesis that the information content of macro variables and the term structure of interest rates as a whole help improving forecasting performance. Our main conclusion is quite simple: if one is interested in computing short-term forecasts, then there is no significant improvement in incorporating information other than the one already present in past observations of the yield at hand; however, if one worries about long-term forecasts (which is frequently the case with pension insurance companies), then the information content of macro variables and the term structure can improve forecasting performance.     

On Iterative Methods with Accelerated Convergence for Solving Systems of Nonlinear Equations

We present a modified method for solving nonlinear systems of equations with order of convergence higher than other competitive methods. We generalize also the efficiency index used in the one-dimensional case to several variables. Finally, we show some numerical examples, where the theoretical results obtained in this paper are applied.     

One-pass heuristics for large-scale unconstrained binary quadratic problems

Many significant advances have been made in recent years for solving unconstrained binary quadratic programs (UQP). As a result, the size of problem instances that can be efficiently solved has grown from a hundred or so variables a few years ago to 2000 or 3000 variables today. These advances have motivated new applications of the model which, in turn, have created the need to solve even larger problems. In response to this need, we introduce several new “one-pass” heuristics for solving very large versions of this problem. Our computational experience on problems of up to 9000 variables indicates that these methods are both efficient and effective for very large problems. The significance of problems of this size is that they not only open the door to solving a much wider array of real world problems, but also that the standard linear mixed integer formulations of the nonlinear models involve over 40,000,000 variables and three times that many constraints. Our approaches can be used as stand-alone solution methods, or they can serve as procedures for quickly generating high quality starting points for other, more sophisticated methods.     

Direct versus indirect credit scoring classifications

We introduce a new approach to assigning bank account holders to ‘good’ or ‘bad’ classes based on their future behaviour. Traditional methods simply treat the classes as qualitatively distinct, and seek to predict them directly, using statistical techniques such as logistic regression or discriminant analysis based on application data or observations of previous behaviour. We note, however, that the ‘good’ and ‘bad’ classes are defined in terms of variables such as the amount overdrawn at the time at which the classification is required. This permits an alternative, ‘indirect’, form of classification model in which, first, the variables defining the classes are predicted, for example using regression, and then the class membership is derived deterministically from these predicted values. We compare traditional direct methods with these new indirect methods using both real bank data and simulated data. The new methods appear to perform very similarly to the traditional methods, and we discuss why this might be. Finally, we note that the indirect methods also have certain other advantages over the traditional direct methods.     

Variable selection methods for multi-class classification using signomial function

We develop several variable selection methods using signomial function to select relevant variables for multi-class classification by taking all classes into consideration. We introduce a \(\ell _{1}\)-norm regularization function to measure the number of selected variables and two adaptive parameters to apply different importance weights for different variables according to their relative importance. The proposed methods select variables suitable for predicting the output and automatically determine the number of variables to be selected. Then, with the selected variables, they naturally obtain the resulting classifiers without an additional classification process. The classifiers obtained by the proposed methods yield competitive or better classification accuracy levels than those by the existing methods.     

The exploratory analysis of qualitative variables by means of three-way analysis of two types of quantification matrices

A comparison is made between a number of techniques for the exploratory analysis of qualitative variables. The paper mainly focuses on a comparison between multiple correspondence analysis (MCA) and Gower's principal co-ordinates analysis (PCO), applied to qualitative variables. The main difference between these methods is in how they deal with infrequent categories. It is demonstrated that MCA solutions can be dominated by infrequent categories, and that, especially in such cases, PCO is a useful alternative to MCA, because it tends to downweight the influence of infrequent categories. Apart from studying the difference between MCA and PCO, other alternatives for the analysis of qualitative variables are discussed, and compared to MCA and PCO.     

Discrete random bounds for general random variables and applications to reliability

We here propose some new algorithms to compute bounds for (1) cumulative density functions of sums of i.i.d. nonnegative random variables, (2) renewal functions and (3) cumulative density functions of geometric sums of i.i.d. nonnegative random variables. The idea is very basic and consists in bounding any general nonnegative random variable X   by two discrete random variables with range in hN$h\mathbb{N}$, which both converge to X as h goes to 0. Numerical experiments are lead on and the results given by the different algorithms are compared to theoretical results in case of i.i.d. exponentially distributed random variables and to other numerical methods in other cases.     

An accessible,justifiable and transferable pedagogical approach for the differential equations of simple harmonic motion

Recently, Gauthier introduced a method to construct solutions to the equations of motion associated with oscillating systems into the mathematics education research literature. In particular, Gauthier's approach involved certain manipulations of the differential equations; and drew on the theory of complex variables.

Motivated by the work of Gauthier, we construct an alternative pedagogical approach for the learning and teaching of solution methods to these equations. The innovation lies in drawing on factorization techniques of differential equations and harmonizing them with Gauthier's approach of the theory of complex variables. When blended together to form a new approach, the significance lies in its accessibility, justifiability and transferrability to other problems.

We pedagogically ground our approach in the educational development theory of Piaget, with the results informing the learning and teaching of solution methods to differential equations for lecturers, teachers and learners within universities, colleges, polytechnics and schools around the world.     

Applications of fuzzy linear programming with generalized LR flat fuzzy parameters

In this paper, the limitations of existing methods to solve the problems of fuzzy assignment, fuzzy travelling salesman and fuzzy generalized assignment are pointed out. All these problems can be formulated in linear programming problems wherein the decision variables are represented by real numbers and other parameters are represented by fuzzy numbers. To overcome the limitations of existing methods, a new method is proposed. The advantage of proposed method over existing methods is demonstrated by solving the problems mentioned above which can or cannot be solved by using the existing methods.     

A conversion of an SDP having free variables into the standard form SDP

This paper deals with a semidefinite program (SDP) having free variables, which often appears in practice. To apply the primal–dual interior-point method, we usually need to convert our SDP into the standard form having no free variables. One simple way of conversion is to represent each free variable as a difference of two nonnegative variables. But this conversion not only expands the size of the SDP to be solved but also yields some numerical difficulties which are caused by the non-existence of a primal–dual pair of interior-feasible solutions in the resulting standard form SDP and its dual. This paper proposes a new conversion method that eliminates all free variables. The resulting standard form SDP is smaller in its size, and it can be more stably solved in general because the SDP and its dual have interior-feasible solutions whenever the original primal–dual pair of SDPs have interior-feasible solutions. Effectiveness of the new conversion method applied to SDPs having free variables is reported in comparison to some other existing methods.