An iterative scheme for solving nonlinear equations with monotone operators

An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with monotone operators is proposed and its convergence is proved. A discrepancy principle is proposed and justified. A priori and a posteriori stopping rules for the iterative scheme are formulated and justified. AMS subject classification (2000)  47J05, 47J06, 47J35, 65R30     

The full-information best choice problem with a random number of observations

The full-information best choice problem with a random number of observations is considered. N i.i.d. random variables with a known continuous distribution are observed sequentially with the object of selecting the largest. Neither recall nor uncertainty of selection is allowed and one choice must be made. In this paper the number N of observations is random with a known distribution. The structure of the stopping set is investigated. A class of distributions of N (which contains in particular the uniform, negative-binomial and Poisson distributions) is determined, for which the so-called “monotone case” occurs. The theoretical solution for the monotone case is considered. In the case where N is geometric the optimal solution is presented and the probability of winning worked out. Finally, the case where N is uniform is examined. A simple asymptotically optimal stopping rule is found and the asymptotic probability of winning is obtained.     

NONDESCENT SUBGRADIENT METHOD FOR NONSMOOTH CONSTRAINED MINIMIZATION

A kind of nondecreasing subgradient algorithm with appropriate stopping rule has been proposed for nonsmooth constrained minimization problem. The dual theory is invoked in dealing with the stopping rule and general global minimiizing algorithm is employed as a subroutine of the algorithm. The method is expected to tackle a large class of nonsmooth constrained minimization problem.     

Confident estimation for density of a biological population based on line transect sampling

Line transect sampling is a very useful method in survey of wildlife population. Confident interval estimation for density D of a biological population is proposed based on a sequential design. The survey area is occupied by the population whose size is unknown. A stopping rule is proposed by a kernel-based estimator of density function of the perpendicular data at a distance. With this stopping rule, we construct several confidence intervals for D by difference procedures. Some bias reduction techniques are used to modify the confidence intervals. These intervals provide the desired coverage probability as the bandwidth in the stopping rule approaches zero. A simulation study is also given to illustrate the performance of this proposed sequential kernel procedure.     

Sequential Estimation of the Maximum in a Model for Corrosion Data

One method of monitoring corrosion in an underground storage tank involves placing a sensor in the tank and running it around the tank's interior. As it runs, the sensor records the local thickness of the tank. In this paper we consider the problem of estimating the maximum pit depth by providing a confidence interval that achieves both a specified confidence level and a specified degree of precision. A particular model, the three-parameter beta, is considered, and a stopping rule for determining the sample size is proposed. It is shown that the stopping rule achieves the desired confidence level and precision, asymptotically as the precision requirement becomes increasingly stringent. Moreover, the stopping rule is asymptotically efficient in terms of sample size. The limiting distribution of the stopping rule is derived, and simulation results are presented to supplement the asymptotics with finite sample size behavior.     

Monotone stopping rules forstochastic processes in a semimartingale representation with applications

A monotone stopping problem is considered for stochastic processes in a semimartingale representation. Such a representation allows a direct infinitesimal characterization of the optimal stopping time. Transformations of such processes are investigated, which leave the semimartingale property unchanged. One of these transformations is a change of tiltration which leads to the stopping problem with partial information. Findly an application is discussed.     

AIR Tools — A MATLAB package of algebraic iterative reconstruction methods

We present a MATLAB package with implementations of several algebraic iterative reconstruction methods for discretizations of inverse problems. These so-called row action methods rely on semi-convergence for achieving the necessary regularization of the problem. Two classes of methods are implemented: Algebraic Reconstruction Techniques (ART) and Simultaneous Iterative Reconstruction Techniques (SIRT). In addition we provide a few simplified test problems from medical and seismic tomography. For each iterative method, a number of strategies are available for choosing the relaxation parameter and the stopping rule. The relaxation parameter can be fixed, or chosen adaptively in each iteration; in the former case we provide a new “training” algorithm that finds the optimal parameter for a given test problem. The stopping rules provided are the discrepancy principle, the monotone error rule, and the NCP criterion; for the first two methods “training” can be used to find the optimal discrepancy parameter.     

On a new stopping rule for stochastic approximation

Summary A new stopping rule for the Robbins-Monro process, based on an F-statistic criterion is proposed and its asymptotic behavior established. On the basis of evidence obtained through experimental sampling, the procedure seems to work well over a wide variety of situations. A two-stage procedure, coupling the new rule with an earlier one proposed by Sielken [1973] is recommended for practical use.     

Goldstein, Levitin-Polyak投影法中的一个可行步长准则

１引　言设Ω是Ｒｎ空间的一个非空的凸闭紧子集,Ｆ是Ｒｎ→Ｒｎ的算子．我们考虑变分不等式问题： 变分不等式问题在数学规划中起着很重要的作用,因此,长期以来一直受到广泛的重视．求解变分不等式问题的方法中,有一类投影迭代方法,例如［１,４,６,９］．在所有的投影迭代方法中,Ｇｏｌｄｓｔｅｉｎ［６］,Ｌｅｖｉｔｉｎ－Ｐｏｌｙａｋ［９］所提出的方法;是最简单的．这里,ＰΩ（ｘ）是ｘ在　上的投影,即　的唯一解． 我们称算子Ｆ在集合Ω上是单调的,若在用Ｇｏｌｄｓｔｅｉｎ,Ｌｅｖｉｔｉｎ－Ｐｏｌｙａｋ方法（２）求…     

A POTENTIAL REDUCTION ALGORITHM FOR MONOTONE VARIATIONAL INEQUALITY PROBLEMS

A potential reduction algorithm is proposed for the solution of monotone variational inequality problems. At each step of the algorithm, a system of linear equations is solved to get the search direction and the Armijo's rule is used to determine the stepsize.It is proved that the algorithm is globally convergent. Computational results are reported.     

Sequential stopping rules for the multistart algorithm in global optimisation

In this paper a sequential stopping rule is developed for the Multistart algorithm. A statistical model for the values of the observed local maxima of an objective function is introduced in the framework of Bayesian non-parametric statistics. A suitablea-priori distribution is proposed which is general enough and which leads to computationally manageable expressions for thea-posteriori distribution. Sequential stopping rules of thek-step look-ahead kind are then explicitly derived, and their numerical effectiveness compared.     

On the use of fuzzy inference techniques in assessment models: part II: industrial applications

In this paper, we study the applicability of the monotone output property and the output resolution property in fuzzy assessment models to two industrial Failure Mode and Effect Analysis (FMEA) problems. First, the effectiveness of the monotone output property in a single-input fuzzy assessment model is demonstrated with a proposed fuzzy occurrence model. Then, the usefulness of the two properties to a multi-input fuzzy assessment model, i.e., the Bowles fuzzy Risk Priority Number (RPN) model, is assessed. The experimental results indicate that both the fuzzy occurrence model and Bowles fuzzy RPN model are able to fulfill the monotone output property, with the derived conditions (in Part I) satisfied. In addition, the proposed rule refinement technique is able to improve the output resolution property of the Bowles fuzzy RPN model.     

Contraction behaviour of iteration–discretization based on gradient type projections

There is a wide range of iterative methods in infinite dimensional spaces to treat variational equations or variational inequalities. As a rule, computational handling of problems in infinite dimensional spaces requires some discretization. Any useful discretization of the original problem leads to families of problems over finite dimensional spaces. Thus, two infinite techniques, namely discretization and iteration are embedded into each other. In the present paper, the behaviour of truncated iterative methods is studied, where at each discretization level only a finite number of steps is performed. In our study no accuracy dependent a posteriori stopping criterion is used. From an algorithmic point of view, the considered methods are of iteration–discretization type. The major aim here is to provide the convergence analysis for the introduced abstract iteration–discretization methods. A special emphasis is given on algorithms for the treatment of variational inequalities with strongly monotone operators over fixed point sets of quasi-nonexpansive mappings.     

The monotone case approach for the solution of certain multidimensional optimal stopping problems

This paper studies explicitly solvable multidimensional optimal stopping problems of sum- and product-type in discrete and continuous time using the monotone case approach. It gives a review on monotone case stopping using the Doob decomposition, resp. Doob–Meyer decomposition in continuous time, also in its multiplicative versions. The approach via these decompositions leads to explicit solutions for a variety of examples, including multidimensional versions of the house-selling and burglar’s problem, the Poisson disorder problem, and an optimal investment problem.     

Support Recovery from Noisy Measurement via Orthogonal Multi-Matching Pursuit

In this paper, a new stopping rule is proposed for orthogonal multi-matching pursuit (OMMP). We show that, for $ℓ_2$ bounded noise case, OMMP with the new stopping rule can recover the true support of any $K$-sparse signal $x$ from noisy measurements $y = Φx + e$ in at most $K$ iterations, provided that all the nonzero components of $x$ and the elements of the matrix $Φ$ satisfy certain requirements. The proposed method can improve the existing result. In particular, for the noiseless case, OMMP can exactly recover any $K$-sparse signal under the same RIP condition.     

关于外梯度法的步长规则

１．引言 设为Ｒｎ中的一个非空闭凸集,Ｆ（ｘ）为Ｒｎ　Ｒｎ中的一个连续向量函数．变分不等式问题（Ｆ,）就是：找一向量ｘ　使得当 ＝Ｒ时,（１．１）退化成非线性互补问题。在这篇文章中总假定：（Ｈ１） ,这里表示（１．１）的解集;（Ｈ２）Ｆ（ｘ）是单调的,即对,（ｘ－ｙ）（Ｆ（ｘ）－Ｆ（ｘ）－Ｆ（ｙ））． 这类问题出现在工程物理、经济管理等领域,有着极为广泛的应用．因此,其数值解近年来受到重视,提出许多有效算法,见综述［１, ２］．在现有的算法中, Ｋｏｒｐｅｌｅｖｉｃｈ的外梯度法［３］（何炳生称它为投影…     

连续型单参数指数族参数的经验Bayes检验函数的收敛速度

对独立同分布样本情形的连续型单参数指数族的单边假设检验问题,在线性损失下 导出了单调的Bayes检验函数,构造了相应的经验Bayes(EB)检验函数. 在一定条件下, 获得的经验Bayes检验函数的收敛速度可任意接近$O(n^{-1})$.最后给出了满足定理条件的两个例子.     

A general method for finding the optimal threshold in discrete time

We develop an approach for solving one-sided optimal stopping problems in discrete time for general underlying Markov processes on the real line. The main idea is to transform the problem into an auxiliary problem for the ladder height variables. In case that the original problem has a one-sided solution and the auxiliary problem has a monotone structure, the corresponding myopic stopping time is optimal for the original problem as well. This elementary line of argument directly leads to a characterization of the optimal boundary in the original problem. The optimal threshold is given by the threshold of the myopic stopping time in the auxiliary problem. Supplying also a sufficient condition for our approach to work, we obtain solutions for many prominent examples in the literature, among others the problems of Novikov-Shiryaev, Shepp-Shiryaev, and the American put in option pricing under general conditions. As a further application we show that for underlying random walks (and Lévy processes in continuous time), general monotone and log-concave reward functions g lead to one-sided stopping problems.     

Sequential Fixed-Width Confidence Interval for the Product of Two Means

For the product of two population means, the problem of constructing a fixed-width confidence interval with preassigned coverage probability is considered. It is shown that the optimal sample sizes which minimize the total sample size and at the same time guarantee a fixed-width confidence interval of desired coverage depend on the unknown parameters. In order to overcome this, a fully sequential procedure consisting of a sampling scheme and a stopping rule are proposed. It is then shown that the sequential confidence interval is asymptotically consistent and the stopping rule is asymptotically efficient, as the width goes to zero. Furthermore, a second order result for the difference between the expected stopping time and the (total) optimal fixed sample size is established. The theoretical results are supported by appropriate simulations.     

A stopping rule for choosing the best of three coins

A recently proposed stopping rule for choosing the best of three coins is presented as the first exit time from a certain polygonal region by a two-dimensional random walk. A diffusion approximation to the expected value of this exit time is developed, using techniques of conformal mapping. An algebraic method for calculating the exact value of this expectation is also described.