A New Class of Semismooth Newton-Type Methods for Nonlinear Complementarity Problems

We introduce a new, one-parametric class of NCP-functions. This class subsumes the Fischer function and reduces to the minimum function in a limiting case of the parameter. This new class of NCP-functions is used in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations. We present a detailed investigation of the properties of the equation operator, of the corresponding merit function as well as of a suitable semismooth Newton-type method. Finally, numerical results are presented for this method being applied to a number of test problems.     

On NCP-Functions

In this paper we reformulate several NCP-functions for the nonlinear complementarity problem (NCP) from their merit function forms and study some important properties of these NCP-functions. We point out that some of these NCP-functions have all the nice properties investigated by Chen, Chen and Kanzow [2] for a modified Fischer-Burmeister function, while some other NCP-functions may lose one or several of these properties. We also provide a modified normal map and a smoothing technique to overcome the limitation of these NCP-functions. A numerical comparison for the behaviour of various NCP-functions is provided.     

Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems

In this paper, we propose a new family of NCP-functions and the corresponding merit functions, which are the generalization of some popular NCP-functions and the related merit functions. We show that the new NCP-functions and the corresponding merit functions possess a system of favorite properties. Specially, we show that the new NCP-functions are strongly semismooth, Lipschitz continuous, and continuously differentiable; and that the corresponding merit functions have SC¹$S{C}^1$ property (i.e., they are continuously differentiable and their gradients are semismooth) and LC¹$L{C}^1$ property (i.e., they are continuously differentiable and their gradients are Lipschitz continuous) under suitable assumptions. Based on the new NCP-functions and the corresponding merit functions, we investigate a derivative free algorithm for the nonlinear complementarity problem and discuss its global convergence. Some preliminary numerical results are reported.     

A penalized Fischer-Burmeister NCP-function

We introduce a new NCP-function in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations. This new NCP-function turns out to have stronger theoretical properties than the widely used Fischer-Burmeister function and other NCP-functions suggested previously. Moreover, numerical experience indicates that a semismooth Newton method based on this new NCP-function performs considerably better than the corresponding method based on the Fischer-Burmeister function. Received: March 10, 1997 / Accepted: February 15, 2000?Published online May 12, 2000     

A regularization semismooth Newton method based on the generalized Fischer–Burmeister function for -NCPs

We consider a regularization method for nonlinear complementarity problems with F being a P₀-function which replaces the original problem with a sequence of the regularized complementarity problems. In this paper, this sequence of regularized complementarity problems are solved approximately by applying the generalized Newton method for an equivalent augmented system of equations, constructed by the generalized Fischer–Burmeister (FB) NCP-functions φ_p with p>1. We test the performance of the regularization semismooth Newton method based on the family of NCP-functions through solving all test problems from MCPLIB. Numerical experiments indicate that the method associated with a smaller p, for example p[1.1,2], usually has better numerical performance, and the generalized FB functions φ_p with p[1.1,2) can be used as the substitutions for the FB function φ₂.     

A non-interior continuation algorithm for the CP based on a generalized smoothing function

Based on the generalized CP-function proposed by Hu et al. [S.L. Hu, Z.H. Huang, J.S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math. 230 (2009) 69-82], we introduce a smoothing function which is a generalization of several popular smoothing functions. By which we propose a non-interior continuation algorithm for solving the complementarity problem. The proposed algorithm only needs to solve at most one system of linear equations at each iteration. In particular, we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions. The preliminary numerical results demonstrate that the algorithm is effective.     

Modelling class hierarchies in the fuzzy object-oriented data model

In this paper we describe a fuzzy logic based approach to modelling uncertainty in class hierarchies. It is shown that the traditional view of class hierarchies is subsumed in this model as a special case. The problem of multiple inheritance in class hierarchies is discussed and analyzed. The membership value derivations in the inheritance hierarchy reflects the degree of fuzziness existing in the data values and the semantics of the situation being modelled. Thus a more realistic modelling of the universe of discourse is possible through this approach. This model is compatible with existing object-oriented data models.     

Two Approaches for State Space Realization of NARMA Models: Bridging the Gap

This paper studies the necessary and sufficient conditions for observable realization of a general class of nonlinear high-order input-output difference equations. In particular, it proves the equivalence of the two seemingly different existing approaches in the literature. The paper also provides a subclass of NARMA input-output models that are guaranteed to have an observable realization. It is shown that this class covers several important subclasses of existing NARMA models.     

A note on the iterative solution of recurrence relations

Summary A class of Gauss-Seidel iteration schemes suitable for the stable generation of non-dominant solutions of certain third order linear recurrence relations is developed. The algorithms derived have two main advantages over existing algorithms which generally re-formulate the problem as the solution of a system of algebraic equations. Firstly, unlike existing algorithms, the algorithms developed in this paper automatically determine the size of the system to be solved in all cases and secondly they may be extended directly to an important class of nonlinear recurrence relations.     

On Markov-Additive Jump Processes

In 1995, Pacheco and Prabhu introduced the class of so-called Markov-additive processes of arrivals in order to provide a general class of arrival processes for queueing theory. In this paper, the above class is generalized considerably, including time-inhomogeneous arrival rates, general phase spaces and the arrival space being a general vector space (instead of the finite-dimensional Euclidean space). Furthermore, the class of Markov-additive jump processes introduced in the present paper is embedded into the existing theory of jump processes. The best known special case is the class of BMAP arrival processes.     

A two-parameter of weighted exponential distributions

In this paper, a class of distributions called the two-parameter weighted exponential distributions is introduced (TWE). This new class of distributions generalizes the ones-weighted exponential distributions (WE)-proposed by Gupta and Kundu (2009). The main properties of this new class of distributions are investigated. Several statistical properties and statistical inferences are then obtained and studied. Two real data sets of which one is a right censored data set were analyzed, and it is shown that in both two cases our model fits much better than WE or some other existing models.     

A CLASS OF GENERALIZED MULTISPLITTING RELAXATION METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

In this paper,a class of generalized parallel matrix multisplitting relaxation methods for solving linear complementarity problems on the high-speed multiprocessor systems is set up. This class of methods not only includes all the existing relaxation methods for the linear complementarity problems ,but also yields a lot of novel ones in the sense of multisplittlng. We establish the convergence theories of this class of generalized parallel multisplitting relaxation methods under the condition that the system matrix is an H-metrix with positive diagonal elements.     

Optimal control of discrete systems involving delays

For a class of discrete systems with delays and additional state-dependent constraints, a maximum principle is derived. The existing general results in the field of discrete optimal control theory make it possible to deal with this class of optimization problems in a straight-forward way. The formulation of the results presented is fairly general and includes a number of cases when the constraints are specified in more detail.     

Optimizing System Availability Under Minimal Repair with Non-Negligible Repair and Replacement Times

Availability measures are given for a repairable system under minimal repair with constant repair times. A new policy and an existing replacement policy for this type of system are discussed. Each involves replacement at the first failure after time T, with T representing total operating time in the existing model and total elapsed time (i.e. operating time + repair time) in the new model. Optimal values of T are found for both policies over a wide range of parameter values. These results indicate that the new and administratively easier policy produces only marginally smaller optimal availability values than the existing policy.     

Nonparametric Estimation of Extreme Conditional Quantiles with Functional Covariate

Estimation of the extreme conditional quantiles with functional covariate is an important problem in quantile regression. The existing methods, however, are only applicable for heavy-tailed distributions with a positive conditional tail index. In this paper, we propose a new framework for estimating the extreme conditional quantiles with functional covariate that combines the nonparametric modeling techniques and extreme value theory systematically. Our proposed method is widely applicable, no matter whether the conditional distribution of a response variable Y given a vector of functional covariates X is short, light or heavy-tailed. It thus enriches the existing literature.     

Generalized Sampling and Infinite-Dimensional Compressed Sensing

We introduce and analyze a framework and corresponding method for compressed sensing in infinite dimensions. This extends the existing theory from finite-dimensional vector spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary by demonstrating that existing finite-dimensional techniques are ill suited for solving a number of key problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. A conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. Central to this work is the introduction of two novel concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize an infinite-dimensional problem.     

一类环上HX环的结构

自李洪兴１９９１年提出了ＨＸ环以来，人们一直有这么一个问题没解决，就是是否存在非平凡的ＨＸ环的例子？但至今既没找到非平凡的ＨＸ环，也没有证明任一环Ｒ仅存在平凡的ＨＸ环。针对这个问题，本文提出并证明了一类环仅有平凡ＨＸ环，还给出了一系列的结构定理。这样，既为证明任一环Ｒ仅有平凡的ＨＸ环的猜想有新的启示，也为人们指明无须在这一类环上寻找非平凡ＨＸ环。     

Kernel quantile estimator with ICI adaptive bandwidth selection technique

In this article, we consider a class of kernel quantile estimators which is the linear combi- nation of order statistics. This class of kernel quantile estimators can be regarded as an extension of some existing estimators. The exact mean square error expression for this class of estimators will be provided when data are uniformly distributed. The implementation of these estimators depends mostly on the bandwidth selection. We then develop an adaptive method for bandwidth selection based on the intersection confidence intervals （ICI） principle. Monte Carlo studies demonstrate that our proposed approach is comparatively remarkable. We illustrate our method with a real data set.     

Positive Solutions of a Nonlinear Neutral Equation with Positive and Negative Coefficients

In this paper, a class of nonlinear neutral equations with positive and negative coefficients is considered. Several conditions for the existence and nonexistence of eventually positive solutions are obtained. These conditions extend and improve some of the existing criteria in the literature.