Successive Optimization Method via Parametric Monotone Composition Formulation*

In this paper a successive optimization method for solving inequality constrained optimization problems is introduced via a parametric monotone composition reformulation. The global optimal value of the original constrained optimization problem is shown to be the least root of the optimal value function of an auxiliary parametric optimization problem, thus can be found via a bisection method. The parametric optimization subproblem is formulated in such a way that it is a one-parameter problem and its value function is a monotone composition function with respect to the original objective function and the constraints. Various forms can be taken in the parametric optimization problem in accordance with a special structure of the original optimization problem, and in some cases, the parametric optimization problems are convex composite ones. Finally, the parametric monotone composite reformulation is applied to study local optimality.     

Spectral sequences and parametric normal forms

This paper extends recent developments on spectral sequence approach on the simplest normal form theory to parametric cases. In this paper, the method is first applied to two simple examples (parametric single zero and parametric resonant saddle singularities) and then, to parametric generalized Hopf singularity. We provide the simplest parametric normal form for these cases.     

时滞影响下受控斜拉索的参数振动稳定性

研究了轴向激励作用下受控斜拉索系统主参数共振的时滞效应,考虑了拉索垂度和几何非线性的影响,基于Hamilton变分原理建立了受控斜拉索系统轴向激励下的非线性参数振动方程,利用Galerkin方法得到时滞动力系统,运用多尺度法对受控系统的主参数共振进行了分析,得到了不同时滞值、控制增益时参数振动稳定域和受控拉索的时程曲线.研究表明,时滞影响下斜拉索振动控制系统的效果变差,参数共振的稳定域发生偏移,对受控斜拉索系统的控制效果随着时滞的增大而变差,从而对控制系统的参数设计起到指导作用.     

A unifying theory of exactness of linear penalty functions II: parametric penalty functions

In this article, we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.     

参数变分不等式的灵敏性分析

本文在所给函数和映射均不可微的前提下，通过建立参数变分不等式和参数Wiener-Hopf方程的等价性，分析了Hilbert空间中参数变分不等式的局部唯一解的灵敏性。文中所用方法是N.D.Yen之方法的改进，使用这一方法可大大简化N.D.Yen一文中主要结果（引理2.1)的证明。     

参外激励下强非线性广义Van der Pol方程的亚谐共振解（英文）

In this paper a modified L-P method and multiple scale method are used to solve sub-harmonic resonance solutions of strong and nonlinear resonance of general Van der Pol equation with parametric and external excitations by parametric transformation. Bifurcation response equation and transition sets of sub-harmonic resonance with strong nonlinearity of general Van der Pol equation with parametric and external excitation are worked out.Besides, transition sets and bifurcation graphs are drawn to help to analysis the problems theoretically. Conclusions show that the transition sets of general and nonlinear Van der Pol equation with parametric and external excitations are more complex than those of general and nonlinear Van der Pol equation only with parametric excitation, which is helpful for the qualitative and quantitative reference for engineering and science applications.     

一类参数二次规划参数延拓的作用集法和稳定性分析

本文用作用集法考虑一类参数二次规划的参数延拓问题，并研究稳定性问题。     

基于中国地震的损失模型(英文)

对损失分布的估计一直是保险公司的重要问题. 有多种参数方法以及非参数方法拟合损失分布. 本文作者提出了结合参数和非参数的方法来解决损失分布拟合问题. 首先通过超额均值图确定大小损失之间的阈限,再利用广义Pareto分布拟合阈值以上损失, 转换后的核密度估计拟合阈值以下损失. 最后, 通过实证分析将该方法和其他方法进行了误差分析比较, 取得了理想的结果.     

Digital contracts-driven method for pricing complex derivatives

In this paper, we propose a hybrid method of nonparametric and parametric methods, that is a digital contracts-driven (DCD) method, for pricing various complex options. Differing from general nonparametric data-driven methods, in which usually the observed data are used as training data directly, in the DCD method the European-style digital contracts of the underlying assets are used as basic inputs for a learning network. The digital contracts calculated from the observed data based upon the parametric method are used as hints in the learning process, and then enable the DCD method to have superior pricing accuracy to the common data-driven method in practical applications. Some Monte Carlo simulation experiments are performed and the results demonstrate that the proposed hybrid method not only has the advantages of generality and superior accuracy as the nonparametric method, but also the robust property to financial data with noise as the parametric method.     

Covariate selection for semiparametric hazard function regression models

We study a flexible class of nonproportional hazard function regression models in which the influence of the covariates splits into the sum of a parametric part and a time-dependent nonparametric part. We develop a method of covariate selection for the parametric part by adjusting for the implicit fitting of the nonparametric part. Asymptotic consistency of the proposed covariate selection method is established, leading to asymptotically normal estimators of both parametric and nonparametric parts of the model in the presence of covariate selection. The approach is applied to a real data set and a simulation study is presented.     

Using the parametric approach to solve the continuous-time linear fractional max–min problems

A numerical algorithm based on parametric approach is proposed in this paper to solve a class of continuous-time linear fractional max-min programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as a parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this algorithm.     

Direct and iterative methods for interval parametric algebraic systems producing parametric solutions

This paper deals with interval parametric linear systems with general dependencies. Motivated by the so‐called parameterized solution introduced by Kolev, we consider the enclosures of the solution set in a revised affine form. This form is advantageous to a classical interval solution because it enables us to obtain both outer and inner bounds for the parametric solution set and, thus, intervals containing the endpoints of the hull solution, among others. We propose two solution methods, a direct method called the generalized expansion method and an iterative method based on interval‐affine Krawczyk iterations. For the iterative method, we discuss its convergence and show the respective sufficient criterion. For both methods, we perform theoretical and numerical comparisons with some other approaches. The numerical experiments, including also interval parametric linear systems arising in practical problems of structural and electrical engineering, indicate the great usefulness of the proposed methodology and its superiority over most of the existing approaches to solving interval parametric linear systems.     

Likelihood Based Confidence Intervals for the Tail Index

For the estimation of the tail index of a heavy tailed distribution, one of the well-known estimators is the Hill estimator (Hill, 1975). One obvious way to construct a confidence interval for the tail index is via the normal approximation of the Hill estimator. In this paper we apply both the empirical likelihood method and the parametric likelihood method to obtaining confidence intervals for the tail index. Our limited simulation study indicates that the normal approximation method is worse than the other two methods in terms of coverage probability, and the empirical likelihood method and the parametric likelihood method are comparable.     

On the nearest parametric approximation of a fuzzy number

Many nearest parametric approximation methods of fuzzy sets are proposed in the literature. It is clear that the specific approximations may lead to the loss of information about fuzziness. To overcome this problem, most of these methods rely on the minimization of the distance between the original fuzzy set and its approximation. But these approximations mostly are not flexible to the decision maker's choice. Hence, in this paper, we offer a parametric fuzzy approximation method based on the decision maker's strategy as an extension of trapezoidal approximation of a fuzzy number. This method comprises the selection of the form of the parametric membership function and its evaluation.     

A Generalized Time–Cost Trade-Off Transportation Problem

In the classical transportation problem if the unit costs and transportation durations are considered, the time-cost trade-off solutions can be determined by the well-known threshold approach assuming that all the transportations are permitted to be simultaneous in time. If all the unit costs are linear functions of time over a specified interval of time, a parametric technique can be applied for identifying all the time-cost trade-off solutions pertaining to this interval. In this paper, the unit costs considered are piecewise linear non-increasing functions of time and transportations are allowed to be simultaneous. It is shown that a parametric method involving a finite sequence of parametric transportation problems reveals all the time-cost trade-off solutions of this generalized trade-off problem. Computational experience is included. If the transportation problem has considerable degeneracy, the parametric approach may pose some computational difficulty. This difficulty can be reduced by using an alternative method involving the bicriteria optimization approach of Aneja and Nair. Also, a direct method is outlined for the case where a finite set of discrete alternatives of unit cost-time pairs is available.     

Exact traveling wave solutions for a modified Camassa-Holm equation

In this paper, the bifurcation method of planar dynamical systems is utilized to investigate a modified Camassa-Holm equation. After dividing the parametric space, some explicit parametric conditions are derived for the existence of traveling wave solutions. Several exact traveling solutions are also obtained.     

Explicit peakon and solitary wave solutions for the modified Fornberg-Whitham equation

In this paper, the modified Fornberg-Whitham equation is studied by using the bifurcation theory and the method of phase portraits analysis. In some parametric conditions, some peakons and solitary waves are found and their exact parametric representations in explicit form are obtained.     

周期参数扰动的T混沌系统周期轨道分析

本文研究了一类周期参数扰动的T混沌系统的周期轨道问题.利用次谐波Melnikov方法,获得了具有广义Hamilton结构的周期参数扰动的慢变系统的振荡周期轨道和旋转周期轨道.     

Strong Consistency of the Maximum Product of Spacings Estimates with Applications in Nonparametrics and in Estimation of Unimodal Densities

Distributions with unimodal densities are among the most commonly used in practice. However, for many unimodal distribution families the likelihood functions may be unbounded, thereby leading to inconsistent estimates. The maximum product of spacings (MPS) method, introduced by Cheng and Amin and independently by Ranneby, has been known to give consistent and asymptotically normal estimators in many parametric situations where the maximum likelihood method fails. In this paper, strong consistency theorems for the MPS method are obtained under general conditions which are comparable to the conditions of Bahadur and Wang for the maximum likelihood method. The consistency theorems obtained here apply to both parametric models and some nonparametric models. In particular, in any unimodal distribution family the asymptotic MPS estimator of the underlying unimodal density is shown to be universally L¹ consistent without any further conditions (in parametric or nonparametric settings).     

Parametric methods in integer linear programming

In contrast to methods of parametric linear programming which were developed soon after the invention of the simplex algorithm and are easily included as an extension of that method, techniques for parametric analysis on integer programs are not well known and require considerable effort to append them to an integer programming solution algorithm.The paper reviews some of the theory employed in parametric integer programming, then discusses algorithmic work in this area over the last 15 years when integer programs are solved by different methods. A summary of applications is included and the article concludes that parametric integer programming is a valuable tool of analysis awaiting further popularization.