A solvability theorem and minimax fractional programming

This paper presents a stable solvability theorem for general inequality systems under a local closedness condition. It is shown how this mild regularity condition can be characterized by the validity of the solvability theorem for all local perturbations. Based on this solvability theorem zero duality gap and stability are established for general minimax fractional programming problems.The research was initiated while the first named author was a visitor at the University of New South Wales and was completed while the second named author was a visitor at the Technische Hochschule Darmstadt.     

带梯度障碍的抛物型变分不等式解的存在性唯一性和正则性

本文讨论带梯度障碍的抛物型变分不等式解的存在性、唯一性和正则性问题．通过证明一类带梯度障碍的问题的求解等价于解某个双边障碍的问题，并利用双边障碍问题解的存在性、唯一性和正则性，得到了带梯度障碍的问题的相应结果．这一方法将有助于对具有梯度约束的非线性以及完全非线性抛物型方程解的正则性的研究．     

四元数矩阵的右特征值的虚部的估计

指出四元数矩阵A的右特征的虚部的模值是唯一确定的，得到了一些关于A的右特征值的虚部模的估计，推广改进了屠伯埙教授的相应结果．     

四元数矩阵的右特征值的实部估计

本文给出了关于四元数矩阵的右特征值的实部估计的一些不等式，这些不等式推广、改进了屠伯埙教授，Ｐａｂｌｏ Ｔａｒａｚａｇａ，Ｈ．Ｗｏｌｋｏｗｉｃｚ和Ｇ．Ｐ．Ｈ．Ｓｔｙａｎ的相应结果。     

拟常曲率流形中有平行平均曲率向量的子流形

研究了拟常曲率流形中具有平行平均曲率向量的子流形,给出了两个积分不等式.     

Augmented Lagrangian Objective Penalty Function

Augmented Lagrangian function is one of the most important tools used in solving some constrained optimization problems. In this article, we study an augmented Lagrangian objective penalty function and a modified augmented Lagrangian objective penalty function for inequality constrained optimization problems. First, we prove the dual properties of the augmented Lagrangian objective penalty function, which are at least as good as the traditional Lagrangian function's. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker condition. This is especially so when the Karush-Kuhn-Tucker condition holds for convex programming of its saddle point existence. Second, we prove the dual properties of the modified augmented Lagrangian objective penalty function. For a global optimal solution, when the exactness of the modified augmented Lagrangian objective penalty function holds, its saddle point exists. The sufficient and necessary stability conditions used to determine whether the modified augmented Lagrangian objective penalty function is exact for a global solution is proved. Based on the modified augmented Lagrangian objective penalty function, an algorithm is developed to find a global solution to an inequality constrained optimization problem, and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the modified augmented Lagrangian objective penalty function is proved for a local solution. An algorithm is presented in finding a local solution, with its convergence proved under some conditions.     

Inequality constrained ridge regression estimator

We carry out the idea of inequality constrained least squares (ICLS) estimation of Liew (1976) to the inequality constrained ridge regression (ICRR) estimation. We propose ICRR estimator by reducing the primal–dual relation to the fundamental problem of Dantzig and Cottle, 1967, Cottle and Dantzig, 1974 with Lemke (1962) algorithm. Furthermore, we conduct a Monte Carlo experiment.     

Simulation-Based Simultaneous Confidence Bands in Multiple Linear Regression With Predictor Variables Constrained in Intervals

This article presents a method for the construction of a simultaneous confidence band for the normal-error multiple linear regression model. The confidence bands considered have their width proportional to the standard error of the estimated regression function, and the predictor variables are allowed to be constrained in intervals. Past articles in this area gave exact bands only for the simple regression model. When there is more than one predictor variable, only conservative bands are proposed in the statistics literature. This article advances this methodology by providing simulation-based confidence bands for regression models with any number of predictor variables. Additionally, a criterion is proposed to assess the sensitivity of a simultaneous confidence band. This criterion is defined to be the probability that a false linear regression model is excluded from the band at least at one point and hence this false linear regression model is correctly declared as a false model by the band. Finally, the article considers and compares several computational algorithms for obtaining the confidence band.     

On a class of discontinuous vector-valued functions and the associated quasi-variational inequalities

We study in detail a class of discontinuous vector-valued functions defined on a closed convex subset of Rn, which was introduced by B. Ricceri [7] and which is very useful in the theory of variational inequalities. The results are used to give a new proof for the existence theorem due to P. Cubiotti [3]. The proof allows us to have a better understanding of quasi-variational inequalities associated with the abovementioned class of functions.