Complete Solutions and Extremality Criteria to Polynomial Optimization Problems

This paper presents a set of complete solutions to a class of polynomial optimization problems. By using the so-called sequential canonical dual transformation developed in the author’s recent book [Gao, D.Y. (2000), Duality Principles in Nonconvex Systems: Theory, Method and Applications, Kluwer Academic Publishers, Dordrecht/Boston/London, xviii + 454 pp], the nonconvex polynomials in can be converted into an one-dimensional canonical dual optimization problem, which can be solved completely. Therefore, a set of complete solutions to the original problem is obtained. Both global minimizer and local extrema of certain special polynomials can be indentified by Gao-Strang’s gap function and triality theory. For general nonconvex polynomial minimization problems, a sufficient condition is proposed to identify global minimizer. Applications are illustrated by several examples.     

Approximating Data in \mathfrak{R}^{n} by a Quadratic Underestimator with Specified Hessian Minimum and Maximum Eigenvalues

The problem of approximating m data points (x _i , y _i ) in , with a quadratic function q(x, p) with s parameters, m ≥ s, is considered. The parameter vector is to be determined so as to satisfy three conditions: (1) q(x, p) must underestimate all m data points, i.e. q(x _i , p) ≤ y _i , i=1,...,m. (2) The error of the approximation is to be minimized in the L1 norm. (3) The eigenvalues of H are to satisfy specified lower and upper bounds, where H is the Hessian of q(x, p) with respect to x. This is called the Quadratic Underestimator with Bounds on Eigenvalues (QUBE) problem. An algorithm for its solution (QUBE algorithm) is given and justified, and computational results presented. The QUBE algorithm has application to finding the global minimum of a basin (or funnel) shaped function with a large number of local minima. Such problems arise in computational biology where it is desired to find the global minimum of an energy surface, in order to predict native protein-ligand docking geometry (drug design) or protein structure. Computational results for a simulated docking energy surface, with n=15, are presented. It is shown that specifying a small condition number for H improves the ability of the underestimator to correctly predict the global minimum point.     

Use of thermo‐Raman spectroscopy and chemometric analysis to identify dehydration steps of hydrated inorganic samples—application to copper sulfate pentahydrate

In situ thermo‐Raman spectroscopy (TRS) measurements were performed in order to investigate solid‐phase transformation of the copper sulfate pentahydrate from room temperature up to 300 °C. Band‐target entropy minimization (BTEM), a blind‐source separation algorithm, was employed in order to identify and reconstruct the pure component spectra of the species involved in the dehydration process. In spite of low signal‐to‐noise ratio and elevated baseline spectral data, BTEM was successfully utilized to identify and reconstruct four pure component spectra of copper sulfate pentahydrate, trihydrate, monohydrate, and anhydrate, which were formed during this thermally induced process. Subsequent mapping of these four pure component spectral estimates back onto the preprocessed spectra yielded the relative concentrations of each individual species. Finally, the transition temperatures of each dehydration step could be unambiguously deduced from the obtained concentration profile. The current study shows that combined thermo‐Raman spectroscopy and chemometric analysis provides an effective tool to determine the dehydration temperatures as well as to identify the structures of each individual species involved in a solid‐phase dehydration process. Copyright © 2009 John Wiley & Sons, Ltd.     

电子结构计算的数值方法与理论

第一原理电子结构计算已成为探索与研究物质机理、理解与预测材料性质的重要手段和工具.虽然第一原理电子结构计算取得了巨大的成功,但是如何利用高性能计算机又快又好地计算大规模体系,如何从数学角度理解电子结构模型的合理性与计算的可靠性和有效性,依然充满各种挑战.基于密度泛函理论的第一原理电子结构计算的核心数学模型为Kohn-Sham方程或相应的Kohn-Sham能量泛函极小问题.近年来,人们分别从非线性算子特征值问题的高效离散及Kohn-Sham能量泛函极小问题的最优化方法设计两个方面对电子结构计算的高效算法设计及分析展开了诸多研究.本文重点介绍我们小组在电子结构计算的方法与理论方面的一些进展,同时简单介绍该领域存在的困难与挑战.     

A degenerate elliptic-parabolic system arising in competitive contaminant transport

The objective of this work is to study a coupled system of degenerate and nonlinear partial differential equations governing the transport of reactive solutes in groundwater. We show that this system admits a unique weak solution provided the nonlinear adsorption isotherm associated with the reaction process satisfies certain physically reasonable structural conditions, by addressing a more general problem. In addition, we conclude, that the solute concentrations stay non-negative if the source term is componentwise non-negative and investigate numerically the finite speed of propagation of compactly supported initial concentrations, in a two-component test case.     

Convergence Theorem for Variational Inequality in Hilbert Spaces with Applications

In this paper, we study variational inequality over the set of the common fixed points of a countable family of quasi-nonexpansive mappings. To tackle this problem, we propose an algorithm and use it to prove a strong convergence theorem under suitable conditions. As applications, we study variational inequality over the solution set of different nonlinear or linear problems, like minimization problems, split feasibility problems, convexly pseudoinverse problems, convex linear inverse problems, etc.     

An alternative theorem for set-valued maps via set relations and its application to robustness of feasible sets

Abstract

This paper contains a generalized Gordan-type alternative theorem for set-valued maps which characterizes set relations without any convexity assumptions using certain evaluation functions. As a direct consequence and as a good example, we discuss robustness (or stability) of linear programming problems for modelling error. Moreover, this theorem can be utilized for that of general vector optimization problems in special cases due to reformation of the evaluation functions.     

求解低秩矩阵填充的改进的交替最速下降法

矩阵填充是指利用矩阵的低秩特性而由部分观测元素恢复出原矩阵,在推荐系统、信号处理、医学成像、机器学习等领域有着广泛的应用。采用精确线搜索的交替最速下降法由于每次迭代计算量小因而对大规模问题的求解非常有效。本文在其基础上采用分离地精确线搜索,可使得每次迭代下降更多但计算量相同,从而可望进一步提高计算效率。本文分析了新算法的收敛性。数值结果也表明所提出的算法更加有效。     

学习理论中回归问题的误差估计(英文)

本文在一般的损失函数ψ(y-f(x))下,当ψ(z)连续时,讨论了学习理论中回归问题的误差估计.