极大值函数的一类光滑逼近函数的性质研究

针对极大值函数的一类光滑逼近——凝聚函数,对其作进一步研究.指出凝聚函数的一阶导数对光滑参数取极限时恰好得到极大值函数的一个次梯度,从而凝聚函数不仅可以一致逼近极大值函数,而且该函数富含极大值函数的一阶信息,可很好的刻画极大值函数的一阶特征.进一步,对光滑逼近函数的光滑参数做简单分析,得到的结果揭示了光滑参数的变动对凝聚函数的影响.并分别以正值函数及绝对值函数为例,对所得到的结果给出几何说明.     

New filled functions for nonsmooth global optimization

The filled function method is an effective approach to find a global minimizer for a general class of nonsmooth programming problems with a closed bounded domain. This paper gives a new definition for the filled function, which overcomes some drawbacks of the previous definition. It proposes a two-parameter filled function and a one-parameter filled function to improve the efficiency of numerical computation. Based on these analyses, two corresponding filled function algorithms are presented. They are global optimization methods which modify the objective function as a filled function, and which find a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. Numerical results obtained indicate the efficiency and reliability of the proposed filled function methods.     

Fractal Continuation

A fractal function is a function whose graph is the attractor of an iterated function system. This paper generalizes analytic continuation of an analytic function to continuation of a fractal function.     

周期函数和非平稳周期函数的小波变换

探讨了三角函数、周期函数以及一类非平稳周期函数小波变换的一些性质,发现周期函数的小波能谱的峰高和峰宽均正比于信号的周期.提出了一个新的只利用与信号周期有关的一个尺度小波变换系数的重构公式,它可准确地重构三角函数,对一般周期函数的重构结果优于其Fourier级数中的任何一项,对一类均值和振幅变化的非平稳周期函数的重构结果与信号非常吻合.     

求解箱式约束全局优化问题的新的F-C函数（英文）

To solve the global optimization problems which have several local minimizers,a new F-C function is proposes by combining a filled function and a cross function. The properties of the F-C function are discussed and the corresponding algorithm is given in this paper. F-C function has the same local minimizers with the objective function.Therefore, the F-C function method only needs to minimize the objective function once in the first iteration. Numerical experiments are performed and the results show that the proposed method is very effective.     

A filled function method with one parameter for unconstrained global optimization

The filled function method is considered as an efficient approach to solve the global optimization problems. In this paper, a new filled function method is proposed. Its main idea is as follows: a new continuously differentiable filled function with only one parameter is constructed for unconstrained global optimization when a minimizer of the objective function is found, then a minimizer of the filled function will be found in a lower basin of the objective function, thereafter, a better minimizer of the objective function will be found. The above process is repeated until the global optimal solution is found. The numerical experiments show the efficiency of the proposed filled function method.     

A filled function method applied to nonsmooth constrained global optimization

The filled function method is an effective approach to find a global minimizer. In this paper, based on a new definition of the filled function for nonsmooth constrained programming problems, a one-parameter filled function is constructed to improve the efficiency of numerical computation. Then a corresponding algorithm is presented. It is a global optimization method which modify the objective function as a filled function, and which find a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. Illustrative examples are provided to demonstrate the efficiency and reliability of the proposed filled function method.     

Integrals of psi—function

To the author's knowledge, among the so—called special functions, the gamma function is a unique one which is defined by a linear difference equation and is a hyper—transcendental function. There exists an another well—known hyper—transcendental function called the psi function, which is merely the logarithmic derivative of the gamma function. In this paper the author consider an extension of the gamma function and then obtain a series of integrals of the psi function.     

Multifunctional assessment and zoning of crop production system based on set pair analysis-A comparative study of 31 provincial regions in mainland China

In this study, we present a multifunctional indicator system for the performance evaluation of crop production system by set pair analysis method. Five functions were summarized to represent the multifunctionality of crop production system, including production function, supply function, ecological function, security function and economic function. Setting a case study of 31 provincial regions in mainland China, this paper conducted a comparison of each function in different regions, divided into 9 groups by cluster analysis. The results show that: the levels of multifunction in most regions are under a low degree balance; the production function has a high coordination with the economic function and security function in China; the supply function is lowly correlated with the other functions, especially the economic function has negative correlation with the supply function to some extent; some relevant policies and suggestions are deduced for multifunctional improvement. It is concluded that the multifunctional indicators and the set pair analysis method can serve as an effective method for the assessment of crop production system.     

全局最优化问题的一个无参数的填充函数算法

本文研究了全局最优化问题.利用构造填充函数的方法,提出了一个新的无参数填充函数,它是目标函数的一个明确表达式.得到了一个新的无参数填充函数算法,数值试验结果表明该填充函数算法是有效的,从而推广了填充函数算法在求解全局最优化问题方面的应用.     

Unconstrained Optimization Reformulations of Variational Inequality Problems

Recently, Peng considered a merit function for the variational inequality problem (VIP), which constitutes an unconstrained differentiable optimization reformulation of VIP. In this paper, we generalize the merit function proposed by Peng and study various properties of the generalized function. We call this function the D-gap function. We give conditions under which any stationary point of the D-gap function is a solution of VIP and conditions under which it provides a global error bound for VIP. We also present a descent method for solving VIP based on the D-gap function.     

An Algorithm for Finding Global Minimum of Nonlinear Integer Programming

A filled function is proposed by R.Ge[2] for finding a global minimizer of a function of several continuous variables. In [4], an approach for finding a global integer minimizer of nonlinear function using the above filled function is given. Meanwhile a major obstacle is met: if $ρ > 0$ is small, and $||x_I-\overset{*}{x}_I||$ is large, where $x_I$ - an integer point, $\overset{*}{x}_I$ - a current local integer minimizer, then the value of the filled function almost equals zero. Thus it is difficult to recognize the size of the value of the filled function and can not find the global integer minimizer of nonlinear function. In this paper, two new filled functions are proposed for finding global integer minimizer of nonlinear function, and the new filled function improves some properties of the filled function proposed by R. Ge [2].Some numerical results are given, which indicate the new filled function (4.1) to find global integer minimizer of nonlinear function is efficient.     

基于功能不可加系统的系统优化

为了揭示系统的某一功能达到最优与其所有子系统为实现系统这一功能而提供的功能之间的关系,基于系统功能具有可加和不可加的特性,探讨了功能不可加系统的功能优化方法,以子系统功能的实现度为决策变量,以系统功能发挥最大效用为目标函数,以实现功能所需的资源为约束条件,建立了功能不可加系统优化的规划模型,最后辅以算例,并通过EXCEL Solver求解进行了分析.     

Airy函数——二阶微分方程f″-zf=0的特解

本文讨论了一个用含变量的积分表示的函数 ,即 Airy函数 .证明了 Airy函数是整函数 ,并且是二阶微分方程 f″-zf=0的一个特解 ,进一步给出了 Airy函数的麦克劳林展开式 .     

Designing a majorization scheme for the recourse function in two-stage stochastic linear programming

We discuss issues pertaining to the domination from above of the second-stage recourse function of a stochastic linear program and we present a scheme to majorize this function using a simpler sublinear function. This majorization is constructed using special geometrical attributes of the recourse function. The result is a proper, simplicial function with a simple characterization which is well-suited for calculations of its expectation as required in the computation of stochastic programs. Experiments indicate that the majorizing function is well-behaved and stable.     

无约束全局优化的一个可调参数的填充函数

A filled function with adjustable parameters is suggested in this paper for finding a global minimum point of a general class of nonlinear programming problems with a bounded and closed domain. This function has two adjustable parameters. We will discuss the properties of the proposed filled function. Conditions on this function and on the values of parameters are given so that the constructed function has the desired properties of traditional filled function.     

A convex representation of totally balanced games

We analyze the least increment function, a convex function of n variables associated to an n-person cooperative game. Another convex representation of cooperative games, the indirect function, has previously been studied. At every point the least increment function is greater than or equal to the indirect function, and both functions coincide in the case of convex games, but an example shows that they do not necessarily coincide if the game is totally balanced but not convex. We prove that the least increment function of a game contains all the information of the game if and only if the game is totally balanced. We also give necessary and sufficient conditions for a function to be the least increment function of a game as well as an expression for the core of a game in terms of its least increment function.     

Criteria for generalized invex monotonicities

In this paper, under appropriate conditions, we establish that (i) if the gradient of a function is (strictly) pseudo-monotone, then the function is (strictly) pseudo-invex; (ii) if the gradient of a function is quasi-monotone, then the function is quasi-invex; and (iii) if the gradient of a function is strong pseudo-monotone, then the function is strong pseudo-invex.     

Solution of the control problem for a linear descriptor system with rectangular coefficient matrix

We pose a control problem for linear stationary algebro-differential system whose coefficients are rectangular matrices. We obtain a controllability criterion for the partially given output function and the boundary values of the state function. We construct a control function and a state function under minimal smoothness requirements on the output function.     

基于二次多项式的积极集光滑化max函数及其在无约束minimax问题中的应用

本文基于分段二次多项式方程,构造了一种积极集策略的光滑化max函数.通过给出与光滑化max函数相关的分量函数指标集的直接计算方法,将分段二次多项式方程转化为一般二次多项式方程.利用二次多项式方程根的性质,给出了该光滑化max函数的稳定计算策略,证明了其具有一阶光滑性,其梯度函数具有局部Lipschitz连续性和强半光滑性.该光滑化max函数仅与函数值较大的分量函数相关,适用于含分量函数较多且复杂的max函数的问题.为了验证其效率,本文基于该函数构造了一种解含多个复杂分量函数的无约束minimax问题的光滑化算法,数值实验表明了该光滑化max函数的可行性及有效性.