The applications of shannon entropy and kullback ' s cross - entropy as perturbations are discussed . by maximizing the perturbed lagrangians in dual space , we obtain the exponential penalty function and exponential multiplier penalty function , respectively , for inequality constrained nlps 文中分别以shannon熵函数和kullback叉熵函数作为摄动函数，导出其对偶函数分别是原问题的指数罚函数和指数乘子罚函数。
In efgm , in order to get a numerical solution for a partial differential equation , the shape function is constructed by moving least squares ( mls ) , the control equation is derived from the weak form of variational equation and essential boundary conditions are imposed by penalty function method 它采用移动最小二乘法构造形函数，从能量泛函的弱变分形式中得到控制方程，并用罚函数法施加本质边界条件，从而得到偏微分方程的数值解。
In the modeling , the paper has replaced the original single - factor model with multi - factors expenses model ; in the solution , the paper has used the spsa and 2spsa algorithm , and proposed one kind " punish function method with absolute value " to transform optimization problems 在建模方面,提出一种多元费用模型,代替了原有的一元模型;在求解方面,采用了最新的spsa算法和2spsa算法,并提出了一种“绝对值罚函数法”用来转化最优化问题。
Secondly , the penalty coefficient may converge to infinity in many situations when the iterative point is closely near the bound of feasible set , while the parameters are bounded if the solution set of constrained optimization is nonempty , which is available for numerical computation 另外在很多情况下，罚函数法中的罚因子当迭代点接近可行域边界时趋于无穷大，而参数控制算法中，只要约束优化问题有最优解，则参数是有界的，这对数值计算是有利的。
To date , in the study of the common linear bilevel programming prob - lem to admit an exact penalty function formulation , it is required usually that the optimal value achieves at some , vertex of a permissible set p , the set p consisted of the vectors which conformed to the leader ' s and the follow ' s constraints 迄今为止，对于一般的线性双层规划问题的恰当罚函数的研究，通常都要求问题具有最优值可以在允许集p的某个极点处达到这一性质。这里允许集p是指满足问题( lfbp )上下两层中所有的不等式约束的变量的集合。
To assure astringency , some technologies have been used such as iterative penality function methods , assemblage mass matrix , reduced integration algoritlun , newton iteration method with parameters for non - linear equation set , introducing relax factors and double steps solution and so on , and an algorithin for solving the nonlinear equation set of flow field by fem has been presented 基于有限元法建立了流场求解列式，为保证其收敛性，采用了迭代罚函数法，集中质量矩阵，缩减积分计算，带参数的newton迭代求解，引入松驰因子及双层解法等技术，提出了一套适合流场有限元方程计算的非线性方程求解方法。
The general nonlinear programming problem and the basic assumptions under which our convergence results hold are introduced in chapter 2 . in chapter 3 , we give the definition of the mpf which our method is based on , the mpf method and the trust region algorithm . the convergence results for the mpf method , some aspects concerning the practical implementation and some concluded remarks of the method are discussed in chapter 4 第一章为绪论部分，第二章介绍本文收敛性理论所需要的一般非线性规划问题的最优性条件和基本的假设条件，第三章给出修改的罚函数的定义，修改的罚函数法，以及第步迭代所用的信赖域算法，第四章讨论修改的罚函数法的收敛理论，数值实验和由修改的罚函数法得出的一些结论。
Such methods are generally decreasing method , such as , feasible direction methods , constrained variable metric methods , etc . another class is sub - problems method , which approximates the optimal solution by solving a series of simple sub - problems , such as penalty function methods , trust region methods , and successive quadratic programming sub - problems , etc . the same property of two classes of methods is that they determine whether the next iterative point is " good " or " bad " by comparing the objective function value or merit function value at the current point and next iterative point 另一类叫做子问题算法，这种算法是通过一系列简单子问题的解来逼近原问题的最优解，如罚函数法、信赖域算法、逐步二次规划算法等。这两类算法的一个共同特点是，通过比较当前点和下一个迭代点的目标函数值或评价函数值来确定迭代点的“优”或“劣” ，若迭代点比当前点“优”则该迭代点可以被接受，否则须继续搜索或调整子问题。
The second chapter reveals the mathematical essence of entropy regularization method for the finite min - max problem , through exploring the relationship between entropy regularization method and exponential penalty function method . the third chapter extends maximum entropy method to a general inequality constrained optimization problem and establishes the lagrangian regularization approach . the fourth chapter presents a unified framework for constructing penalty functions by virtue of the lagrangian regularization approach , and illustrates it by some specific penalty and barrier function examples 第一章为绪论，简单描述了熵正则化方法与罚函数法的研究现状；第二章，针对有限极大极小问题，通过研究熵正则化方法与指数(乘子)罚函数方法之间的关系，揭示熵正则方法的数学本质；第三章将极大熵方法推广到一般不等式约束优化问题上，建立了拉格朗日正则化方法；第四章利用第三章建立的拉格朗日正则化方法，给出一种构造罚函数的统一框架，并通过具体的罚和障碍函数例子加以说明。
