【作者】 叶彩鸿；

【导师】 何炳生； 杨超；

【作者基本信息】 华中科技大学 ， 管理科学与工程， 2007， 博士

【摘要】 在经济管理领域中,“黑箱”问题是普遍存在的。例如,政府在制定相关政策时,并不能准确预知该政策实施后会给社会带来什么样的影响,而只能在政策真正实施后对政策所带来的影响进行观察,此时,政府制定的政策对社会的影响过程是一个“黑箱”,其输入变量是“政策”,输出变量是“社会影响”。同样,企业在进行定价决策时,往往不能准确预知新价格可能会带来的市场反应,而只能在价格公开后去观察客户与竞争对手的反应。此时,企业价格策略的作用过程也是一个“黑箱”,其输入变量是“价格”,其输出变量是“客户与竞争对手的反应”。这些管理中的“黑箱”,我们可以知道其作用过程的一些规律和机制,却没有量化的函数表达式。由于“黑箱”现象的普遍性,在管理优化过程中,人们经常需要面对由“黑箱”现象所带来的困扰。当管理优化模型中存在黑箱函数时,对优化决策系统输入的控制变量x1 , x2 ,…, xn,我们只能观测到黑箱函数f ( x1 , x2 ,…, xn)的值(信息)并利用观测到的信息进行求解。然而,这种观测(信息的获得)往往代价不菲,或者观测次数受到限制。因此,如何在黑箱函数条件下建立管理优化模型并给出行之有效的求解方法是一个有实际应用背景的问题。尽管黑箱函数条件给管理优化问题的求解带来了一定的障碍,但并非无法求解。华罗庚教授推广普及的黄金分割法就是一种一维单峰黑箱函数条件下的最优化方法,在管理实践中已被广泛接受与应用,成为黑箱条件下一维优化的有力工具。本文对多维单峰黑箱函数条件下的一些极值问题与多维单调黑箱函数条件下的一些平衡问题进行了讨论,是对华罗庚推广普及黄金分割法的继承与发扬。论文首先给出了选题的背景,认为黑箱是管理中的普遍现象,人们在管理优化中必然需要面对黑箱函数障碍。当管理优化模型中存在黑箱函数时,寻求一些只用函数值的管理优化方法是很有必要的。本研究的思路是将一些管理优化问题写为变分不等式模型,从而可以用一些求解变分不等式的只用函数值的方法对问题进行求解。在介绍了一些管理优化问题的变分不等式形式以及变分不等式在管理优化中的一些应用之后,提出本文的意义在于其可以为目标函数表达式未知时或对? f ( x) =0的求解非常困难时的管理优化决策提供参考,并给出了本文研究的主要方法、目标与内容。其次,论文总结了相关的变分不等式算法,介绍了不同条件、不同结构变分不等式问题的各类求解方法,包括基本投影法、预测-校正方法、结构变分不等式的求解方法与结构可分变分不等式的求解方法。对于结构可分的变分不等式,本文在交替方向法的基础上提出了一种下降方法,并给出了相应的算法收敛性证明与数值试验结果。第三,论文给出了管理优化目标为a≤f ( u )≤b时的管理优化变分不等式模型。以物流运输服务价格调整模型与能源价格调整模型为例,当管理优化的目标为a≤f ( u )≤b且f ( u )为黑箱函数时,利用决策变量与相关表达式之间的互补关系建立了隐式互补模型。在一定的假设之下,论文证明了函数f ( u )的单调性与连续性,从而可以利用一些直接迭代方法(仅需函数f ( u )的观测结果)进行求解。对给出的算例,论文给出了不同算法的计算结果并进行了比较。第四,论文给出了管理优化目标为max( or min) f ( x )时的Box约束管理优化模型。以一类基于空间价格均衡的物流配送量优化模型与一类价格调整模型为例,在一定的假设条件下,分析了目标函数最大或最小时决策变量与黑箱函数之间的互补关系,从而将其改写为互补模型,并对模型的假设条件与求解方法等进行了讨论。对给定的算例,论文给出了计算结果。第五,论文给出了在线性约束条件下管理优化目标为max( or min) f ( x )时的管理优化模型。以物流配送量优化问题为例,分别讨论了存在配送中心能力限制、边容量限制、配送中心与边容量双重限制条件下的配送量优化模型。对模型的假设条件与求解方法等进行了讨论。对给定的算例,论文给出了不同限制条件下的优化结果。最后对全文内容及研究结论和创新之处进行了总结,并对文中有待进一步深入研究的地方提出日后继续研究的方向和展望。更多还原

【Abstract】 In the economical and managerial field,‘black box’is a common phenomenon. For example, when a government is making some policy, it can’t predict exactly what effect the policy will make on the society and can only observe the effect after the policy is implemented. So the affecting process of the policy made by the government on the society is a‘black box’. The input variable is‘policy’and the output variable is‘social effect’. Similarly, when an enterprise is making a pricing policy, it can’t know exactly what social reaction the policy will lead to and can only observe the reactions of customers and competitors after the price is announced. Then the affecting process of the pricing policy is also a‘black box’. The input variable is‘price’and the output variable is‘the reactions of customers and competitors’. We can know some law of those‘black boxes’. In spite of that, the exact operation mechanisms are unknown.Because the universality of the‘black box’, managers have to face the troubles brought by the phenomenon when optimizing the managerial problems. When an optimizations model has a‘black box’function, to a control variable of the optimal policy, we can only solve the model by observing the value of the‘black box’. However, the observation is an expensive process or the number of such observation is limited. Therefore, how to build management optimization models which are based on‘black box’and how to present an effective algorithm are issues with practical applications.‘Black box’brings some obstacles to the solving processes. In spite of that, the problems are not insurmountable. The golden section method presented by Luogen Hua is an optimization method which is used under the conditions involving one dimension, one hump, and‘black box’. The golden section method, a powerful tool for one-dimensional optimization, has been widely accepted. In this dissertation, we carried down and developed the idea of golden section and focused on two kinds of problems, one is management optimization models and algorithms under the conditions involving multi dimensions, one hump, and‘black box’, the other is equilibrium models and algorithms under the conditions involving multi dimensions, monotonous, and‘black box’. Firstly, the thesis introduces the backgrounds, the significance, the innovative achievements and the motivations of choosing this topic.‘Black box’is a common phenomenon in the economical and managerial field and managers have to face the troubles brought by‘black box’. It is highly necessary to find some some solutions without derivatives when there are‘black boxes’. The thesis introduces some applications of variational inequality in optimization and the equivalent variational inequality forms of some optimizations. The significance or contribution of this thesis is to provide a reference for the decision when some functions are black boxes or it is difficult to solve the system of equations ? f ( x) = 0.Secondly, the thesis reviews some algorithms which are used to solve variational inequalities and introduces various kinds of algorithms to different condition and structure, including the basic projection methods, the Prediction-Correction methods, the methods for structured variational inequalities, and the methods for variational inequalities with separate structures. To variational inequalities with separate structures, the thesis presents a descend method based on the alternating directions method, and some numerical results demonstrate that the new method is effective in practice.Thirdly, the thesis presents some variational inequality models whose objectives of optimization are a≤f ( u )≤b. Take the process of the logistics provider readjusting the prices and the government readjusting the resource prices for instance, when the objective is a≤f ( u )≤b and f ( u ) is a‘black box’, we can build an implicit complementarity model according to the complement relationship between decision variables and some expressions. Under the given assumptions, the thesis proves that f ( u ) is continuous and monotone, thereby the direct iterative algorithm, which only needs the value of f ( u ), can be used in the process of solving. To the given numerical experiments, the thesis provides the computation results of various algorithms and makes a comparison.Fourthly, the thesis presents some variational inequality models when the objective of management optimization is max( or min) f ( x )and there is no restriction. Take the models of a kind of distribution optimizationof the models of a kind of distribution optimizations based on spatial price equilibrium and the models of a kind of price readjusting for instance, the thesis analyzes the complement relationship between the decision variables and the black-box functions when the objective function reaches the maximum or minimum point. And then an implicit complementarity model can be build.Finally the thesis provides the computation results of the given numerical experiments. Fifthly, the thesis presents some variational inequality models with linear constraints whose objectives of management optimization are max( or min) f ( x ). Take the models of distribution optimization for instance, the thesis presents the distribution models separately with capacity constraints from the centers of distribution, links, and both of them. The assumptions and algorithms are discussed. To the given numerical experiments, the optimal distribution plan with various capacity constraints is given.Finally, the thesis gives conclusions to the contents and innovative achievements of the research, and presents the future scope, purpose and prospect of this topic in further studies.更多还原