| 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. |
