委托代理双层规划问题和不动点问题的同伦算法
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摘要
委托代理问题是信息经济学研究的一个重要问题,但是现有研究大多关注于模型的建立、一阶方法的有效性条件等理论方面的讨论,求解方法较少,尤其在连续情形下的委托代理模型本质上是一个非凸的无穷维双层优化问题,其全局收敛的有效算法更少.不动点问题是非线性分析的一个重要研究课题,目前被广泛应用于偏微分方程、控制论、经济平衡理论及对策理论等领域.同伦方法是求解非线性方程组、变分不等式和非线性规划等问题的一种具有全局收敛性的快速有效方法,本文考虑了求解委托代理双层规划问题和不动点问题的同伦方法,主要工作如下：
在第一章,主要介绍了委托代理模型和求解不动点、非线性规划等非线性问题的同伦方法研究现状,以及预备知识.
在第二章,考虑了基于连续分布情形的标准的委托代理双层规划模型,设计了分片线性契约函数,对能够直接积分的同时满足单调似然率条件和分布函数凸性条件的特殊分布函数与典型的风险规避效用函数,通过一阶方法将该模型转化为等价的有限维单层非凸规划问题,提出了求解该单层非凸规划问题的改进的动约束同伦方法,在移动可行集有界且内部非空、边界正则性和初始可行集满足法锥的较弱条件下证明了光滑同伦路径的存在性和全局收敛性.用Matlab实现算法,和Matlab自带函数fmincon,以及常用的求解非线性规划问题的软件LOQO和MINOS一起做了一些数值测试.数值结果显示：在获得相同目标值时,与设计分片常数契约的离散委托代理模型相比,设计分片线性契约函数仅需求解一个较低维的优化问题和需要很少的计算时间,改进的动约束同伦方法不仅可以算fmincon, LOQO和MINOS可以算的问题而且可以算其不可以算的问题,且数值经验说明了改进的动约束同伦方法是可行的、有效的和鲁棒的.
在第三章,考虑了另一类委托代理双层规划模型,设计了分片线性契约函数,使用复化的Simpson公式数值求积,提出了求解与该模型等价的有限维单层非凸优化问题的改进的非可行内点同伦方法,并在移动可行集有界且内部非空、边界正则性和只与不等式约束有关的移动集满足边界1法锥的较弱条件下证明了光滑同伦路径的存在性和全局收敛性.用Matlab实现算法,和Matlab自带函数fmincon,以及软件LOQO和MINOS一起做了一些数值测试,并给出了带分片线性契约函数的委托代理问题最优目标值随着离散片数逐渐增加将趋于一个常数的变化趋势图.数值结果显示：设计分片线性契约函数明显优于设计分片常数契约函数,改进的非可行内点同伦方法不仅可以算fmincon, LOQO和MINOS可以算的问题而且可以算其不可以算的问题,且数值经验说明了改进的非可行内点同伦方法是可行的、有效的和鲁棒的.
在第四章,考虑了非凸约束区域上自映射不动点问题,在带等式约束与不等式约束但不假定有界的一般非凸约束区域上,提出了求解自映射不动点问题的改进的同伦方法,并在较弱的伪锥条件下证明了光滑同伦路径的存在性与全局收敛性.该方法与可行集满足伪锥条件下求解一般非凸规划的同伦相比,所构造的同伦仅与毛发映射有关而与约束函数的梯度无关,且同伦中约束函数的乘子仅是一次的而不需要平方,以及毛发映射不要求与约束区域具有相容性.用Matlab实现算法,并进行了数值试验,数值结果显示改进的同伦方法是可行的和有效的.
The principal-agent problem has been an important problem of information economics, but almost all of the existing results are the theoretical discussion on the construction of the model, the validity of the first-order approach and so on. The solution methods are fewer, especially the effective globally convergent methods are much fewer for solving the principal-agent model in the continuous case which is an infinite-dimensional bilevel nonconvex optimization problem in essence. Fixed point problem is an important research problem of nonlinear analysis and has been widely applied to partial differential equations, control theory, economic equilibrium theory, game theory and so on. Homotopy method for solving nonlinear equations, variational inequalities, nonlinear programming and other nonlinear problems has been a fast and effective method with global convergence. In this dissertation, homotopy algorithms for solving principal-agent bilevel programming problem and fixed point problem are considered, and the main work is listed as follows.
In Chapter1, the principal-agent model, the research on homotopy method for solving fixed point, nonconvex programming and other nonlinear problems, and some preliminaries are introduced.
In Chapter2, the standard principal-agent bilevel programming model with continuous distribution is considered. To design a piecewise linear contractual function, for some typi-cal risk averse utility functions and the typical distribution functions which simultaneously satisfy monotone likelihood ratio condition and convexity of the distribution function con-dition, the model is transformed to an equivalent finite-dimensional single-level nonconvex programming, a modified constraint shifting homotopy method for solving the single-level nonconvex programming is proposed, and the existence and global convergence of a smooth homotopy pathway are proven under the mild conditions of the boundedness and nonempty of the shifted constraint set, boundary regularity and initial feasible set satisfying normal cone condition. Some numerical tests are done by our homotopy method implemented by Matlab, as well as by using fmincon in Matlab, LOQO and MINOS. Numerical tests show that:when almost the same optimal objective values are obtained, in comparison with the principal-agent bilevel programming model with designing piecewise constant contract in corresponding discrete case, designing a piecewise linear contract needs only to solve a much lower dimensional optimization problem and hence needs much less computing time, and the modified constraint shifting homotopy method can compute the problems which can not only be computed but also not be computed by fmincon, LOQO and MINOS. And the numerical experiences also show that the modified constraint shifting homotopy method is feasible, effective and robust.
In Chapter3, the other principal-agent bilevel programming model with continuous distribution is considered, in which the integration is approximately computed by the com-posite Simpson's rule. To design a piecewise linear contractual function, a modified infeasi-ble interior point homotopy method for solving a finite-dimensional single-level nonconvex programming which is equivalent to the model is proposed, and the existence and global convergence of a smooth homotopy pathway are also proven under the mild conditions of boundedness and nonempty of the shifted constraint set, boundary regularity and the initial shifted set which is only with respect to inequality constraint functions satisfying normal cone condition. Some numerical tests are done by the modified infeasible interior point homotopy method which is implemented by Matlab, fmincon, LOQO and MINOS, and the change trend of the optimal objective value is drawn. Numerical tests show that:in com-parison with a piecewise constant contract, a piecewise linear contract is much better, and the modified infeasible interior point homotopy method can compute the problems which can not only be computed but also not be computed by fmincon, LOQO and MINOS. And the numerical experiences also show that the modified infeasible interior point homotopy method is feasible, effective and robust.
In Chapter4, the fixed point problem for self-mapping in nonconvex region is consid-ered. A modified homotopy method for solving fixed point problems of self-mapping is pro-posed in the general nonconvex region with equality constraints and inequality constraints which needn't be bounded, and the existence and global convergence of a smooth homotopy path is proven under weaker conditions. In comparison with the homotopy method for solv-ing nonconvex programming under the pseudo cone condition, the modified homotopy needs only the hair mapping not gradients of the constraint functions and needs only once not twice of the lagrange multipliers, and the hair mappings need not be consistent mappings of the constraint set. The modified homotopy method is implemented by Matlab and some numerical tests are done. The numerical results show that the modified homotopy method is feasible and effective.
在第一章,主要介绍了委托代理模型和求解不动点、非线性规划等非线性问题的同伦方法研究现状,以及预备知识.
在第二章,考虑了基于连续分布情形的标准的委托代理双层规划模型,设计了分片线性契约函数,对能够直接积分的同时满足单调似然率条件和分布函数凸性条件的特殊分布函数与典型的风险规避效用函数,通过一阶方法将该模型转化为等价的有限维单层非凸规划问题,提出了求解该单层非凸规划问题的改进的动约束同伦方法,在移动可行集有界且内部非空、边界正则性和初始可行集满足法锥的较弱条件下证明了光滑同伦路径的存在性和全局收敛性.用Matlab实现算法,和Matlab自带函数fmincon,以及常用的求解非线性规划问题的软件LOQO和MINOS一起做了一些数值测试.数值结果显示：在获得相同目标值时,与设计分片常数契约的离散委托代理模型相比,设计分片线性契约函数仅需求解一个较低维的优化问题和需要很少的计算时间,改进的动约束同伦方法不仅可以算fmincon, LOQO和MINOS可以算的问题而且可以算其不可以算的问题,且数值经验说明了改进的动约束同伦方法是可行的、有效的和鲁棒的.
在第三章,考虑了另一类委托代理双层规划模型,设计了分片线性契约函数,使用复化的Simpson公式数值求积,提出了求解与该模型等价的有限维单层非凸优化问题的改进的非可行内点同伦方法,并在移动可行集有界且内部非空、边界正则性和只与不等式约束有关的移动集满足边界1法锥的较弱条件下证明了光滑同伦路径的存在性和全局收敛性.用Matlab实现算法,和Matlab自带函数fmincon,以及软件LOQO和MINOS一起做了一些数值测试,并给出了带分片线性契约函数的委托代理问题最优目标值随着离散片数逐渐增加将趋于一个常数的变化趋势图.数值结果显示：设计分片线性契约函数明显优于设计分片常数契约函数,改进的非可行内点同伦方法不仅可以算fmincon, LOQO和MINOS可以算的问题而且可以算其不可以算的问题,且数值经验说明了改进的非可行内点同伦方法是可行的、有效的和鲁棒的.
在第四章,考虑了非凸约束区域上自映射不动点问题,在带等式约束与不等式约束但不假定有界的一般非凸约束区域上,提出了求解自映射不动点问题的改进的同伦方法,并在较弱的伪锥条件下证明了光滑同伦路径的存在性与全局收敛性.该方法与可行集满足伪锥条件下求解一般非凸规划的同伦相比,所构造的同伦仅与毛发映射有关而与约束函数的梯度无关,且同伦中约束函数的乘子仅是一次的而不需要平方,以及毛发映射不要求与约束区域具有相容性.用Matlab实现算法,并进行了数值试验,数值结果显示改进的同伦方法是可行的和有效的.
The principal-agent problem has been an important problem of information economics, but almost all of the existing results are the theoretical discussion on the construction of the model, the validity of the first-order approach and so on. The solution methods are fewer, especially the effective globally convergent methods are much fewer for solving the principal-agent model in the continuous case which is an infinite-dimensional bilevel nonconvex optimization problem in essence. Fixed point problem is an important research problem of nonlinear analysis and has been widely applied to partial differential equations, control theory, economic equilibrium theory, game theory and so on. Homotopy method for solving nonlinear equations, variational inequalities, nonlinear programming and other nonlinear problems has been a fast and effective method with global convergence. In this dissertation, homotopy algorithms for solving principal-agent bilevel programming problem and fixed point problem are considered, and the main work is listed as follows.
In Chapter1, the principal-agent model, the research on homotopy method for solving fixed point, nonconvex programming and other nonlinear problems, and some preliminaries are introduced.
In Chapter2, the standard principal-agent bilevel programming model with continuous distribution is considered. To design a piecewise linear contractual function, for some typi-cal risk averse utility functions and the typical distribution functions which simultaneously satisfy monotone likelihood ratio condition and convexity of the distribution function con-dition, the model is transformed to an equivalent finite-dimensional single-level nonconvex programming, a modified constraint shifting homotopy method for solving the single-level nonconvex programming is proposed, and the existence and global convergence of a smooth homotopy pathway are proven under the mild conditions of the boundedness and nonempty of the shifted constraint set, boundary regularity and initial feasible set satisfying normal cone condition. Some numerical tests are done by our homotopy method implemented by Matlab, as well as by using fmincon in Matlab, LOQO and MINOS. Numerical tests show that:when almost the same optimal objective values are obtained, in comparison with the principal-agent bilevel programming model with designing piecewise constant contract in corresponding discrete case, designing a piecewise linear contract needs only to solve a much lower dimensional optimization problem and hence needs much less computing time, and the modified constraint shifting homotopy method can compute the problems which can not only be computed but also not be computed by fmincon, LOQO and MINOS. And the numerical experiences also show that the modified constraint shifting homotopy method is feasible, effective and robust.
In Chapter3, the other principal-agent bilevel programming model with continuous distribution is considered, in which the integration is approximately computed by the com-posite Simpson's rule. To design a piecewise linear contractual function, a modified infeasi-ble interior point homotopy method for solving a finite-dimensional single-level nonconvex programming which is equivalent to the model is proposed, and the existence and global convergence of a smooth homotopy pathway are also proven under the mild conditions of boundedness and nonempty of the shifted constraint set, boundary regularity and the initial shifted set which is only with respect to inequality constraint functions satisfying normal cone condition. Some numerical tests are done by the modified infeasible interior point homotopy method which is implemented by Matlab, fmincon, LOQO and MINOS, and the change trend of the optimal objective value is drawn. Numerical tests show that:in com-parison with a piecewise constant contract, a piecewise linear contract is much better, and the modified infeasible interior point homotopy method can compute the problems which can not only be computed but also not be computed by fmincon, LOQO and MINOS. And the numerical experiences also show that the modified infeasible interior point homotopy method is feasible, effective and robust.
In Chapter4, the fixed point problem for self-mapping in nonconvex region is consid-ered. A modified homotopy method for solving fixed point problems of self-mapping is pro-posed in the general nonconvex region with equality constraints and inequality constraints which needn't be bounded, and the existence and global convergence of a smooth homotopy path is proven under weaker conditions. In comparison with the homotopy method for solv-ing nonconvex programming under the pseudo cone condition, the modified homotopy needs only the hair mapping not gradients of the constraint functions and needs only once not twice of the lagrange multipliers, and the hair mappings need not be consistent mappings of the constraint set. The modified homotopy method is implemented by Matlab and some numerical tests are done. The numerical results show that the modified homotopy method is feasible and effective.
引文
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