Homotopy Algorithms For Solving Principal-Agent Bilevel Programming Problem And Fixed Point Problem

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.