基于分式模型的信赖域优化算法研究

In recent years, nonlinear optimization has been widely applied to various fields. In the nonlinear optimization problem, for some nonquadratic behavior is strong, more severe curvature change function, we often using non-quadratic models such as conic models to approximate. .This project studies the trust region optimization algorithm based on the fractional model. This project intends to construct a class of fractional models with multiple parameter vectors, discuss its properties, and deeply study the trust region method with fractional model. Firstly, by discussing the Newton point and the steepest descent point of the fractional model trust region subproblem of the unconstrained optimization, a simple dogleg method for solving the subproblem is established, and the global convergence of the algorithm is studied and applied to solve the general constrained optimization problem. Secondly, by cyclically fixing the fractional coefficient part of the fractional approximation function, the fractional model trust region subproblem is transformed into a simple one-dimensional quadratic model subproblem, and a new approximate solution method for solving the subproblem is obtained. Thirdly, we search for an approximate solution to the new conic trust region subproblem in two or more mutually orthogonal directions, and then a new fractional model trust region algorithm based on alternating direction method for the unconstrained optimization problem is proposed. last, we study a hybrid algorithm of line search method, quasi-Newton method and fractional model trust region method. .The fractional model method is a new type of non-quadratic model method. The design of the algorithm has many changes and has a lot of research space. The completion of this project will enrich and develop the theory and algorithm of nonlinear programming, and also provide a new numerical calculation method for engineers and technicians.

近年来，非线性最优化被广泛地应用到各个领域。对于一些非二次性态强、曲率变化剧烈的函数，经常采用锥模型等非二次模型来逼近。本项目研究基于分式模型的信赖域优化算法。.本项目拟构造一类含有多个参向量的分式模型，讨论其性质，深入研究分式模型信赖域算法。首先，通过讨论无约束优化分式模型信赖域子问题的牛顿点和最速下降点，建立一种求解子问题的简单折线法，研究算法的全局收敛性，并应用于求解一般约束优化问题。其次，通过循环固定分式模型的分式系数部分,得到求解子问题的新的近似解方法。然后，拟在相互正交的几个方向上多步搜索求得子问题的一个近似解。最后，研究与线搜索法、拟牛顿法相结合的分式模型信赖域混合算法。.分式模型法是一类新的非二次模型法，算法的设计变化多，具有很大的研究空间。本项目的完成将丰富非线性规划的理论和算法，也可为工程技术人员提供新的数值计算方法。

信赖域方法是非线性优化的一类重要的数值计算方法，该方法有很好的稳定性和很强的收敛性。传统的信赖域算法主要是利用二次模型来逼近目标函数，然而对于非二次性态强、曲率变化较为剧烈的函数，逼近的效果往往不是很好。针对这一缺陷，Davidon首先提出了锥函数。使用锥模型去逼近的效果可能好于二次模型，但其水平向量参数只有一个，这会影响其搜索方向的选择。因此，我们考虑二次模型和锥模型的推广形式—分式模型，它含有三个水平参量，在充分利用以前迭代过程中的函数信息基础上，可以通过恰当选择这三个水平向量使其满足更多的插值条件，从而更好地逼近原目标函数。当迭代点接近极小点时，分式模型退化为二次模型，从而保留了二次模型在极小点附近收敛快的优点。本项目通过构造一类含有多个参量的分式模型，讨论其性质，通过控制参量的选择简化分式模型信赖域子问题，然后分别利用简单折线法、循环迭代的近似解方法、交替方向搜索法求解子问题，研究算法的全局收敛性。本项目通过深入研究分式模型信赖域算法并将其应用于求解一般约束优化问题，在理论上丰富了非二次模型法，为非线性优化问题提供新的理论和算法，也为一些实际问题的求解提供了理论基础。