非线性等式与不等式系统的快速算法研究

Systems of nonlinear equalities and inequalities widely appear in the field of applied mathematics such as nonlinear optimization, complementarity problem, variational inequality, then, the study of the algorithm is an important part of applied mathematics. The meanings of the research are that: the new equivalent reformulation of systems of nonlinear equalities and inequalities will be given, and a fast and stable algorithm will be proposed by combining nonmonotone technique, derivative-free strategy and Barzilai-Borwein steplength selection. The innovations are as follows: 1. Based on derivative-free method and nonmonotone technique, a nonmonotone derivative-free method will be proposed which does not need derivative information and has a wide range application for problems especially for the large scale problems or the one whose derivatives are extremely expensive to compute; 2. By using slack variables and minimum function，a new equivalent reformulation of systems of nonlinear equalities and inequalities will be established, and an efficient non-monotone inexact smoothing algorithm will be proposed; 3. Based on Barzilai-Borweinsteplength selection and nonmonotone technique, a fast and stable projected algorithm will be proposed for the bound constrained systems of nonlinear equalities and inequalities.

非线性等式与不等式系统广泛出现于非线性优化、互补问题、变分不等式等应用数学领域,因而对其算法的研究是应用数学的重要部分。本项目的研究意义在于：给出了非线性等式与不等式系统的新的等价转化形式,并结合非单调技术、无导数策略、Barzilai-Borwei步长选择等优化技巧，提出了稳定性更好的快速算法。创新点如下：1.结合无导数算法和非单调技术，提出了非单调无导数算法，算法不需要计算导数，实用性更广，尤其适合于导数难计算或大规模问题；2.通过引入松弛变量以及利用min-函数的性质，建立非线性等式与不等式系统的一种新的转化形式，并提出了求解这种非线性系统的非单调不精确光滑算法；3.针对带简单界约束的非线性等式与不等式系统问题，提出基于Barzilai-Borwei步长选择技巧和非单调技术的稳定性更好的快速投影算法。

非线性等式与不等式系统广泛地应用于数据分析、集合分类、计算机辅助设计、图像重构等一系列领域。因此，探讨能够有效求解等式与不等式系统的算法具有十分重要的理论研究和应用价值。首先，通过引入松弛变量和利用NCP函数的性质，将非线性等式与不等式系统等价地转化为非光滑方程组；然后基于光滑重构的思想，利用min函数的光滑函数将非光滑方程组转化为含参数的光滑方程组，并提出了一个求解光滑方程组的修正的非内点连续算法，证明了算法是适定的，且是全局收敛的。数值试验的结果也显示了算法的有效性。