非线性约束全局优化的新方法研究

Nonlinear constrained global optimization is a frontier topic in optimization theory, with extensive applications in financial economy, engineering design, production management, transportation and national defense industry, etc. As auxiliary function method, filter technique and canonical dual method being important solving methods in optimization theory, they can be integrated to construct new mothods on nonlinear constrained global optimization. This project is aiming at this job. Beyond constructing new methods, we will also discuss these methods' theoretical properties, searching technique, convergence property and termination rules. We will combine modern filter technique with auxiliary function method, using filters' filtration character to solve the difficulty caused by the minimizer on the border. We will also absorb canonical dual method to study termination rules or optimal approximation conditions for some nonlinear constrained global optimization, putting forward canonical inverse differential equation and canonical dual function by using KKT conditions. This project can not only enrich the nonlinear constrained global optimization theory and corresponding algorithms, but also can provide theoretical basis for solving practical problems.

非线性约束全局优化在金融经济、工程设计、生产管理、交通运输和国防工业等部门都有广泛应用，是最优化理论的前沿课题。辅助函数方法、滤子技术及Canonical对偶方法是解决最优化问题的重要方法。本项目将辅助函数方法、滤子方法和Canonical对偶方法有机结合，探讨非线性约束全局优化的若干新方法，讨论其理论性质、搜索技巧、收敛性质和终止准则等。以辅助函数方法为基础，结合滤子的过滤性质，改善新方法的算法收敛性，克服全局优化问题局部极小点出现在边界时的求解困难；借助Canonical对偶方法，利用KKT条件提出Canonical倒向微分方程，构造Canonical对偶函数和Canonical 对偶规划，给出算法在某些非线性约束全局优化问题的终止准则或最优性逼近条件。本项目的开展有助于丰富和完善非线性约束全局优化的理论与算法。

全局优化问题面临两个困难，一个是如何跳出当前的局部最优解而找到更好的局部最优解，另外一个是如何确定找到的点就是全局最优解。本项目着力研究非线性约束全局优化问题的新方法，讨论其理论性质、搜索技巧、收敛性质以及终止准则等。将辅助函数方法有机的结合滤子方法和Canonical对偶方法，提出更有效的求解非线性约束连续全局优化的新算法。同时，为将非线性约束连续优化问题中的一些新方法推广到离散以及混合整数型问题中，结合结合离散问题的连续化、NCP函数及QP-free方法等，提出新的算法。分析并完善这些方法的理论体系和算法收敛性条件，为全局优化问题的求解提供有效的新思路、新技术，是对全局优化的理论与算法的丰富和完善。