最优化问题过滤方法的理论研究与应用

项目名称： 最优化问题过滤方法的理论研究与应用

项目编号： No.10871130

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 建筑科学

项目作者： 朱德通

作者单位： 上海师范大学

项目金额： 26万元

中文摘要： 本项目使用过滤的线搜索技术和过滤的信赖域策略,结合序贯二次规划方法和完全投影正割方法研究非线性约束优化问题。将技巧性地结合预条件法、共轭梯度法与Lanczos方法等以及微分方程思想构造各种新的仿射路径解仿射内点信赖域子问题，以期拓展于等式/不等式约束的优化问题，获得新的理论结果和有效的数值实现算法。依据各类不等式起作用约束集指示函数的特定条件,解决退化的有界变量约束非线性规划与缺乏严格互补性条件的非线性等式/不等式约束优化问题,将研究与发展新的辨别指示函数的技巧和手段及其方法，推广于解决退化的非线性互补性问题与退化的变分不等式问题。发展过滤方法的理论研究与数值计算实践解决约束的非线性(半光滑)方程组与约束的非线性互补性问题以及约束的变分不等式问题等。将过滤方法和双水平规划思想与方法分别发展用于金融投资与风险调控问题。

中文关键词： 过滤法；信赖域方法；非线性互补性问题；变分不等式；投资与风险调控组合模型。

英文摘要： This project proposes filter line search technique and filter trust region strategy in association with the projected reduced Hessian methods and the full secant algorithms for nonlinear constrained optimization, respectively. The various curvilinear paths such as preconditional path, optimal path, modified gradient path, conjugate gradient path, Lanczos path and differential system path are presented for solving the affine scaling trust region problems. Furthermore, these paths can be applied and developed to solve the equality/inequality constrained optimization problems. The global convergence and fast local convergent rate of the proposed algorithms will be established and their performances and numerical results will be illustrated to show the effectiveness. Employing an identification function of the active constraints and the new improving affine scaling matrix, the project proposes two families of new affine scaling trust region algorithms with a nonmonotonic interior point backtracking technique which improve the classical affine scaling interior trust region algorithms for bound-constrained nonlinear optimization and inequality constrained optimization in the degenerate case where the bound constraints and inequality constrains do not satisfy the strict complementarity, respectively. Based on applications of new identification function techniques and reformulations of some differentiable merit functions, the project will extend and develep the designing and the implementation of some new algorithms for solving degenerate nonlinear complementarity problems and degenerate variational inequality problems. The project proposes and analyzes the filter methods for solving constrained semismooth equations under local error bound condition, constrained nonlinear complementarity problems and constrained variational inequality problems, respectively. In this project, the new filter methods and bilevel programming methods will be developed and applied to research and solve risk control and asset return portfolio optimization model.

英文关键词： filter methods;trust region methods; nonlinear complementarity problems; variational inequality; risk control and asset return portfolio optimization.