约束无导数最优化问题的理论与方法及其应用

This project is devoted to the study of derivative-free algorithms and the numerical analysis of constrained optimization. The line search technique and trust region strategy in association with the projected reduced Hessian methods and the full secant algorithms are proposed for solving constrained derivative-free optimization, respectively. 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. The various curvilinear paths such as affine scaling path, residual path instead of conjugate gradient path, Lanczos path，Krylov subspace method and path of finite-difference equation are presented for solving derivative-free trust region subproblems. Furthermore, these paths can be applied and developed to solve the equality/inequality constrained derivative-free optimization. Descent in direct-search methods is guaranteed away from stationarity by combining such mechanisms with a possible reduction of the corresponding step size parameter such as a mesh size parameter, a simplex diameter, a line-search parameter, or a trust-region radius. The proposed algorithms will guarantee some form of control of the geometry of the sample sets by examples of measures of geometry where the function is evaluated. The derivative-free algorithms will drive the step size parameter to zero by the best stopping criteria based on some form of stationarity. Employing some new identification functions of the active constraints, the project will extend and develep the designing and the implementation of improved algorithms for solving bound-constrained derivative-free optimization and inequality constrained derivative-free optimization in the degenerate case where the bound constraints and inequality constrains do not satisfy the strict complementarity, respectively. The project also proposes and analyzes filter line search technique and filter trust region methods for solving constrained derivative-free optimization. Furthermore, the filter methods will be extended and develeped to solve derivative-free semismooth equations under local error bound condition, derivative-free nonlinear complementarity problems and derivative-free variational inequality problems.

本项目将提供约束无导数优化问题的理论研究和方法及其数值分析。将技巧地使用线搜索技术/信赖域策略，结合序贯二次规划方法和完全投影正割方法分别研究无导数算法的整体收敛性和局部收敛速率。结合仿射变换、残差替代共轭梯度法、Lanczos法与Krylov子空间法以及差分方程等思想构造各种新的路径解信赖域子问题，以期拓展于等式/不等式的约束无导数优化问题，获取新的理论分析和数值算法。以调整参数，误差分析和网状参数以及引入清晰滤子等方法解决多项式插值或者退化的信赖域模型和搜索方向，寻求新的样本集合形式的几何控制以确保计算函数时稳定性和适定性。构建约束区域外的搜索方向及步长，以使有下降度，判定准则与有效可行性。提供新的辨别指示函数的技巧和手段，推广于解决退化的约束无导数优化问题。发展过滤法的理论与数值计算解决约束无导数优化问题，并推广于解约束无导数的非线性方程组和非线性互补问题以及无导数的变分不等式问题。

本项目提供约束无导数优化问题的理论研究和方法及其数值分析，技巧地使用线搜索技术/信赖域策略, 分别结合（不精确）的序贯二次规划方法和完全投影正割方法研究无导数算法的整体收敛性和局部收敛速率。结合（仿射变换）残差替代（仿射）共轭梯度法、Lanczos法以及Krylov子空间法等构造各种新的路径解类信赖域子问题，拓展于解等式/不等式的约束无导数优化问题，获取所提供方法的整体收敛性和局部超线性收敛速率和数值实现。给出Hölder条件下非线性方程系统的拟Guass-Newton方法，理论分析获得方法的R-局部超线性收敛速率。. 以调整参数，误差分析和网状参数以及引入渐弱过滤等方法解决多项式插值或者退化的信赖域模型和搜索方向，获取新样本集合形式的几何控制以确保计算函数时稳定性和适定性。构建约束区域外有下降度的搜索方向及步长，判定准则与有效可行性。基于新辨别指示函数的技巧和手段，推广于解决退化的约束无导数优化问题，即变量有界约束和线性不等式约束不满足严格互补性。项目中研究了线搜索技术结合立方正则法解变量有界约束的无导数优化问题，获得了算法的整体收敛性和局部超线性收敛速率，数值结果表明算法的有效性和可行性。推广于线搜索结合立方正则法解线性不等式约束的优化问题和无导数线性方程组。发展（渐弱）过滤法的理论与数值计算解决约束无导数优化问题，进一步，推广于解约束无导数的非线性方程组和非线性互补问题以及无导数的变分不等式问题。