求解非线性规划的自适应滤子方法研究

With the advent of the era of Big Data, the size of many practical optimization problems have increased, and some special difficulties or structures (such as, difficulties for computing Jacobian) are often involved in these problems, which brings a very big challenge to methods for solving nonlinear programing. Filter methods, as an important part of nonlinear programing, always attract attention of researchers. This project mainly investigates new adaptive filter methods for large-scale nonlinear optimization problems. First, making use of linear algebra technique (say Krylov subspace method), SSLE technique, Quasi-Newton technique and algorithmic differentiation, we devise new filter algorithms which compute Jacobian information of the nonlinear optimization problems approximately and reduce the dimension of the subproblems, so that it can solve large-scale nonlinear optimization problems (probably having some special structure). Second, based on NCP functions, we improve the filter test conditions, and construct new filter algorithms so that both theoretical results and practical efficiency can be improved. Third, under some weak conditions (say CRSC, CPG conditions), we study the global and local convergence. Finally, utilizing parallel computing and scaling techniques, we write efficient Matlab codes to implement these new filter algorithms, and conduct numerical experiments on large-scale problems.

随着大数据时代的到来，许多实际优化问题的规模不断增加，而且时常伴有特殊困难或结构（如Jacobian计算困难），这给求解优化问题的非线性规划方法提出了很大的挑战。滤子方法作为非线性规划的一个重要组成部分，一直受到学术界关注。本项目主要研究新的可求解大规模问题的自适应滤子方法。首先，运用数值代数技巧(如Krylov子空间方法)、序列线性方程组技术、拟牛顿技术以及算法微分技术，对约束优化问题的Jacobian进行逼近，子问题进行降维，设计新的自适应滤子方法，求解大规模优化问题；其次，利用非线性互补函数，改进滤子测试条件，设计新的滤子方法以改善滤子算法的理论和计算效果；再次，研究新的滤子方法在较弱的条件下（如在CRSC, CPG条件下）的全局收敛性、局部收敛速度。最后，利用并行、调比等各种技术，编写高效的Matlab程序实现上述各类新算法，并对大规模问题进行测试。

本项目研究求解非线性规划问题的滤子方法及其应用。我们基本完成了项目申请书中关于非线性规划的理论、算法和应用的研究，也在这些方面取得了系列成果。在项目经费的支持下，我们开展了一系列创新性的工作，主要研究成果分为四个方面：一是运用多维滤子方法求解约束非线性规划问题(NLP)，并证明了算法在CPG较弱条件下具有全局收敛性。二是利用Anderson加速技术对外梯度方法进行加速以求解非线性互补问题，从理论上证明了新算法的收敛性，并在数值实验上表明该算法有明显的稳定的加速效果。三是研究了最小二乘半正定规划问题的L-BFGS算法，证明了所提算法的全局收敛性，并在数值上验证了所提算法的有效性。四是将优化算法与理论应用到能源经济中，求解了ESCO利益最大化视角的合同参数决策模型。本项目共发表学术论文4篇，其中SCI论文2篇, EI论文1篇。