微分代数方程动态优化问题的快速求解策略研究
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摘要
随着计算机技术的发展与非线性规划求解技术的日趋成熟,动态优化方法在过程工业、机械控制等研究领域受到越来越多的关注。联立法由于采用了将连续动态优化问题离散为非线性规划(NLP)问题进行求解的思路,使得该方法简单易行并具有很高的求解效率,逐渐成为当前最具发展前景的动态优化求解策略之一。但是,在一些对计算的实时性有较高要求的应用环境中,仍然需要对现阶段的基于联立法的动态优化工具不断改进来提高优化的效率。本文围绕动态优化问题的快速求解策略展开研究,共分析讨论了4个方面的内容：
1.对利用有限元正交配置法(OCFE)离散微分代数方程(DAE)优化问题的一般形式进行了深入分析,通过严格证明得到了直接离散格式与间接离散格式的等价条件。在此基础上,对多种常用的配置方式进行了模拟与分析,并发现了间接Lobatto配置在诸多方面均可获得更好的效果。此外,进一步深入讨论了配置法离散的本质,提出基函数离散的思想,并以径向基函数(RBF)离散为例验证了非多项式离散的可行性。
2.对记忆增强优化方法(MEO)的本质进行了讨论,提出根据历史经验数据生成高等初值点(ASP)的实质为多元散乱数据拟合的观点,并成功应用RBF插值方法改进了MEO方法,使应用效果大幅提升。在此基础上,提出两层MEO的思想,并设计出效果更好的具有误差修正的RBF-MEO方法。
3.为克服RBF插值方法存在的先天不足,提出采用支持向量机(SVM)替代RBF来实现MEO的功能。同时,还针对SVM中QP求解效率不高的现状,提出了Hesse阵稀疏化方法及多元输出问题的联立求解策略,成功地提升了SVM方法的执行效率。进一步将SVM-MEO应用于解决动态参数化优化问题,提出了高效的“部分MEO"策略。
4.针对动态优化研究平台缺少的现状,构建了DAE优化框架,并在Matlab与AMPL环境中成功实施。该平台采用模块化设计思路,将离散后的NLP问题划分为模型部分与方法部分。在此平台上,可进行离散方法、模型开发与求解工具等方面的研究。该部分内容还包括如何将不同的动态优化模型进行标准化描述,以及根据离散结构的特点而设计的快速求解方法。
With the development of computer technology and the increasingly perfecting of non-linear programming (NLP) solution technology, dynamic optimization methods have at-tracted more and more attention in the process industries, machine control and other re-search areas. Since the simultaneous strategies adopt the idea that the continuous dynamic optimization problems are discretized into NLP problems and solved successively, the strategies can be executed easily and have extremely solution efficiency. The simultane-ous strategies gradually become more prospective. However, current simultaneous meth-ods still can not meet much application requirements, so need to be improved further to enhance the efficiency and practicality. This work mainly focuses on fast solving strategies for dynamic optimization with differential-algebraic equations (DAEs), including the next four aspects:
1. It is analyzed deeply that the orthogonal collocation on finite element (OCFE) meth-ods are used to discretize the DAE optimization problems, and the equivalent con--ditions between direct and indirect discretization structures are proved theoretically. On this basis, combined with the specific collocation formulae for analysis and verifi-cation, it is concluded that the Lobatto formula has the best or better effects on many solution aspects. In addition, the nature of collocation discretization methods is dis-cussed further, and then the idea of basis function discretization are proposed. The feasibility of non-polynomial discretization is exemplified by radial basis function (RBF) methods.
2. The nature of mnemonic enhancement optimization (MEO) is discussed, and it is pro-posed that the problems of advanced starting point (ASP) generated from empirical data are multivariate scatter data fitting problems in fact. Thus, the RBF interpolation principle is used to improve the MEO method, and the application results are signifi- cantly improved. Furthermore, an improved RBF-MEO method with error correction is proposed.
3. In order to avoid the inherent deficiencies of RBF method, the support vector machine (SVM) is employed as an alternative to realize the functions of MEO. According to the fact that QP problems can not be solved quickly enough in the SVM, the Hessian matrix sparseness methods and the simultaneous strategy of multi-output problems are put forward, successfully raising the solution efficiency of the SVM. The SVM-MEO method is applied to solve the parametric dynamic optimization problems, and an efficient "partial MEO" is proposed.
4. Since scarcity of research platform for dynamic optimization, a DAE optimization framework is built and successfully implemented in MATLAB and AMPL environ-ment. The platform adopts the modular design concept and divides the discrete for-mula into the model part and the method part. On this platform, the further research of discretization methods, model development and solver tools can be distributed. This section also includes standardization of dynamic optimization models and fast solution strategy based on discretization structure.
1.对利用有限元正交配置法(OCFE)离散微分代数方程(DAE)优化问题的一般形式进行了深入分析,通过严格证明得到了直接离散格式与间接离散格式的等价条件。在此基础上,对多种常用的配置方式进行了模拟与分析,并发现了间接Lobatto配置在诸多方面均可获得更好的效果。此外,进一步深入讨论了配置法离散的本质,提出基函数离散的思想,并以径向基函数(RBF)离散为例验证了非多项式离散的可行性。
2.对记忆增强优化方法(MEO)的本质进行了讨论,提出根据历史经验数据生成高等初值点(ASP)的实质为多元散乱数据拟合的观点,并成功应用RBF插值方法改进了MEO方法,使应用效果大幅提升。在此基础上,提出两层MEO的思想,并设计出效果更好的具有误差修正的RBF-MEO方法。
3.为克服RBF插值方法存在的先天不足,提出采用支持向量机(SVM)替代RBF来实现MEO的功能。同时,还针对SVM中QP求解效率不高的现状,提出了Hesse阵稀疏化方法及多元输出问题的联立求解策略,成功地提升了SVM方法的执行效率。进一步将SVM-MEO应用于解决动态参数化优化问题,提出了高效的“部分MEO"策略。
4.针对动态优化研究平台缺少的现状,构建了DAE优化框架,并在Matlab与AMPL环境中成功实施。该平台采用模块化设计思路,将离散后的NLP问题划分为模型部分与方法部分。在此平台上,可进行离散方法、模型开发与求解工具等方面的研究。该部分内容还包括如何将不同的动态优化模型进行标准化描述,以及根据离散结构的特点而设计的快速求解方法。
With the development of computer technology and the increasingly perfecting of non-linear programming (NLP) solution technology, dynamic optimization methods have at-tracted more and more attention in the process industries, machine control and other re-search areas. Since the simultaneous strategies adopt the idea that the continuous dynamic optimization problems are discretized into NLP problems and solved successively, the strategies can be executed easily and have extremely solution efficiency. The simultane-ous strategies gradually become more prospective. However, current simultaneous meth-ods still can not meet much application requirements, so need to be improved further to enhance the efficiency and practicality. This work mainly focuses on fast solving strategies for dynamic optimization with differential-algebraic equations (DAEs), including the next four aspects:
1. It is analyzed deeply that the orthogonal collocation on finite element (OCFE) meth-ods are used to discretize the DAE optimization problems, and the equivalent con--ditions between direct and indirect discretization structures are proved theoretically. On this basis, combined with the specific collocation formulae for analysis and verifi-cation, it is concluded that the Lobatto formula has the best or better effects on many solution aspects. In addition, the nature of collocation discretization methods is dis-cussed further, and then the idea of basis function discretization are proposed. The feasibility of non-polynomial discretization is exemplified by radial basis function (RBF) methods.
2. The nature of mnemonic enhancement optimization (MEO) is discussed, and it is pro-posed that the problems of advanced starting point (ASP) generated from empirical data are multivariate scatter data fitting problems in fact. Thus, the RBF interpolation principle is used to improve the MEO method, and the application results are signifi- cantly improved. Furthermore, an improved RBF-MEO method with error correction is proposed.
3. In order to avoid the inherent deficiencies of RBF method, the support vector machine (SVM) is employed as an alternative to realize the functions of MEO. According to the fact that QP problems can not be solved quickly enough in the SVM, the Hessian matrix sparseness methods and the simultaneous strategy of multi-output problems are put forward, successfully raising the solution efficiency of the SVM. The SVM-MEO method is applied to solve the parametric dynamic optimization problems, and an efficient "partial MEO" is proposed.
4. Since scarcity of research platform for dynamic optimization, a DAE optimization framework is built and successfully implemented in MATLAB and AMPL environ-ment. The platform adopts the modular design concept and divides the discrete for-mula into the model part and the method part. On this platform, the further research of discretization methods, model development and solver tools can be distributed. This section also includes standardization of dynamic optimization models and fast solution strategy based on discretization structure.
引文
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