大规模过程系统非线性优化的简约空间理论与算法研究
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摘要
非线性规划(NLP)是过程系统工程的重要工具。计算机技术与数值方法的发展使过程工程师可以考虑求解更大规模、更复杂的系统。如何实现大规模NLP问题的高效、稳定求解成为研究重点。简约空间方法适合过程系统优化命题的高维、低自由度特征,并可利用问题结构和近似二阶信息求解,是该类优化命题的良好求解途径。本论文的目标是设计并实现适于大规模NLP问题求解的简约空间算法,特别是为实现稳定求解以及权衡计算代价与求解精度提出解决方案,在理论上分析其特性,并通过各类NLP问题求解测试其应用性能。
本文讨论的简约空间算法包括简约空间SQP算法(rSQP)和简约空间内点法(rIPOPT)。提出了这两种算法可用的求解策略。为了高效求解空间分解系统,引入一种近似求解方法并相应调整求解计算顺序。在不满秩系统求解中,为实现系统分解和数值稳定性,提出必要的维数变化方法。此外,在求解过程中还需要检测系统秩变化及(局部)不可行性。这可以通过高效应用稀疏系统求解器来实现。为了促进算法全局收敛,本文提出基于障碍法和投影梯度法的两种可行性恢复算法,并对无可行性恢复阶段的鲁棒算法进行了探讨。基于收敛深度的收敛准则能够权衡计算代价与求解精度,为用户提供可接受近似解,控制优化进程适时终止。
本文的理论探讨包括不满秩(特别是不相容)系统的障碍法可行性恢复问题求解方法;投影梯度可行性恢复算法的全局收敛性及收敛速度分析;无可行性恢复阶段的鲁棒算法全局收敛性分析;基于收敛深度的收敛准则性质证明等。
作为通用NLP问题求解器,本文通过CUTE/COPS/MITT标准算例库求解,测试了MATLAB环境下rSQP算法的性能,并将其与MINOS和SNOPT两个求解器的性能进行了比较。在过程系统优化应用中,乙烯生产过程联塔系统优化显示出简约空间算法的优越性。FORTRAN环境下的rIPOPT算法在不满秩系统求解、全局收敛性测试和不可行性识别中表现出良好性能。收敛深度控制准则的有效性,通过rSQP算法应用于标准算例库求解、变负荷联塔系统优化、催化剂混合优化等数值实验得到体现。
Nonlinear programming (NLP) is an essential tool in process engineering. The advances in computer technology and numerical methods enable engineers to consider more large-scale and complex systems. Emphasis is laid on solving large NLPs efficiently and stably. Reduced-space methods have advantages in process engineering. For one thing, most problems in this category have only few degrees of freedom compared to the total number of variables. For another, reduced-space approaches allow to exploit problem-dependent structure and approximate second derivative information. The objective of this dissertation is the design and implementation of reduced-space algorithms appropriate to large NLPs, especially the strategies necessary for NLP solvers to solve problems stably and balance between computational cost and precision. Properties of the strategies are analysed, and performance of the new solvers is evaluated by tests on a large variety of NLPs.
The algorithms discussed include reduced-space SQP (rSQP) and interior point method (rIPOPT). Strategies which can be applied to both of the algorithms are proposed. In order to solve the decomposed system efficiently, an approximate method is introduced, accordingly, the computation order is changed. When solving rank deficient systems, dimension change is necessary for achieving system decomposition and numerical stability. Additionally, checking on rank change and (local) infeasibility during the process of optimization is needed, which can be implemented efficiently by appropriate application of a sparse system solver. Global convergence is guaranteed by feasibility restoration algorithms, i.e. the algorithms based on barrier method and projected gradient method respectively. A robust algorithm without feasibility restoration phase is discussed as well. The trade-off between computational cost and precision is implemented in criteria taking advantage of convergence depth, which try to detect acceptable approximate solution, and discover the proper time to terminate the optimization algorithm.
Theoretical discussion includes the solution of barrier feasibility restoration problem corresponding to a rank deficient system, especially an inconsistent system; global convergence and convergence rate of the feasibility restoration algorithm following projected gradient methods; global convergence of the robust algorithm without feasibility restoration phase; and proofs of the properties of convergence criteria based on convergence depth.
The practical performance of rSQP algorithm coded in MATLAB as a general purpose NLP solver is tested on various NLP problems from CUTE, COPS, and MITT. The performance is compared with that of MINOS and SNOPT. In the application to process engineering, the advantage of reduced-space methods is demonstrated in the optimization of distillation columns in ethylene production. rIPOPT in FORTRAN performs very well in the solution of rank deficient problems and tests on global convergence and infeasibility identification. The effectiveness of criteria based on convergence depth control is shown by the rSQP algorithm in the solution of problems from CUTE, distillation columns under changing feed conditions, and catalyst mixing problem.
本文讨论的简约空间算法包括简约空间SQP算法(rSQP)和简约空间内点法(rIPOPT)。提出了这两种算法可用的求解策略。为了高效求解空间分解系统,引入一种近似求解方法并相应调整求解计算顺序。在不满秩系统求解中,为实现系统分解和数值稳定性,提出必要的维数变化方法。此外,在求解过程中还需要检测系统秩变化及(局部)不可行性。这可以通过高效应用稀疏系统求解器来实现。为了促进算法全局收敛,本文提出基于障碍法和投影梯度法的两种可行性恢复算法,并对无可行性恢复阶段的鲁棒算法进行了探讨。基于收敛深度的收敛准则能够权衡计算代价与求解精度,为用户提供可接受近似解,控制优化进程适时终止。
本文的理论探讨包括不满秩(特别是不相容)系统的障碍法可行性恢复问题求解方法;投影梯度可行性恢复算法的全局收敛性及收敛速度分析;无可行性恢复阶段的鲁棒算法全局收敛性分析;基于收敛深度的收敛准则性质证明等。
作为通用NLP问题求解器,本文通过CUTE/COPS/MITT标准算例库求解,测试了MATLAB环境下rSQP算法的性能,并将其与MINOS和SNOPT两个求解器的性能进行了比较。在过程系统优化应用中,乙烯生产过程联塔系统优化显示出简约空间算法的优越性。FORTRAN环境下的rIPOPT算法在不满秩系统求解、全局收敛性测试和不可行性识别中表现出良好性能。收敛深度控制准则的有效性,通过rSQP算法应用于标准算例库求解、变负荷联塔系统优化、催化剂混合优化等数值实验得到体现。
Nonlinear programming (NLP) is an essential tool in process engineering. The advances in computer technology and numerical methods enable engineers to consider more large-scale and complex systems. Emphasis is laid on solving large NLPs efficiently and stably. Reduced-space methods have advantages in process engineering. For one thing, most problems in this category have only few degrees of freedom compared to the total number of variables. For another, reduced-space approaches allow to exploit problem-dependent structure and approximate second derivative information. The objective of this dissertation is the design and implementation of reduced-space algorithms appropriate to large NLPs, especially the strategies necessary for NLP solvers to solve problems stably and balance between computational cost and precision. Properties of the strategies are analysed, and performance of the new solvers is evaluated by tests on a large variety of NLPs.
The algorithms discussed include reduced-space SQP (rSQP) and interior point method (rIPOPT). Strategies which can be applied to both of the algorithms are proposed. In order to solve the decomposed system efficiently, an approximate method is introduced, accordingly, the computation order is changed. When solving rank deficient systems, dimension change is necessary for achieving system decomposition and numerical stability. Additionally, checking on rank change and (local) infeasibility during the process of optimization is needed, which can be implemented efficiently by appropriate application of a sparse system solver. Global convergence is guaranteed by feasibility restoration algorithms, i.e. the algorithms based on barrier method and projected gradient method respectively. A robust algorithm without feasibility restoration phase is discussed as well. The trade-off between computational cost and precision is implemented in criteria taking advantage of convergence depth, which try to detect acceptable approximate solution, and discover the proper time to terminate the optimization algorithm.
Theoretical discussion includes the solution of barrier feasibility restoration problem corresponding to a rank deficient system, especially an inconsistent system; global convergence and convergence rate of the feasibility restoration algorithm following projected gradient methods; global convergence of the robust algorithm without feasibility restoration phase; and proofs of the properties of convergence criteria based on convergence depth.
The practical performance of rSQP algorithm coded in MATLAB as a general purpose NLP solver is tested on various NLP problems from CUTE, COPS, and MITT. The performance is compared with that of MINOS and SNOPT. In the application to process engineering, the advantage of reduced-space methods is demonstrated in the optimization of distillation columns in ethylene production. rIPOPT in FORTRAN performs very well in the solution of rank deficient problems and tests on global convergence and infeasibility identification. The effectiveness of criteria based on convergence depth control is shown by the rSQP algorithm in the solution of problems from CUTE, distillation columns under changing feed conditions, and catalyst mixing problem.
引文
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