基于自动微分算法的过程系统优化
摘要
随着计算机技术的飞速发展和企业自动化程度的不断提高，过程系统优化已
经从纯学术的理论发展成为能对工业起到巨大推动作用的技术力量，成为过程工
业企业保持竞争力、在激烈的市场竞争中立于不败之地的主要技术手段。过程系
统优化的核心是数学规划算法。当前的高性能数学规划算法大都依赖于导数计算
以快速收敛到最优点，所以对求导精度的要求比较高；而导数计算所消耗的时间
占优化时间的比例又非常的大。所以，求导问题是优化问题中一个非常重要的方
面，值得过程系统优化的研究人员进行深入的分析。
在过程系统优化中，一个理想的求导算法应该能够快速求取程序模块的解析
导数。近年来发展迅速的自动微分算法比较好地满足了这个条件，并已在优化中
得到了广泛的应用。但是，鉴于过程系统优化中模型的复杂性，现有的自动微分
算法还有许多不足之处，需要得到进一步地改进。本文基于自动微分算法的过程
系统优化研究，就是通过结合过程建模和优化的特点，改进现有的自动微分算法，
并将其和优化算法结合以达到提高过程系统优化总体效率的目的。
本文的研究工作主要有以下几点：
在阐述过程系统优化的各类任务及其对应的数学规划问题的求解方法的基
础上，分析和总结了求导和优化之间的关系，指出在优化中使用自动微分的
优势。通过系统阐述自动微分算法的理论和相关技术，分析其固有的优点和
缺点，为算法的改进提供了理论基础并指明了研究方向。
通过比较和分析符号微分和自动微分，提出了一种新型的符号求导算法--
符号自动微分算法。符号自动微分具有可对子程序形式的函数求导、可以充
分利用模型的稀疏性、无需辅助操作的优点，非常适合针对结构相对简单、
计算式以多项式为主的过程系统模型求导。开发了能同时实现数值自动微分
和符号自动微分的扩展的自动微分工具--XADMAT，并将其成功地应用在
一个典型的过程系统优化问题中。
研究了过程建模与优化、求导的关系。分析和比较了当前两大基本建模方法
--联立方程法和序贯模块法对优化效率的影响，指出结合这两种方法优点
的复合建模方法在优化中的优势，并通过一个精馏塔稳态优化问题验证了这
一结论。分析了建模方法对求导效率的影响，指出不同的求导算法适用于不
同结构的过程模型，提出对于复杂模型求导最好的方法是根据其不同模块的
特点采用合适的求导算法。
11 浙江大学硕士学位论文
．提出了一种结合现有的求导算法对复杂模型求导以大幅提高求导效率的方
法——面向模块的自动微分算法。该方法沿袭前向自动微分拆分元函数和利
用链式求导法则组合元偏导数的求导思路，将模型拆分为子模块逐次处理。
讨论了子模块连接的两类方法——直接连接法和间接连接法，从理论上证明
了直接连接方法的优势。面向模块的自动微分可以更加灵活和充分地利用模
型的结构特点，这个优势在精馏塔稳态优化和缩聚反应釜动态优化这两个典
型的工业过程系统优化问题中得到了充分的体现。
．研究了面向模块的自动微分的核心问题——模型的分割和子模块求导算法
的选择。提出了以最小函数为基本单元，并惜助模型稀疏信息和导数计算复
杂程度的信息来分割模型的总体思路。提出通过计算模型变量导数的稀疏模
式矩阵来获得模型的稀疏性结构。定义了模块复杂度的概念来表征求导算法
对模块求导所需的单元标量计算次数，并以此为依据来为模块选择最合适的
求导算法。通过精馏塔稳态优化和缩聚反应釜动态优化问题验证了上述分析
方法的有效性。
Process system optimization (PSO) has become a major technology that helps companies in process industry to remain competitive. Numerical derivatives play an important role in mathematical programming, which is the core area in PSO. On one hand, numerical derivatives provide proper search directions in optimization, therefore their accuracy is of great importance for fast convergency. On the other hand, derivative evaluation is one of the most time-consuming steps in optimization. Improving the efficiency of the evaluation of derivatives is an effective way to reduce the time for optimization, and is of significant help to real-time optimization.
Automatic differentiation (AD), which is well recognized as the most promising differentiation algorithm in PSO and has been developed rapidly in recent twenty years, is a technique for augmenting computer programs with derivative computations. It exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations such as additions or elementary functions such as exponential function. By applying the chain rule of derivative calculus repeatedly to these operations, derivatives of arbitrary order can be computed automatically, and accurate to working precision. However, AD has some limitations when applied to complicated process models. The research work in this thesis is focused on AD-based differentiation methodology that can fully exploit the structure of the objective model to greatly reduce the time for Jacobian evaluation and improve the efficiency of optimization. The main contributions are as follows:
Applications of optimization in process engineering are introduced and the mathematical programming problems related to them are indicated. Then solution methods of the major types of optimization problems are reviewed, based on which the importance of derivative evaluation in optimization is analyzed and summarized. After that, the principle and technique of AD are systematically discussed meanwhile the advantages and disadvantages of AD are pointed out to give hint on more efficient differentiation approach.
Symbolic Differentiation (SD) and AD are compared in details and then a novel SD algorithm, Symbolic Automatic Differentiation (SAD), is presented. SAD is able to differentiate models in form of subroutines, fully exploit the sparsity of the models and need few additional operations. It is concluded that SAD is very
suitable for the process models that have relatively simple computational structure and consist mainly of polynomials. An extended AD tool XADMAT, which is derived from ADMAT, the operator-overloading AD tool in MATLAB, is developed to realize both numerical and symbolical AD. XADMAT is successfully applied to an alkylation process optimization problem.
The influence of process modeling approaches on optimization and differentiation is studied. It is concluded that a combination of the two main modeling approaches, the sequential modular approach and the equation-oriented approach, can keep a good balance between flexibility and efficiency. So-obtained process model, called composite model in this paper, contains hidden variables that can not be observed externally. Accordingly, traditional algorithms are not able to fully exploit the structure of complex composite model so as to highly improve the efficiency of optimization. It is pointed out that applying different differentiation algorithm to different parts of the model might be the recipe to conquer the difficulty.
Module-oriented Automatic Differentiation (MAD), a new AD approach that can employ current differentiation algorithms flexibly to differentiate each module of the process model, is presented to achieve more efficient Jacobian evaluation in optimization. In MAD, the model of interest is partitioned into a series of modules, just as in AD the function of interest is divided into a sequence of elementary functions. Two modes for accumulating the Jacobians of the modules, direct mode and indirect mode are discusse
经从纯学术的理论发展成为能对工业起到巨大推动作用的技术力量，成为过程工
业企业保持竞争力、在激烈的市场竞争中立于不败之地的主要技术手段。过程系
统优化的核心是数学规划算法。当前的高性能数学规划算法大都依赖于导数计算
以快速收敛到最优点，所以对求导精度的要求比较高；而导数计算所消耗的时间
占优化时间的比例又非常的大。所以，求导问题是优化问题中一个非常重要的方
面，值得过程系统优化的研究人员进行深入的分析。
在过程系统优化中，一个理想的求导算法应该能够快速求取程序模块的解析
导数。近年来发展迅速的自动微分算法比较好地满足了这个条件，并已在优化中
得到了广泛的应用。但是，鉴于过程系统优化中模型的复杂性，现有的自动微分
算法还有许多不足之处，需要得到进一步地改进。本文基于自动微分算法的过程
系统优化研究，就是通过结合过程建模和优化的特点，改进现有的自动微分算法，
并将其和优化算法结合以达到提高过程系统优化总体效率的目的。
本文的研究工作主要有以下几点：
在阐述过程系统优化的各类任务及其对应的数学规划问题的求解方法的基
础上，分析和总结了求导和优化之间的关系，指出在优化中使用自动微分的
优势。通过系统阐述自动微分算法的理论和相关技术，分析其固有的优点和
缺点，为算法的改进提供了理论基础并指明了研究方向。
通过比较和分析符号微分和自动微分，提出了一种新型的符号求导算法--
符号自动微分算法。符号自动微分具有可对子程序形式的函数求导、可以充
分利用模型的稀疏性、无需辅助操作的优点，非常适合针对结构相对简单、
计算式以多项式为主的过程系统模型求导。开发了能同时实现数值自动微分
和符号自动微分的扩展的自动微分工具--XADMAT，并将其成功地应用在
一个典型的过程系统优化问题中。
研究了过程建模与优化、求导的关系。分析和比较了当前两大基本建模方法
--联立方程法和序贯模块法对优化效率的影响，指出结合这两种方法优点
的复合建模方法在优化中的优势，并通过一个精馏塔稳态优化问题验证了这
一结论。分析了建模方法对求导效率的影响，指出不同的求导算法适用于不
同结构的过程模型，提出对于复杂模型求导最好的方法是根据其不同模块的
特点采用合适的求导算法。
11 浙江大学硕士学位论文
．提出了一种结合现有的求导算法对复杂模型求导以大幅提高求导效率的方
法——面向模块的自动微分算法。该方法沿袭前向自动微分拆分元函数和利
用链式求导法则组合元偏导数的求导思路，将模型拆分为子模块逐次处理。
讨论了子模块连接的两类方法——直接连接法和间接连接法，从理论上证明
了直接连接方法的优势。面向模块的自动微分可以更加灵活和充分地利用模
型的结构特点，这个优势在精馏塔稳态优化和缩聚反应釜动态优化这两个典
型的工业过程系统优化问题中得到了充分的体现。
．研究了面向模块的自动微分的核心问题——模型的分割和子模块求导算法
的选择。提出了以最小函数为基本单元，并惜助模型稀疏信息和导数计算复
杂程度的信息来分割模型的总体思路。提出通过计算模型变量导数的稀疏模
式矩阵来获得模型的稀疏性结构。定义了模块复杂度的概念来表征求导算法
对模块求导所需的单元标量计算次数，并以此为依据来为模块选择最合适的
求导算法。通过精馏塔稳态优化和缩聚反应釜动态优化问题验证了上述分析
方法的有效性。
Process system optimization (PSO) has become a major technology that helps companies in process industry to remain competitive. Numerical derivatives play an important role in mathematical programming, which is the core area in PSO. On one hand, numerical derivatives provide proper search directions in optimization, therefore their accuracy is of great importance for fast convergency. On the other hand, derivative evaluation is one of the most time-consuming steps in optimization. Improving the efficiency of the evaluation of derivatives is an effective way to reduce the time for optimization, and is of significant help to real-time optimization.
Automatic differentiation (AD), which is well recognized as the most promising differentiation algorithm in PSO and has been developed rapidly in recent twenty years, is a technique for augmenting computer programs with derivative computations. It exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations such as additions or elementary functions such as exponential function. By applying the chain rule of derivative calculus repeatedly to these operations, derivatives of arbitrary order can be computed automatically, and accurate to working precision. However, AD has some limitations when applied to complicated process models. The research work in this thesis is focused on AD-based differentiation methodology that can fully exploit the structure of the objective model to greatly reduce the time for Jacobian evaluation and improve the efficiency of optimization. The main contributions are as follows:
Applications of optimization in process engineering are introduced and the mathematical programming problems related to them are indicated. Then solution methods of the major types of optimization problems are reviewed, based on which the importance of derivative evaluation in optimization is analyzed and summarized. After that, the principle and technique of AD are systematically discussed meanwhile the advantages and disadvantages of AD are pointed out to give hint on more efficient differentiation approach.
Symbolic Differentiation (SD) and AD are compared in details and then a novel SD algorithm, Symbolic Automatic Differentiation (SAD), is presented. SAD is able to differentiate models in form of subroutines, fully exploit the sparsity of the models and need few additional operations. It is concluded that SAD is very
suitable for the process models that have relatively simple computational structure and consist mainly of polynomials. An extended AD tool XADMAT, which is derived from ADMAT, the operator-overloading AD tool in MATLAB, is developed to realize both numerical and symbolical AD. XADMAT is successfully applied to an alkylation process optimization problem.
The influence of process modeling approaches on optimization and differentiation is studied. It is concluded that a combination of the two main modeling approaches, the sequential modular approach and the equation-oriented approach, can keep a good balance between flexibility and efficiency. So-obtained process model, called composite model in this paper, contains hidden variables that can not be observed externally. Accordingly, traditional algorithms are not able to fully exploit the structure of complex composite model so as to highly improve the efficiency of optimization. It is pointed out that applying different differentiation algorithm to different parts of the model might be the recipe to conquer the difficulty.
Module-oriented Automatic Differentiation (MAD), a new AD approach that can employ current differentiation algorithms flexibly to differentiate each module of the process model, is presented to achieve more efficient Jacobian evaluation in optimization. In MAD, the model of interest is partitioned into a series of modules, just as in AD the function of interest is divided into a sequence of elementary functions. Two modes for accumulating the Jacobians of the modules, direct mode and indirect mode are discusse
引文
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Tanartkit, P. and Biegler, L. T. (1995) . Stable decomposition for dynamic optimization. Ind. Eng. Chem. Res., 34,1253-1266．
Tolsma J. E. and Barton P. I. (1998) . On computational differentiation. Computers Chem. Engng., 22(4/5) , 475-490．
Tolsma, J. E. and Barton, P. I. (1999) . Efficient calculation of sparse jacobians. SIAM Journal on Scientific Computing, 20,2282-2296．
Tolsma, J. E., Clabaugh, J. A. and Barton, P. I. (2001) . Symbolic incorporation of external procedures into process modeling environments. Industrial & Engineering Chemistry Research, submitted.
Tolsma, J. E. and Barton, P. I. (2002) . Hidden discontinuities and parametric sensitivity calculations. SIAM Journal on Scientific Computingv23,1861-1874．
Vasantharajan, S., Biegler, L. T. (1988) , Large-scale decomposition for successive quadratic programming. Computers Chem. Engng., 12(11) : 1087-1101．
Vassiliadis, V. S., Sargent, R. W. H. and Pantelides, C. C. (1994a), Solution of a class of multistage dynamic optimization problem 1, Problems without path constraints. Ind. Eng. Chem. Res., 33,2111-2122．
Vassiliadis, V. S., Sargent, R. W. H., Pantelides, C. C. (1994b), Solution of a class of multistage dynamic optimization problem 2, Problems with path constraints. Ind. Eng. Chem. Res., 33, 2123-2133．
Verma, A. (1998) . Structured automatic differentiation. Ph.D. Thesis, Cornell University.
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Westerlund, T. and Pettersson, F. (1995) , A cutting plane method for solving convex MINLP problems. Computers Chem. Engng., 19,131-136．
Wolbert, D., Joulia, X., Koehret, B. and Biegler, L. T. (1994) . Flowsheet optimization and optimal sensitivity analysis using analytical derivatives. Computers Chem. Engng., 18,1083-1095．
Wright, S. J. (1996) , Primal-Dual Interior Point Methods. SIAM, Philadelphia.
Yuan, X., Zhang, S., Piboleau, L. and Domenech, S. (1988) . Une Methods d'optimisation nonlineare en Variables mixtes pour la conception de precedes, RAIRO.22,331．
Zhong, W, Shao; Z., Zhang, Y. and Qian, J. (2000) . Applying analytical derivative and sparse matrix techniques to large-scale process optimization problems. Chinese Journal of Chemical Engineering, 8,212-217．
Zhong, W., Wang, W, Shao, Z., Zhang, Y, Qian, J. (2001) , Opimization of an industrial batch polycondensation reactor. Proceedings of the American Control Conference 2001, 368-373．
陈宗海(1997) ．过程系统建模与仿真，中国科学技术大学出版社，合肥．
钱积新，鲍立威，邵之江，张余岳(1998a)．生产过程综合自动化中的在线优化 (Ⅰ)．石油化工自动化，4，2-4．
钱积新，鲍立威，邵之江，张余岳(1998b)．生产过程综合自动化中的在线优化 (Ⅱ)．石油化工自动化，5，2-4．
邵之江(1997) ．连续工业过程的在线优化，浙江大学博士学位论文．
沈静珠(1994) ．过程系统优化，清华大学出版社，北京．
王世怀，徐亦方，沈复(1998) ．序列二次规划法在石油化工过程优化中的应用及 其改进．石油学报(石油加工)，14(4) ，57-63．
徐博文(1997) ．过程建模与优化．化工自动化及仪表，24(5) ，52～55．
杨冀宏，麻德贤(1989) ．过程系统工程导论，烃加工出版社，北京．
杨友麒(1995) ．过程流程模拟．计算机与应用化学，12(1) ，1-6．
姚平经(1992) ．化工过程系统工程，大连理工大学出版社，大连．
张余岳(1998) ．化工过程模拟与优化，浙江大学博士学位论文．
仲卫涛(2001) ．过程系统优化的大规模优化问题研究，浙江大学博士学位论文．
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