两种随机优化算法的改进及其化工应用研究
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摘要
在过去的30年中，能源价格持续增长，环境控制日益严格，产品竞争趋于全球化，面对这些压力，优化技术是企业降低成本提高效益的一个有效技术。从产品设计到供应链管理，优化技术可以应用于化工过程的每一个层次。然而物质能量转化过程内在的非线性、以及装置操作中的离散性使得化工过程优化存在诸多困难。面对诸多实际问题，经典数学规划法已显无能为力，因此对随机的、智能的优化技术的需求日益迫切。
随机优化方法，如遗传算法、模拟退火、禁忌搜索、蚁群算法和粒子群算法等在解决现实问题中显示了强大的搜索能力，它们可在合理的时间内逼近问题的最优解，这些算法涉及人工智能、统计热力学、生物进化论以及仿生学，所以又被称为智能优化算法。随机优化算法不受应用问题结构束缚，对问题的数学解析性质要求低，无需函数导数，甚至不需要显式的目标函数，既可处理连续问题也可以处理离散问题，并能以较大概率找到全局最优解，算法容易引入启发式逻辑规则，算法原理直观易于编码实现，这些优点已使随机优化算法成功应用于许多化工优化问题。本文以两种随机优化算法，经典的遗传算法和新颖的粒子群算法为研究对象，针对具体化工应用问题，对它们进行改进研究，提高算法解决具体问题的效率。因粒子群优化算法具有算法简单、收敛速度快的优点，成为近年随机优化领域热点之一，它是本文的重点研究对象。
本文首先根据化工优化中存在的困难和确定性优化算法内在的缺点，分析了随机优化算法的重要性，并提出研究随机优化算法应注意的问题；其次，将遗传算法应用于两个数据驱动建模问题，一为组合优化问题，一为混合整数优化问题；再次，从粒子群优化算法的基本结构、运动行为、改进方法做了系统的研究：最后，将提出的两种改进粒子群优化算法应用于相平衡计算问题，为非凸全局优化问题。本文的主要研究成果可归纳如下：
1．光谱分析是化工中常用的分析方法，波长选择是一种重要的光谱分析预处理步骤，通过筛选特征波长点，可以得到建模变量的最优组合，使所建模型的预测性能达到最佳。近红外光谱波长范围宽，波长选择可达2~(1000)种组合，其规模甚大。该问题的优化变量为0—1变量，目标函数无显式表达式，优化有一定难度。为此，本文提出移动窗口—迭代遗传算法(MW—IGA)波长选择算法，在移动窗口扫描所得的信息区间基础上，以迭代遗传算法作细化搜索，选出最优波长区间组合。该算法考虑了光谱的连续相关性特点，保留了一定的信息冗余度，使模型更为稳健。MW—IGA亦可用于其他类型光谱波长选择，若原光谱波长点小于200，可直接使刚IGA。该算法已成功应用于感冒液多组分测定的紫外—可见光谱选择和小麦水分测定的近红外光谱选择。
2．人工神经网络常用于建立非线性数据驱动模型，在化工操作优化、过程控制中较为常见。本文提出了一种改进的径向基函数一循环子空间回归(RBF-CSR)模型，它具有标准的网络结构设计方法。以模型预测性能为优化目标，优化变量同时含有实数和整数，为此，本文提出一种优进混合编码遗传算法(EHCGA)训练该模型，不同类型的变量采用不同的编码方式，对整型变量进行二进制编码，对实型变量进行浮点型编码。它采用分段交叉算子和分段变异算子，并引入Powell优进算子加速进化。该方法成功用于回收己内酰胺的脉冲波板填料塔萃取过程建模。EHCGA不仅可用于RBF—CSR模型训练，还可推广到其他混合整数规划问题。
3．为克服粒子群优化算法用于高维问题时容易早熟的缺点，本文提出一种合作粒子群优化算法(CLPSO)，它将粒子种群分成两个部分，一部分粒子负责局部开发，一部分粒子负责全局探测，这两部分粒子分工合作，使种群始终保持多样性，大大提高了算法的全局寻优性能。
4．针对粒子群优化算法运行后期收敛速度减慢的缺点，在合作粒子群算法的基础上，本文提出一种局部加速粒子群算法(LAPSO)，它将引入相对进化度的概念，用以监测种群的进化速度，并引入一些加速规则，应用Nelder-Mead单纯形法对局部区域进行局部精细搜索。粒子群算法探测全局解可能所在的区域，单纯形算法又适时地在该区域内细化搜索，加快了种群收敛速度，并提高解的精度。合作粒子群算法始终维持种群多样性，不会因引入局部算法而导致种群早熟。
5．化工问题中常存在物料守恒、质量守恒、原子守恒这类线性约束，针对粒子群优化算法无法处理约束的缺点，本文提出一种线性约束粒子群算法(LCPSO)，它对粒子群算法的位置更新步骤作了改进，各维分量的速度更新采用同一随机数，使速度更新成为线性操作，进而可以直接处理带有线性约束的非凸优化问题。LCPSO在可行空间内产生初始解，利用算法自身的线性进化算子使种群各粒子始终满足线性约束，是一种高效的保持种群于可行空间的约束优化方法。
6．相稳定性分析可判定所给定的相态是否稳定、相平衡计算结果是否正确等。Gibbs自由能切平面距离法是最常见的相稳定性判定方法，该优化问题的目标函数非凸，且受摩尔分率归一化约束。为此，本文提出采用线性约束粒子群算法LCPSO最小化切平面距离，该方法适用于各种热力学模型，可判定各种分相形式。将LCPSO应用到三类热力学模型，根据热力学原理对每种模型的目标函数作了约简，大大减少目标函数计算量。
7．复杂相平衡体系Gibbs自由能函数存在多个局部极小点，应用局部优化算法难以得到全局解。不含化学反应的相平衡问题存在物料守恒约束，通过引入组分余相分率，可将其转化为无约束优化问题。本文采用LAPSO求解这类相平衡问题，无需考虑体系实际存在的相态，计算不依赖函数导数，收敛至全局解的概率高。含化学反应的相平衡问题受到原子守恒约束问题，采用LCPSO求解该问题，可使种群始终保持在可行空间内运动，计算效率高。将原子守恒改为元素守恒，极大提高了初始可行种群的产生效率，有利于减少随机抽样产生的无效解。
随机优化算法在一些化工问题中的成功应用，确定性全局优化算法对问题数学解析性质要求高以及计算量太大，这些现实会继续促使随机优化算法在化工领域的应用研究，特别是在组合优化类型问题上的研究。
In the last thirty years, the energy price has been increasing, the control of environment has been more rigorous, and the product competition has become worldwide. Facing these pressures, optimization technique is an effective approach that can reduce the cost and increase the revenue of enterprise. From product design to supply chain management, optimization can be applied on any scale of chemical process. But the intrinsic nonlinearity in substance and energy conversion, addition to the discreteness of process operation, results in many difficulties for optimization of chemical process. In front of many practical problems, the classical mathematic programming methods are helpless. So the demand for stochastic and intelligent optimization methods is more urgent.
Stochastic optimization algorithms, such as genetic algorithm, simulated annealing, tabu search, ant colony optimization and particle swarm optimization, have powerful searching ability, and they can approach the true solution of practical problem in reasonable time. These algorithms are usually related to artificial intelligence, statistical thermodynamics, biology evolutionism, and bionics, so they are also called intelligent optimization methods. The stochastic optimization algorithms are not limited to the structure of problem, and have not rigorous restriction on mathematic properties of problem. They needn't the first derivative, and even explicit objective function. They can not only deal with continuous problems, but also discrete problems. Stochastic algorithms can find the global optimum with great probability, and are easy to fuse the heuristic rules. Such advantages have made stochastic optimization algorithms applied to many chemical engineering problems successfully. This dissertation researched on two stochastic algorithms, one of which is genetic algorithm and the other is particle swarm optimization algorithm. Aiming at the specific engineering problems, some modifications have been proposed on the two algorithms, which can improve their efficiency in specific problems. Because particle swarm optimization algorithm is more concise, and it has rapid convergence, which bring it to be a research hotspot in the field of evolutionary computation, so it is an emphasis in this dissertation.
Firstly, according to the difficulties in the optimization of chemical engineering and the intrinsic disadvantage of deterministic optimization algorithms, this work analyzed the importance and advantage of stochastic algorithms, and proposed some important aspects in research on them. Secondly, genetic algorithm was applied to two problems of data driven modeling, one of which was combination problem, the other was mixed integer nonlinear programming. Thirdly, systemic investigations were made on the basic structure, dynamic behavior and modifications of particle swarm optimization. Lastly, two kinds of proposed PSO algorithms were applied on calculation of phase equilibrium, which is nonconvex optimization. The major contributions of this work are summarized as follows.
1. Spectral wavelength selection is an important spectrum preprocessing step, which can get the best combination of modeling variables for the best predictive performance. The near infrared spectrum has wide spectral range, and there are nearly 2~(1000) combinations for wavelength selection. The optimization variables in this problem are binary and the problem hasn't
explicit optimization objective function. So moving window - iterative genetic algorithm (MW-IGA) was proposed to select wavelength, in which moving window scanning finds the information regions. Genetic algorithm selects the best combination of wavelength intervals. This approach considers the correlation of continuous wavelength points, and the resulted wavelengths contain some redundancy that make the model more robust. MW-IGA could be applied to wavelength selection for other spectrum. If the number of wavelength points is less than 200, the step of moving window could be neglected. This method has been applied to UV-Vis spectrum of cough syrup and NIR spectrum of corn.
2. Artificial neural network is used to establish the nonlinear data-driven model, which is very common in operative optimization and process control for chemical process. A modified radical basis function - cyclic subspace regression (RBF-CSR) neural network was proposed, and it has standard network structure. The model contains real and integer variables simultaneously, so eugenic hybrid coding genetic algorithm (EHCGA) was devised to train the neural network. EHCGA adopted different coding methods for different types of variable, which means integer variables adopted binary codes, while real variables adopted float codes. It used different crossing and mutation operator separately for different codes and Powell eugenic operator was introduced to accelerate evolution. RBF-CSR model trained by EHCGA has applied on pulsed extraction process for recover caprolactam successfully. EHCGA can be applied to other MINLP problem.
3. Particle swarm optimization usually converges prematurely for high dimensional problem, so a collaborative PSO (CLPSO) was proposed. The particle population is divided into two parts, one part is responsible for global exploration and the other part is responsible for local exploitation. The two parts work together and maintain the diversity of population, which improves the global searching ability.
4. PSO converges very slowly in the late evolutionary period. Based on CLPSO, a locally accelerated PSO (LAPSO) algorithm was proposed. LAPSO introduced the concept of relative evolutionary extent, which could detect the evolutionary rate of algorithm. Some acceleration rules were introduced into LAPSO and Nelder-Mead simplex algorithm was adopted for local search. PSO finds the area that may include global solution, while simplex searches the solution in the area precisely in time. LAPSO also accelerates the convergence of PSO.
5. In chemical engineering, there exist many linear constraints, such as material balance, mass balance and atom balance. A linear constraint PSO (LCPSO) was devised for this kind of problems. LCPSO modified the velocity and position updating operation, and the random number in every dimension for velocity updating adopted same value. So the new velocity for every particle became the linear combination of old velocity and position. LCPSO can be applied to nonconvex optimization constrained by linear equalities. It generates the initial population in feasible space, and utilizes its intrinsic linear operations to maintain the particle satisfying the constraints. LCPSO is an effective constrained optimization algorithm that deals with linear equalities.
6. Phase stability analysis can determine whether the given phase is stable and whether the result of phase equilibrium calculation is right. Tangent
plane distance function (TPDF) approach is the popular method for phase stability analysis. TPDF is nonconvex and the problem is constrained by the mole fraction summation. LCPSO was utilized to minimize the TPDF, and this method can applied to any thermodynamic model and can detect any type of instability. This work applied LCPSO to three types of thermodynamic model. According to thermodynamic theories, the objective functions were simplified, which greatly decreased the amount of calculation.
7. For complex phase equilibrium system, the Gibbs energy function has several local minima, so it's difficult to get the global minimum by the local optimization algorithms. If chemical reactions don't occur, there are only material balances for phase equilibrium. Component phase fraction was introduced, which converts the original problem to an unconstrained one. LAPSO was utilized to solve the phase equilibrium without reactions, and it need not considering the actual number and type of phases and needn't the derivative. For phase equilibrium problem with chemical reaction, the constraints are atom balances, which are general linear equalities. LCPSO was utilized to compute this kind of equilibrium. LCPSO maintains the particle within feasible space and the computing efficiency is high. When atom balances are converted to element balances, the efficiency of generating initial feasible population is greatly improved.
The facts that stochastic optimization algorithms were applied to chemical engineering successfully and the deterministic algorithms have the intrinsic disadvantages, will promote the research on stochastic algorithm in chemical engineering, especially in combinational and global optimization.
随机优化方法，如遗传算法、模拟退火、禁忌搜索、蚁群算法和粒子群算法等在解决现实问题中显示了强大的搜索能力，它们可在合理的时间内逼近问题的最优解，这些算法涉及人工智能、统计热力学、生物进化论以及仿生学，所以又被称为智能优化算法。随机优化算法不受应用问题结构束缚，对问题的数学解析性质要求低，无需函数导数，甚至不需要显式的目标函数，既可处理连续问题也可以处理离散问题，并能以较大概率找到全局最优解，算法容易引入启发式逻辑规则，算法原理直观易于编码实现，这些优点已使随机优化算法成功应用于许多化工优化问题。本文以两种随机优化算法，经典的遗传算法和新颖的粒子群算法为研究对象，针对具体化工应用问题，对它们进行改进研究，提高算法解决具体问题的效率。因粒子群优化算法具有算法简单、收敛速度快的优点，成为近年随机优化领域热点之一，它是本文的重点研究对象。
本文首先根据化工优化中存在的困难和确定性优化算法内在的缺点，分析了随机优化算法的重要性，并提出研究随机优化算法应注意的问题；其次，将遗传算法应用于两个数据驱动建模问题，一为组合优化问题，一为混合整数优化问题；再次，从粒子群优化算法的基本结构、运动行为、改进方法做了系统的研究：最后，将提出的两种改进粒子群优化算法应用于相平衡计算问题，为非凸全局优化问题。本文的主要研究成果可归纳如下：
1．光谱分析是化工中常用的分析方法，波长选择是一种重要的光谱分析预处理步骤，通过筛选特征波长点，可以得到建模变量的最优组合，使所建模型的预测性能达到最佳。近红外光谱波长范围宽，波长选择可达2~(1000)种组合，其规模甚大。该问题的优化变量为0—1变量，目标函数无显式表达式，优化有一定难度。为此，本文提出移动窗口—迭代遗传算法(MW—IGA)波长选择算法，在移动窗口扫描所得的信息区间基础上，以迭代遗传算法作细化搜索，选出最优波长区间组合。该算法考虑了光谱的连续相关性特点，保留了一定的信息冗余度，使模型更为稳健。MW—IGA亦可用于其他类型光谱波长选择，若原光谱波长点小于200，可直接使刚IGA。该算法已成功应用于感冒液多组分测定的紫外—可见光谱选择和小麦水分测定的近红外光谱选择。
2．人工神经网络常用于建立非线性数据驱动模型，在化工操作优化、过程控制中较为常见。本文提出了一种改进的径向基函数一循环子空间回归(RBF-CSR)模型，它具有标准的网络结构设计方法。以模型预测性能为优化目标，优化变量同时含有实数和整数，为此，本文提出一种优进混合编码遗传算法(EHCGA)训练该模型，不同类型的变量采用不同的编码方式，对整型变量进行二进制编码，对实型变量进行浮点型编码。它采用分段交叉算子和分段变异算子，并引入Powell优进算子加速进化。该方法成功用于回收己内酰胺的脉冲波板填料塔萃取过程建模。EHCGA不仅可用于RBF—CSR模型训练，还可推广到其他混合整数规划问题。
3．为克服粒子群优化算法用于高维问题时容易早熟的缺点，本文提出一种合作粒子群优化算法(CLPSO)，它将粒子种群分成两个部分，一部分粒子负责局部开发，一部分粒子负责全局探测，这两部分粒子分工合作，使种群始终保持多样性，大大提高了算法的全局寻优性能。
4．针对粒子群优化算法运行后期收敛速度减慢的缺点，在合作粒子群算法的基础上，本文提出一种局部加速粒子群算法(LAPSO)，它将引入相对进化度的概念，用以监测种群的进化速度，并引入一些加速规则，应用Nelder-Mead单纯形法对局部区域进行局部精细搜索。粒子群算法探测全局解可能所在的区域，单纯形算法又适时地在该区域内细化搜索，加快了种群收敛速度，并提高解的精度。合作粒子群算法始终维持种群多样性，不会因引入局部算法而导致种群早熟。
5．化工问题中常存在物料守恒、质量守恒、原子守恒这类线性约束，针对粒子群优化算法无法处理约束的缺点，本文提出一种线性约束粒子群算法(LCPSO)，它对粒子群算法的位置更新步骤作了改进，各维分量的速度更新采用同一随机数，使速度更新成为线性操作，进而可以直接处理带有线性约束的非凸优化问题。LCPSO在可行空间内产生初始解，利用算法自身的线性进化算子使种群各粒子始终满足线性约束，是一种高效的保持种群于可行空间的约束优化方法。
6．相稳定性分析可判定所给定的相态是否稳定、相平衡计算结果是否正确等。Gibbs自由能切平面距离法是最常见的相稳定性判定方法，该优化问题的目标函数非凸，且受摩尔分率归一化约束。为此，本文提出采用线性约束粒子群算法LCPSO最小化切平面距离，该方法适用于各种热力学模型，可判定各种分相形式。将LCPSO应用到三类热力学模型，根据热力学原理对每种模型的目标函数作了约简，大大减少目标函数计算量。
7．复杂相平衡体系Gibbs自由能函数存在多个局部极小点，应用局部优化算法难以得到全局解。不含化学反应的相平衡问题存在物料守恒约束，通过引入组分余相分率，可将其转化为无约束优化问题。本文采用LAPSO求解这类相平衡问题，无需考虑体系实际存在的相态，计算不依赖函数导数，收敛至全局解的概率高。含化学反应的相平衡问题受到原子守恒约束问题，采用LCPSO求解该问题，可使种群始终保持在可行空间内运动，计算效率高。将原子守恒改为元素守恒，极大提高了初始可行种群的产生效率，有利于减少随机抽样产生的无效解。
随机优化算法在一些化工问题中的成功应用，确定性全局优化算法对问题数学解析性质要求高以及计算量太大，这些现实会继续促使随机优化算法在化工领域的应用研究，特别是在组合优化类型问题上的研究。
In the last thirty years, the energy price has been increasing, the control of environment has been more rigorous, and the product competition has become worldwide. Facing these pressures, optimization technique is an effective approach that can reduce the cost and increase the revenue of enterprise. From product design to supply chain management, optimization can be applied on any scale of chemical process. But the intrinsic nonlinearity in substance and energy conversion, addition to the discreteness of process operation, results in many difficulties for optimization of chemical process. In front of many practical problems, the classical mathematic programming methods are helpless. So the demand for stochastic and intelligent optimization methods is more urgent.
Stochastic optimization algorithms, such as genetic algorithm, simulated annealing, tabu search, ant colony optimization and particle swarm optimization, have powerful searching ability, and they can approach the true solution of practical problem in reasonable time. These algorithms are usually related to artificial intelligence, statistical thermodynamics, biology evolutionism, and bionics, so they are also called intelligent optimization methods. The stochastic optimization algorithms are not limited to the structure of problem, and have not rigorous restriction on mathematic properties of problem. They needn't the first derivative, and even explicit objective function. They can not only deal with continuous problems, but also discrete problems. Stochastic algorithms can find the global optimum with great probability, and are easy to fuse the heuristic rules. Such advantages have made stochastic optimization algorithms applied to many chemical engineering problems successfully. This dissertation researched on two stochastic algorithms, one of which is genetic algorithm and the other is particle swarm optimization algorithm. Aiming at the specific engineering problems, some modifications have been proposed on the two algorithms, which can improve their efficiency in specific problems. Because particle swarm optimization algorithm is more concise, and it has rapid convergence, which bring it to be a research hotspot in the field of evolutionary computation, so it is an emphasis in this dissertation.
Firstly, according to the difficulties in the optimization of chemical engineering and the intrinsic disadvantage of deterministic optimization algorithms, this work analyzed the importance and advantage of stochastic algorithms, and proposed some important aspects in research on them. Secondly, genetic algorithm was applied to two problems of data driven modeling, one of which was combination problem, the other was mixed integer nonlinear programming. Thirdly, systemic investigations were made on the basic structure, dynamic behavior and modifications of particle swarm optimization. Lastly, two kinds of proposed PSO algorithms were applied on calculation of phase equilibrium, which is nonconvex optimization. The major contributions of this work are summarized as follows.
1. Spectral wavelength selection is an important spectrum preprocessing step, which can get the best combination of modeling variables for the best predictive performance. The near infrared spectrum has wide spectral range, and there are nearly 2~(1000) combinations for wavelength selection. The optimization variables in this problem are binary and the problem hasn't
explicit optimization objective function. So moving window - iterative genetic algorithm (MW-IGA) was proposed to select wavelength, in which moving window scanning finds the information regions. Genetic algorithm selects the best combination of wavelength intervals. This approach considers the correlation of continuous wavelength points, and the resulted wavelengths contain some redundancy that make the model more robust. MW-IGA could be applied to wavelength selection for other spectrum. If the number of wavelength points is less than 200, the step of moving window could be neglected. This method has been applied to UV-Vis spectrum of cough syrup and NIR spectrum of corn.
2. Artificial neural network is used to establish the nonlinear data-driven model, which is very common in operative optimization and process control for chemical process. A modified radical basis function - cyclic subspace regression (RBF-CSR) neural network was proposed, and it has standard network structure. The model contains real and integer variables simultaneously, so eugenic hybrid coding genetic algorithm (EHCGA) was devised to train the neural network. EHCGA adopted different coding methods for different types of variable, which means integer variables adopted binary codes, while real variables adopted float codes. It used different crossing and mutation operator separately for different codes and Powell eugenic operator was introduced to accelerate evolution. RBF-CSR model trained by EHCGA has applied on pulsed extraction process for recover caprolactam successfully. EHCGA can be applied to other MINLP problem.
3. Particle swarm optimization usually converges prematurely for high dimensional problem, so a collaborative PSO (CLPSO) was proposed. The particle population is divided into two parts, one part is responsible for global exploration and the other part is responsible for local exploitation. The two parts work together and maintain the diversity of population, which improves the global searching ability.
4. PSO converges very slowly in the late evolutionary period. Based on CLPSO, a locally accelerated PSO (LAPSO) algorithm was proposed. LAPSO introduced the concept of relative evolutionary extent, which could detect the evolutionary rate of algorithm. Some acceleration rules were introduced into LAPSO and Nelder-Mead simplex algorithm was adopted for local search. PSO finds the area that may include global solution, while simplex searches the solution in the area precisely in time. LAPSO also accelerates the convergence of PSO.
5. In chemical engineering, there exist many linear constraints, such as material balance, mass balance and atom balance. A linear constraint PSO (LCPSO) was devised for this kind of problems. LCPSO modified the velocity and position updating operation, and the random number in every dimension for velocity updating adopted same value. So the new velocity for every particle became the linear combination of old velocity and position. LCPSO can be applied to nonconvex optimization constrained by linear equalities. It generates the initial population in feasible space, and utilizes its intrinsic linear operations to maintain the particle satisfying the constraints. LCPSO is an effective constrained optimization algorithm that deals with linear equalities.
6. Phase stability analysis can determine whether the given phase is stable and whether the result of phase equilibrium calculation is right. Tangent
plane distance function (TPDF) approach is the popular method for phase stability analysis. TPDF is nonconvex and the problem is constrained by the mole fraction summation. LCPSO was utilized to minimize the TPDF, and this method can applied to any thermodynamic model and can detect any type of instability. This work applied LCPSO to three types of thermodynamic model. According to thermodynamic theories, the objective functions were simplified, which greatly decreased the amount of calculation.
7. For complex phase equilibrium system, the Gibbs energy function has several local minima, so it's difficult to get the global minimum by the local optimization algorithms. If chemical reactions don't occur, there are only material balances for phase equilibrium. Component phase fraction was introduced, which converts the original problem to an unconstrained one. LAPSO was utilized to solve the phase equilibrium without reactions, and it need not considering the actual number and type of phases and needn't the derivative. For phase equilibrium problem with chemical reaction, the constraints are atom balances, which are general linear equalities. LCPSO was utilized to compute this kind of equilibrium. LCPSO maintains the particle within feasible space and the computing efficiency is high. When atom balances are converted to element balances, the efficiency of generating initial feasible population is greatly improved.
The facts that stochastic optimization algorithms were applied to chemical engineering successfully and the deterministic algorithms have the intrinsic disadvantages, will promote the research on stochastic algorithm in chemical engineering, especially in combinational and global optimization.
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