基于区间的不确定性优化理论与算法
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摘要
不确定性广泛存在于工程实际问题中,不确定性优化理论和算法的研究对于产品或系统的可靠性设计具有重要意义。随机规划和模糊规划是两类传统的不确定性优化方法,它们需要大量的不确定性信息以构造变量的精确概率分布或模糊隶属度函数。然而,对于很多工程问题,获得足够的不确定性信息往往显得非常困难或成本过高,这便使得两类方法在适用性上具有一定的局限性。区间数优化是一类相对较新的不确定性优化方法,它利用区间描述变量的不确定性,只需要通过较少的信息获得变量的上下界,故在不确定性建模方面体现了很好的方便性和经济性。区间数优化方法的研究近年来开始逐渐受到国内外的重视,有望在未来成为继随机规划和模糊规划之后的第三大不确定性优化方法,并且在工程领域展现了比后两者更强的应用潜力。然而目前区间数优化的研究尚处于初步阶段,特别是非线性区间数优化的研究还刚刚起步,在数学转换模型的建立、两层嵌套优化问题的求解等方面尚存在一系列的技术难点需要解决。
为此,本文将针对非线性区间数优化展开系统的研究,力求在其数学规划理论本身及实用性算法方面做出一些卓有成效的尝试和探索。数学规划理论方面的工作是提出两种非线性区间数优化的转换模型,实现了不确定性优化问题向确定性优化问题的转换,此部分工作是整篇论文的基础;实用性算法方面的工作主要是将目前工程优化领域中的一些求解工具有机引入非线性区间数优化,一定程度上解决因两层嵌套优化造成的低效问题,从而构造出多种具一定工程实用性的高效非线性区间数优化算法。基于此思路,本论文开展和完成了如下研究工作:
(1)针对一般的不确定性优化问题,从数学规划理论层面提出了两种非线性区间数优化的数学转换模型,即区间序关系转换模型和区间可能度转换模型。给出了一种改进的区间可能度构造方法,将不确定不等式约束转换为确定性约束;给出了不确定等式约束的处理方法,最终将之转换为两个确定性约束。两种转换模型采用了上述相同的不确定约束的处理方法,但在不确定目标函数的处理上有所不同,即分别基于区间序关系和区间可能度将不确定目标函数转换为确定性目标函数。通过转换模型,得到一确定性的两层嵌套优化问题。最后,提出一种基于遗传算法的两层嵌套优化方法来求解转换后的确定性优化问题。
(2)给出多网络和单网络两种混合优化算法求解转换后的两层嵌套优化问题,从而构造出两种高效的非线性区间数优化算法。多网络混合优化算法中,通过多个人工神经网络模型建立设计向量与目标函数区间或约束区间之间的映射关系,并且采用遗传算法作为优化求解器;单网络混合优化算法中,只通过单个人工神经网络模型建立设计变量和不确定变量与相应的目标函数值和约束值之间的映射关系,并且采用遗传算法作为内、外层优化求解器。利用混合优化算法对转换后的确定性优化问题进行求解时,不再使用原耗时的真实模型,而是每次调用人工神经网络模型进行快速计算,从而大大提高了非线性区间数优化的计算效率。
(3)对区间结构分析方法进行了扩展,并基于区间结构分析方法发展出了一种高效的非线性区间数优化算法。基于区间集合理论和子区间技术,提出了大不确定性区间结构分析方法,以计算较大的变量不确定性水平下的结构响应边界;将区间结构分析方法引入复合材料弹性波动问题,并与混合数值法相结合提出了一种复合材料层合板的弹性波动区间数值算法,用于计算层合板在不确定材料属性和载荷下的瞬态位移响应边界;利用区间结构分析方法求解不确定目标函数和约束在任一设计向量下的区间,有效地消除了内层优化,原先通过转换模型获得的两层嵌套优化问题变成了一传统的单层优化问题,从而构造出一种高效的非线性区间数结构优化算法。
(4)基于序列线性规划技术,发展出了一种高效的非线性区间数优化算法。每一迭代步,通过一阶泰勒展式建立目标函数和约束关于设计变量和不确定变量的线性近似模型,从而得到一线性区间数优化问题;基于区间分析方法,高效求解不确定目标函数和约束在当前近似优化问题最优解处的边界,并以此判断当前得到的最优设计向量是否为可行下降解,只有可行下降解才得以保留至下一迭代步;另外,根据区间数优化的特点给出多个停止判断准则,保证算法的收敛性。
(5)提出了基于近似模型管理策略的非线性区间数优化算法。整个优化过程由一系列近似不确定性优化问题迭代完成:每一迭代步,通过近似模型技术建立一近似不确定性优化问题,并进一步通过非线性区间数优化的转换模型将之转变为确定性优化问题进行求解;利用信赖域方法对优化过程中的近似模型进行管理,即每一迭代步计算可靠性指标以判断近似模型精度,并对设计向量和信赖域半径矢量进行更新,使得设计空间不断向最优解靠近。
(6)提出了基于局部加密近似模型技术的非线性区间数优化算法。每一迭代步,根据当前近似不确定性优化问题的求解结果,对目标函数和约束的当前样本进行加密,使得下一迭代步中的近似模型在近似空间中对应于响应边界的两个关键区域的局部精度得到提高。该算法只追求近似模型在局部关键区域而非整体近似空间上的精度,所以相比于常规的基于均匀分布样本的近似优化方法,能大大减少所需样本的数量;另外,该算法在提高优化效率的同时还能一定程度上避免因样本过多而造成的近似模型矩阵的奇异。
Uncertainty widely exists in practical engineering problems, and studying the theories and algorithms of uncertain optimization is significant for reliability design of industrial products and systems. Stochastic programming and fuzzy programming are two types of traditional uncertain optimization methodologies, in which a great amount of sampling information on the uncertainty is required to construct the precise probability distributions or fuzzy membership functions. Unfortunately, it often seems very difficult or sometimes expensive to obtain sufficient uncertainty information, and hence these two types of methods will encounter some limitations in applicability. The interval number optimization is a relatively newly-developed uncertain optimization method, in which interval is used to model the uncertainty of a variable. Thus the variation bounds of the uncertain variables are only required, which can be obtained through only a small amount of uncertainty information. In recent years, the interval number optimization has been attracting more and more attentions, and it is expected to become the third major uncertain optimization methodology after stochastic programming and fuzzy programming. Furthermore, the interval number optimization even shows a larger application potential for practical engineering problems than the other two methods. However, it is still at its preliminary stage for the present interval number optimization research, especially for the nonlinear interval number optimization (NINO), studies for which are just getting started. Some key technical difficulties remain, such as creation of the mathematical transformation models, effective solution of the nesting optimization problem etc.
This dissertation conducts a systematical research for the NINO, and aims at contributing some useful researches and trials on mathematical programming theories and practical algorithms. Firstly, two transformation models of NINO are put forward through which the uncertain optimization problem can be transformed into a deterministic optimization problem, and this part of work is the basis of the whole dissertation. On the other hand, some computation tools in the present engineering optimization field are extended into the interval number optimization to solve the lower efficiency problem caused by the nesting optimization and whereby several efficient NINO algorithms are constructed. As a result, the following studies are carried out in this dissertation:
(1) For a general uncertain optimization problem, two mathematical transformation models of NINO are suggested at the level of mathematical programming theory, respectively based on the order relation of interval number and the possibility degree of interval number. A modified construction method is suggested for the possibility degree of interval number based on the probability method, and based on it an uncertain inequality constraint can be transformed into a deterministic constraint; an approach is also suggested to transform an uncertain equality constraint into two deterministic constraints. The two transformation models employ the above same way to deal with the uncertain constraints, while different ways for uncertain objective function, namely adopt the order relation of interval number and the possibility degree of interval number to transform the uncertain objective function into deterministic objective function, respectively. Through the transformation model, a deterministic nesting optimization problem can be finally formulated, which can be solved by a suggested nesting optimization method based on a genetic algorithm (GA).
(2) Two hybrid optimization algorithms respectively based on multiple networks and single network are suggested to solve the transformed nesting optimization problem, and whereby two kinds of NINO methods with high efficiency are constructed. In the multiple-networks hybrid optimization algorithm, several artificial neural network models are required and each one creates the projection relation between the design variables and the bounds of the uncertain objective function or a constraint, and the GA is adopted as optimization solver. In the single-network hybrid optimization algorithm, only one artificial neural network model is required to create the projection relation between the variables (design variables and uncertain variables) and the functional values (uncertain objective function and constraints), and the GA is also used as optimization solver for both of the inner and outer layer optimization. In the optimization process, the actual model is replaced by the efficient artificial neural network model and hence the optimization efficiency of NINO can be improved greatly.
(3) Some extensions are made for the interval structural analysis method, and furthermore an efficient NINO algortithm is suggested based on the interval structural analysis method. Firstly, the interval structural analysis method is extended to compute the response bounds of structures with large uncertainty levels, based on the interval set theory and the subinterval technique. Secondly, the interval structural analysis method is introduced into the elastic wave propagation problem of composite materials, and a kind of interval numerical algorithm of elastic wave is proposed for composite laminated plates based on the hybrid numerical method. Thus the transient displacement response bounds of a composite laminated plate caused by the uncertain load and material property can be computed. Thirdly, in the NINO solving process, the interval structural analysis is adopted to compute the bounds of the uncertain objective function and constraints at each iterative step, and whereby the inner layer optimization can be eliminated successfully. Therefore, the transformed nesting optimization problem becomes a traditional single-layer optimization problem, and hence an NINO algorithm with high efficiency can be constructed.
(4) Based on the sequential linear programming technique, an efficient NINP algorithm is developed. At each iterative step, linear approximation models with respect to the design variables and uncertain variables are created for the uncertain objective function and constraints using the first-order Taylor expansion, and whereby a linear interval number optimization problem is obtained; based on the interval analysis method, bounds of the uncertain objective function and constraints at the optimal design vector of the current approximation optimization problem can be achieved very efficiently, and whereby whether the currently obtained design vector is a feasible and descending point can be judged, as only a feasible and descending design can be remained to the next iterative step; several termination criteria are provided to ensure convergence of the present algorithm.
(5) An efficient NINO algorithm is suggested based on an approximation model management strategy. The whole optimization process consists of a sequence of approximate optimization problems. At each iterative step, an approximate optimization problem can be created through the approximation model technique, and it can be solved by being changed as a deterministic optimization problem using a transformation model of NINO. The trust region method is then employed to manage the approximation models in the optimization process. At each iterative step, a reliability index is computed to judge the precision of the current approximation models, and whereby the design vector and trust region radius vector can be updated. Therefore, the design space can be ensured to keep closing to the actual optimal design vector.
(6) An efficient NINO algorithm is suggested based on a local-densifying approximation model technique. At each iterative step, the current samples of the uncertain objective function and constraints can be densified according to the solution of current approximate optimization problem. Thus the local precision of the approximation models in the two key regions within the approximation space corresponding to the response bounds can be improved. This algorithm aims at improving the precision of the local key regions instead of the whole approximation space, and hence much less samples are required comparing with the conventional approximation optimization methods based on the uniformly distributed samples. Furthermore, the algorithm can also avoid singularity of the involved approximation models caused by the overmany samples a certain extent, as well as improvement of the optimization efficiency.
为此,本文将针对非线性区间数优化展开系统的研究,力求在其数学规划理论本身及实用性算法方面做出一些卓有成效的尝试和探索。数学规划理论方面的工作是提出两种非线性区间数优化的转换模型,实现了不确定性优化问题向确定性优化问题的转换,此部分工作是整篇论文的基础;实用性算法方面的工作主要是将目前工程优化领域中的一些求解工具有机引入非线性区间数优化,一定程度上解决因两层嵌套优化造成的低效问题,从而构造出多种具一定工程实用性的高效非线性区间数优化算法。基于此思路,本论文开展和完成了如下研究工作:
(1)针对一般的不确定性优化问题,从数学规划理论层面提出了两种非线性区间数优化的数学转换模型,即区间序关系转换模型和区间可能度转换模型。给出了一种改进的区间可能度构造方法,将不确定不等式约束转换为确定性约束;给出了不确定等式约束的处理方法,最终将之转换为两个确定性约束。两种转换模型采用了上述相同的不确定约束的处理方法,但在不确定目标函数的处理上有所不同,即分别基于区间序关系和区间可能度将不确定目标函数转换为确定性目标函数。通过转换模型,得到一确定性的两层嵌套优化问题。最后,提出一种基于遗传算法的两层嵌套优化方法来求解转换后的确定性优化问题。
(2)给出多网络和单网络两种混合优化算法求解转换后的两层嵌套优化问题,从而构造出两种高效的非线性区间数优化算法。多网络混合优化算法中,通过多个人工神经网络模型建立设计向量与目标函数区间或约束区间之间的映射关系,并且采用遗传算法作为优化求解器;单网络混合优化算法中,只通过单个人工神经网络模型建立设计变量和不确定变量与相应的目标函数值和约束值之间的映射关系,并且采用遗传算法作为内、外层优化求解器。利用混合优化算法对转换后的确定性优化问题进行求解时,不再使用原耗时的真实模型,而是每次调用人工神经网络模型进行快速计算,从而大大提高了非线性区间数优化的计算效率。
(3)对区间结构分析方法进行了扩展,并基于区间结构分析方法发展出了一种高效的非线性区间数优化算法。基于区间集合理论和子区间技术,提出了大不确定性区间结构分析方法,以计算较大的变量不确定性水平下的结构响应边界;将区间结构分析方法引入复合材料弹性波动问题,并与混合数值法相结合提出了一种复合材料层合板的弹性波动区间数值算法,用于计算层合板在不确定材料属性和载荷下的瞬态位移响应边界;利用区间结构分析方法求解不确定目标函数和约束在任一设计向量下的区间,有效地消除了内层优化,原先通过转换模型获得的两层嵌套优化问题变成了一传统的单层优化问题,从而构造出一种高效的非线性区间数结构优化算法。
(4)基于序列线性规划技术,发展出了一种高效的非线性区间数优化算法。每一迭代步,通过一阶泰勒展式建立目标函数和约束关于设计变量和不确定变量的线性近似模型,从而得到一线性区间数优化问题;基于区间分析方法,高效求解不确定目标函数和约束在当前近似优化问题最优解处的边界,并以此判断当前得到的最优设计向量是否为可行下降解,只有可行下降解才得以保留至下一迭代步;另外,根据区间数优化的特点给出多个停止判断准则,保证算法的收敛性。
(5)提出了基于近似模型管理策略的非线性区间数优化算法。整个优化过程由一系列近似不确定性优化问题迭代完成:每一迭代步,通过近似模型技术建立一近似不确定性优化问题,并进一步通过非线性区间数优化的转换模型将之转变为确定性优化问题进行求解;利用信赖域方法对优化过程中的近似模型进行管理,即每一迭代步计算可靠性指标以判断近似模型精度,并对设计向量和信赖域半径矢量进行更新,使得设计空间不断向最优解靠近。
(6)提出了基于局部加密近似模型技术的非线性区间数优化算法。每一迭代步,根据当前近似不确定性优化问题的求解结果,对目标函数和约束的当前样本进行加密,使得下一迭代步中的近似模型在近似空间中对应于响应边界的两个关键区域的局部精度得到提高。该算法只追求近似模型在局部关键区域而非整体近似空间上的精度,所以相比于常规的基于均匀分布样本的近似优化方法,能大大减少所需样本的数量;另外,该算法在提高优化效率的同时还能一定程度上避免因样本过多而造成的近似模型矩阵的奇异。
Uncertainty widely exists in practical engineering problems, and studying the theories and algorithms of uncertain optimization is significant for reliability design of industrial products and systems. Stochastic programming and fuzzy programming are two types of traditional uncertain optimization methodologies, in which a great amount of sampling information on the uncertainty is required to construct the precise probability distributions or fuzzy membership functions. Unfortunately, it often seems very difficult or sometimes expensive to obtain sufficient uncertainty information, and hence these two types of methods will encounter some limitations in applicability. The interval number optimization is a relatively newly-developed uncertain optimization method, in which interval is used to model the uncertainty of a variable. Thus the variation bounds of the uncertain variables are only required, which can be obtained through only a small amount of uncertainty information. In recent years, the interval number optimization has been attracting more and more attentions, and it is expected to become the third major uncertain optimization methodology after stochastic programming and fuzzy programming. Furthermore, the interval number optimization even shows a larger application potential for practical engineering problems than the other two methods. However, it is still at its preliminary stage for the present interval number optimization research, especially for the nonlinear interval number optimization (NINO), studies for which are just getting started. Some key technical difficulties remain, such as creation of the mathematical transformation models, effective solution of the nesting optimization problem etc.
This dissertation conducts a systematical research for the NINO, and aims at contributing some useful researches and trials on mathematical programming theories and practical algorithms. Firstly, two transformation models of NINO are put forward through which the uncertain optimization problem can be transformed into a deterministic optimization problem, and this part of work is the basis of the whole dissertation. On the other hand, some computation tools in the present engineering optimization field are extended into the interval number optimization to solve the lower efficiency problem caused by the nesting optimization and whereby several efficient NINO algorithms are constructed. As a result, the following studies are carried out in this dissertation:
(1) For a general uncertain optimization problem, two mathematical transformation models of NINO are suggested at the level of mathematical programming theory, respectively based on the order relation of interval number and the possibility degree of interval number. A modified construction method is suggested for the possibility degree of interval number based on the probability method, and based on it an uncertain inequality constraint can be transformed into a deterministic constraint; an approach is also suggested to transform an uncertain equality constraint into two deterministic constraints. The two transformation models employ the above same way to deal with the uncertain constraints, while different ways for uncertain objective function, namely adopt the order relation of interval number and the possibility degree of interval number to transform the uncertain objective function into deterministic objective function, respectively. Through the transformation model, a deterministic nesting optimization problem can be finally formulated, which can be solved by a suggested nesting optimization method based on a genetic algorithm (GA).
(2) Two hybrid optimization algorithms respectively based on multiple networks and single network are suggested to solve the transformed nesting optimization problem, and whereby two kinds of NINO methods with high efficiency are constructed. In the multiple-networks hybrid optimization algorithm, several artificial neural network models are required and each one creates the projection relation between the design variables and the bounds of the uncertain objective function or a constraint, and the GA is adopted as optimization solver. In the single-network hybrid optimization algorithm, only one artificial neural network model is required to create the projection relation between the variables (design variables and uncertain variables) and the functional values (uncertain objective function and constraints), and the GA is also used as optimization solver for both of the inner and outer layer optimization. In the optimization process, the actual model is replaced by the efficient artificial neural network model and hence the optimization efficiency of NINO can be improved greatly.
(3) Some extensions are made for the interval structural analysis method, and furthermore an efficient NINO algortithm is suggested based on the interval structural analysis method. Firstly, the interval structural analysis method is extended to compute the response bounds of structures with large uncertainty levels, based on the interval set theory and the subinterval technique. Secondly, the interval structural analysis method is introduced into the elastic wave propagation problem of composite materials, and a kind of interval numerical algorithm of elastic wave is proposed for composite laminated plates based on the hybrid numerical method. Thus the transient displacement response bounds of a composite laminated plate caused by the uncertain load and material property can be computed. Thirdly, in the NINO solving process, the interval structural analysis is adopted to compute the bounds of the uncertain objective function and constraints at each iterative step, and whereby the inner layer optimization can be eliminated successfully. Therefore, the transformed nesting optimization problem becomes a traditional single-layer optimization problem, and hence an NINO algorithm with high efficiency can be constructed.
(4) Based on the sequential linear programming technique, an efficient NINP algorithm is developed. At each iterative step, linear approximation models with respect to the design variables and uncertain variables are created for the uncertain objective function and constraints using the first-order Taylor expansion, and whereby a linear interval number optimization problem is obtained; based on the interval analysis method, bounds of the uncertain objective function and constraints at the optimal design vector of the current approximation optimization problem can be achieved very efficiently, and whereby whether the currently obtained design vector is a feasible and descending point can be judged, as only a feasible and descending design can be remained to the next iterative step; several termination criteria are provided to ensure convergence of the present algorithm.
(5) An efficient NINO algorithm is suggested based on an approximation model management strategy. The whole optimization process consists of a sequence of approximate optimization problems. At each iterative step, an approximate optimization problem can be created through the approximation model technique, and it can be solved by being changed as a deterministic optimization problem using a transformation model of NINO. The trust region method is then employed to manage the approximation models in the optimization process. At each iterative step, a reliability index is computed to judge the precision of the current approximation models, and whereby the design vector and trust region radius vector can be updated. Therefore, the design space can be ensured to keep closing to the actual optimal design vector.
(6) An efficient NINO algorithm is suggested based on a local-densifying approximation model technique. At each iterative step, the current samples of the uncertain objective function and constraints can be densified according to the solution of current approximate optimization problem. Thus the local precision of the approximation models in the two key regions within the approximation space corresponding to the response bounds can be improved. This algorithm aims at improving the precision of the local key regions instead of the whole approximation space, and hence much less samples are required comparing with the conventional approximation optimization methods based on the uniformly distributed samples. Furthermore, the algorithm can also avoid singularity of the involved approximation models caused by the overmany samples a certain extent, as well as improvement of the optimization efficiency.
引文
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