区间不确定性优化的若干高效算法研究及应用
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摘要
不确定性广泛存在于实际工程优化设计问题中,基于不确定分析理论和算法的研究对于复杂系统的研发设计具有重要意义。概率模型和模糊集模型常被用来描述这些不确定性。概率模型可以很好地描述系统中的随机不确定性,而模糊模型则能有效地描述对系统的认知不确定性或者决策不确定性,但是由于不确定信息不全或者计算成本过高等原因,导致以上两种模型较少应用于实际问题中。区间模型通过上下边界来描述不确定性,并且对不确定分布不做任何假设,因此对于实际问题具有很强的适用性。对区间不确定优化方法的研究近年来开始逐渐受到国内外的重视,并且在工程优化设计领域展现出了较高的应用潜力。然而目前对基于区间模型的不确定优化问题的研究还存在许多有待改进的地方,例如在优化过程中,不确定性度量的数学模型转换导致了两层嵌套优化问题等一系列技术难点,特别是对于计算效率以及结果正确性的高要求,给区间不确定优化理论和方法的发展提出了挑战。
为此,本文将针对区间不确定优化问题的计算效率以及结果有效性等方面展开系统的研究,力求在其理论及实用性算法等方面做出一些卓有成效的尝试和探索。借助于成熟的代理模型技术,根据区间不确定优化本身的特点,结合有效的代理模型管理技术,在一定程度上解决计算效率和结果有效性的矛盾。基于此思路,本论文开展和完成了如下研究工作：
(1)为了有效求解区间不确定优化问题,将其转换为两层嵌套的确定性优化问题,并通过引入代理模型代替真实模型来提高计算效率。为了进一步解决代理模型的近似误差可能导致的优化结果不可靠性,进而提出了局部加密的模型管理方法,保证了不确定优化求解结果的可靠性,达到了同时提高计算效率和保证计算精度的目标。最后将区间不确定优化算法应用于侵彻混凝土复合介质的优化设计问题中,结果表明本文的不确定优化设计方法对侵彻系统中着速和入射角的控制、弹体结构设计、装药设计、引信设计等方面具有重要的参考价值。
(2)为了保证区间不确定优化算法中内层优化的“极值”要求,提出了一种改进的基于代理模型的全局搜索策略,用于解决针对不确定参数的变化范围较大,并且系统非线性程度较强的复杂系统优化设计问题。根据代理模型的特性,以其作为引导,构造出改进的全局搜索策略和有效的算法停止准则,在提高计算效率的同时,保证了区间不确定优化算法的有效性,以及结果的正确性。
(3)为了对两层嵌套优化问题进行解耦求解,采用了不同的鲁棒性和可靠性度量方式。目标函数的鲁棒性通过函数敏感性指标来描述,并提出基于许可度的最小容差圆来度量约束的可靠性。在此基础上通过采用约束偏移解耦方法将两层嵌套优化问题转换为序列优化问题,从而有效地提高了不确定优化方法的计算效率。进一步针对复杂系统的区间不确定可靠性优化问题,通过基于代理模型和估计误差函数的失效面搜索策略,使得近似的失效面逐步逼近精确的失效面,在保证计算效率的同时,进一步提高了区间不确定可靠性优化问题的求解精度。
(4)提出了通过结合高维代理模型技术,有效解决了一般代理模型对于高维问题的描述能力较差等问题。对区间不确定优化问题采用直接解耦方法,统一地处理区间鲁棒性优化问题和区间可靠性优化问题,将两层嵌套优化问题转换为序列优化问题,改进了基于约束偏移解耦的不确定优化算法只有利于解决可靠性优化问题,而不利于求解鲁棒性优化问题的特点。针对一阶高维代理模型非线性描述能力不足而二阶高维代理模型计算量偏大的问题,提出了改进的高维代理模型,找到了高维代理模型在近似能力和计算效率之间的平衡。
Uncertainty widely exists in practical engineering problems. Studies on theories and algorithms of optimization under uncertainty are significant for development of industrial complex products and systems. Uncertainty is commonly quantified by probabilistic model and fuzzy set model. Probabilistic model can easily address the aleatory type of uncertainty, and the fuzzy set model is quite suitable for representing the epistemic type of uncertainty or decision-making uncertainty. These two classes of models, however, are rarely applied in practical engineering problems, due to some limitations and deficiencies, such as low computational efficiency and the inability of handling the uncertainty with incomplete information. Interval models are applicable to problems with incomplete information. It considers only the lower and upper bounds of the uncertain parameters without assuming their precise probability distribution functions. In recent years, the interval-based optimization method has been attracting more and more attentions, and shown a high application potential for practical engineering problems. However, some difficulties in interval-based optimization methods should be undertaken. For instance, uncertainty quantification in interval-based optimization problem results in a two-level deterministic optimization process, thus high requirment for computational efficiency, as well as the validity of optimal result pose a great challenge to the development of interval-based optimization algorithms.
For this reason, this dissertation is dedicated to a systematical research on the computational efficiency and validity of optimal result in interval-based optimization problems, aiming at making some useful contribution and progress to the improvement. According to the features of interval-based optimization problem, this paper tries to deal with the trade-off between computational efficiency and validity of optimal result by using surrogate model technique under the effective model management framework. Based upon this concept, the following studies are carried out in this dissertation:
(1) The general interval-based optimization problem under uncertainty is transformed into a two-level optimization process, such that it can be solved effectively and easily. The surrogate model technique is used to promote the computational efficiency by replacing the time-comsuming simulation model in the two-level optimization process. In order to diminish the untrustworthiness of the optimal result introduced by the approximation error of surrogate model, a local-densifying model management strategy is suggested, based upon which, high computational efficiency and low approximation error can be achieved simultaneously. The interval-based optimization method using surrogate model technique is then applied to an optimization design problem of penetration of multi-layer concrete target. Results indicate that the present method is very encouraging in design of penetration.
(2) An improved global search strategy is proposed to ensure the "extremum" of uncertainty analysis in the lower level optimization. The interval-based optimization method combining with this global search strategy can be used to solve time-consuming problems with highly nonlinearity and large uncertain level. The improved global search strategy is established by using the surrogate model as the sampling guide according to the surrogate model's features. This strategy also gives an effective stopping criterion for the iterations. As a result, an efficienct and trustworthy interval-based optimization method is formed.
(3) Different robust measurement and reliability measurement are suggested to transform the interval-based optimization problem, so that the two-level optimization can be decoupled accordingly. An uncertain sphere with allowable degree and minimum tolerance sphere are used to quantify the reliability of constraints, and the sensitivity information is used to measure the robustness of objective function. Based on the above uncertainty measurement, constraint shift method can decouple the two-level optimization process into a sequential optimization process, so that the computational efficiency would increase significantly.
(4) High dimensional model representation technique supplements the deficiency of ordinary surrogate model for modeling the high dimensional problems. Combined with a direct decoupling method, a new interval-based method is developed to tackle the high dimensional optimization problems under uncertainty. The direct decoupling method can cope with the robustness of objective and reliability of constraints simultaneously by transformed the two-level optimization process into a sequential optimization process. It improves the constraint shift method from the last chapter which is merely suitable for handling the problems of constraint reliability. Last but not least, an improved high dimensional model representation method is suggested to balance the trade-off between approximation capacity and computational efficiency, for the reason that first order high dimensional model representation does not have enough nonlinear modeling ability, while second order high dimensional model representation needs too much amount of computation.
为此,本文将针对区间不确定优化问题的计算效率以及结果有效性等方面展开系统的研究,力求在其理论及实用性算法等方面做出一些卓有成效的尝试和探索。借助于成熟的代理模型技术,根据区间不确定优化本身的特点,结合有效的代理模型管理技术,在一定程度上解决计算效率和结果有效性的矛盾。基于此思路,本论文开展和完成了如下研究工作：
(1)为了有效求解区间不确定优化问题,将其转换为两层嵌套的确定性优化问题,并通过引入代理模型代替真实模型来提高计算效率。为了进一步解决代理模型的近似误差可能导致的优化结果不可靠性,进而提出了局部加密的模型管理方法,保证了不确定优化求解结果的可靠性,达到了同时提高计算效率和保证计算精度的目标。最后将区间不确定优化算法应用于侵彻混凝土复合介质的优化设计问题中,结果表明本文的不确定优化设计方法对侵彻系统中着速和入射角的控制、弹体结构设计、装药设计、引信设计等方面具有重要的参考价值。
(2)为了保证区间不确定优化算法中内层优化的“极值”要求,提出了一种改进的基于代理模型的全局搜索策略,用于解决针对不确定参数的变化范围较大,并且系统非线性程度较强的复杂系统优化设计问题。根据代理模型的特性,以其作为引导,构造出改进的全局搜索策略和有效的算法停止准则,在提高计算效率的同时,保证了区间不确定优化算法的有效性,以及结果的正确性。
(3)为了对两层嵌套优化问题进行解耦求解,采用了不同的鲁棒性和可靠性度量方式。目标函数的鲁棒性通过函数敏感性指标来描述,并提出基于许可度的最小容差圆来度量约束的可靠性。在此基础上通过采用约束偏移解耦方法将两层嵌套优化问题转换为序列优化问题,从而有效地提高了不确定优化方法的计算效率。进一步针对复杂系统的区间不确定可靠性优化问题,通过基于代理模型和估计误差函数的失效面搜索策略,使得近似的失效面逐步逼近精确的失效面,在保证计算效率的同时,进一步提高了区间不确定可靠性优化问题的求解精度。
(4)提出了通过结合高维代理模型技术,有效解决了一般代理模型对于高维问题的描述能力较差等问题。对区间不确定优化问题采用直接解耦方法,统一地处理区间鲁棒性优化问题和区间可靠性优化问题,将两层嵌套优化问题转换为序列优化问题,改进了基于约束偏移解耦的不确定优化算法只有利于解决可靠性优化问题,而不利于求解鲁棒性优化问题的特点。针对一阶高维代理模型非线性描述能力不足而二阶高维代理模型计算量偏大的问题,提出了改进的高维代理模型,找到了高维代理模型在近似能力和计算效率之间的平衡。
Uncertainty widely exists in practical engineering problems. Studies on theories and algorithms of optimization under uncertainty are significant for development of industrial complex products and systems. Uncertainty is commonly quantified by probabilistic model and fuzzy set model. Probabilistic model can easily address the aleatory type of uncertainty, and the fuzzy set model is quite suitable for representing the epistemic type of uncertainty or decision-making uncertainty. These two classes of models, however, are rarely applied in practical engineering problems, due to some limitations and deficiencies, such as low computational efficiency and the inability of handling the uncertainty with incomplete information. Interval models are applicable to problems with incomplete information. It considers only the lower and upper bounds of the uncertain parameters without assuming their precise probability distribution functions. In recent years, the interval-based optimization method has been attracting more and more attentions, and shown a high application potential for practical engineering problems. However, some difficulties in interval-based optimization methods should be undertaken. For instance, uncertainty quantification in interval-based optimization problem results in a two-level deterministic optimization process, thus high requirment for computational efficiency, as well as the validity of optimal result pose a great challenge to the development of interval-based optimization algorithms.
For this reason, this dissertation is dedicated to a systematical research on the computational efficiency and validity of optimal result in interval-based optimization problems, aiming at making some useful contribution and progress to the improvement. According to the features of interval-based optimization problem, this paper tries to deal with the trade-off between computational efficiency and validity of optimal result by using surrogate model technique under the effective model management framework. Based upon this concept, the following studies are carried out in this dissertation:
(1) The general interval-based optimization problem under uncertainty is transformed into a two-level optimization process, such that it can be solved effectively and easily. The surrogate model technique is used to promote the computational efficiency by replacing the time-comsuming simulation model in the two-level optimization process. In order to diminish the untrustworthiness of the optimal result introduced by the approximation error of surrogate model, a local-densifying model management strategy is suggested, based upon which, high computational efficiency and low approximation error can be achieved simultaneously. The interval-based optimization method using surrogate model technique is then applied to an optimization design problem of penetration of multi-layer concrete target. Results indicate that the present method is very encouraging in design of penetration.
(2) An improved global search strategy is proposed to ensure the "extremum" of uncertainty analysis in the lower level optimization. The interval-based optimization method combining with this global search strategy can be used to solve time-consuming problems with highly nonlinearity and large uncertain level. The improved global search strategy is established by using the surrogate model as the sampling guide according to the surrogate model's features. This strategy also gives an effective stopping criterion for the iterations. As a result, an efficienct and trustworthy interval-based optimization method is formed.
(3) Different robust measurement and reliability measurement are suggested to transform the interval-based optimization problem, so that the two-level optimization can be decoupled accordingly. An uncertain sphere with allowable degree and minimum tolerance sphere are used to quantify the reliability of constraints, and the sensitivity information is used to measure the robustness of objective function. Based on the above uncertainty measurement, constraint shift method can decouple the two-level optimization process into a sequential optimization process, so that the computational efficiency would increase significantly.
(4) High dimensional model representation technique supplements the deficiency of ordinary surrogate model for modeling the high dimensional problems. Combined with a direct decoupling method, a new interval-based method is developed to tackle the high dimensional optimization problems under uncertainty. The direct decoupling method can cope with the robustness of objective and reliability of constraints simultaneously by transformed the two-level optimization process into a sequential optimization process. It improves the constraint shift method from the last chapter which is merely suitable for handling the problems of constraint reliability. Last but not least, an improved high dimensional model representation method is suggested to balance the trade-off between approximation capacity and computational efficiency, for the reason that first order high dimensional model representation does not have enough nonlinear modeling ability, while second order high dimensional model representation needs too much amount of computation.
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