半无限规划问题的非光滑束方法的研究

发布时间：2018-03-16 08:40

本文选题：非光滑优化　切入点：束方法　出处：《大连理工大学》2016年博士论文　论文类型：学位论文

【摘要】：半无限规划(Semi-Infinite Programming,简写为SIP)不仅在经济均衡、最优控制、信息技术、工程设计等领域有着广泛的应用,而且对Chebyshev逼近理论、鲁棒优化、模糊集等理论方面的研究也起着重要作用.因此,半无限规划的数值算法有很强的研究价值.束方法被公认为求解非光滑优化的快速的、稳定的算法之一.针对不同问题的特性,束方法已经发展出各类变式,并被广泛的应用于双层规划问题、机会约束问题、最小最大问题、均衡问题等经典优化问题,而且在经济、机械设计、最优控制等实际问题中也有重要应用.本文主要研究半无限规划的非光滑数值算法,包括非光滑凸半无限规划的增量束方法、应用于非光滑非凸半无限规划的非可行迫近束方法、非凸最大特征值优化的束方法.本文的主要内容可以概括如下：1.论文的第三章提出了一个求解非光滑凸半无限规划的非精确增量束方法.该算法主要基于改进函数(improvement function)、增量思想(incremental idea)和非精确数据(inexact oracle)技术.众所周知,SIP问题的主要难点在于具有无限多个约束.本算法使用改进函数,将半无限约束问题转化成一个非光滑无约束优化问题.通过使用增量技术,在构造割平面时仅使用其中一个约束的函数值和次梯度,而不是全部约束的信息.进而,在很大程度上减小了数据存储量和计算量,加快了计算速度.一个新的稳定中心产生后,该算法需要估算出满足一定精度的约束函数的最大函数值.在EMFCQ条件下,分析了该算法的收敛性.最后,通过大量的数值试验,验证了算法的效率和稳定性.2.论文的第四章提出一个解非凸非光滑约束优化的非可行束方法,并将该算法应用到SIP问题.通过定义一个最大值函数,可以将SIP问题转化为一个非光滑非凸优化问题.该问题的目标函数和约束函数是一类特殊的非凸函数,称之为lower-C2函数.基于lower-C2函数的特殊性质,使用再分配技术将迫近参数分成凸化参数和迫近参数两部分.通过使用改进函数,将约束问题转化为一个无约束问题.为了得到迭代点,使用凸化的目标函数和约束函数信息来构造割平面模型.再分配后的迫近参数和凸化参数都是自动更新,且最终都会稳定不变的.在MFCQ条件下,本算法达到了全局收敛性.在EMFCQ条件下,SIP问题的稳定点和非光滑问题的稳定点之间是等价的.数值试验结果表明：该算法即能快速地求解某些非光滑优化问题,又能有效的应用于半无限规划.3.论文的第五章研究一类特殊的半无限优化问题,即非凸最大特征值优化问题,提出一个求解该类问题的回溯迫近束方法.最大特征值优化可转化为一个无约束半无限规划,即是一类特殊的无约束半无限规划.基于最大特征值函数的特殊复合结构,定义了目标函数的一个近似表达,称之为概念模型(conceptual model)该模型由内函数的线性化近似和外函数构成,进而简化割平面模型(cutting-plane model),减少计算过程中的数据存储量.通过使用一个特殊的回溯步(backtracking test),有效地控制概念模型和目标函数的近似程度,从而随着迭代过程优化算法结构.本章给出了该算法的收敛性分析.数值试验结果表明：本章提出的算法既能快速地求解各类最大特征值优化问题,又能有效地应用于反馈控制问题.
[Abstract]:Semi infinite programming (Semi-Infinite Programming, abbreviated as SIP) not only in economic equilibrium, optimal control, information technology, engineering design has been widely used, and the Chebyshev approximation theory, robust optimization, the research of fuzzy set theory also plays an important role. Therefore, the numerical algorithm of semi infinite programming research strong value. Beam method is recognized as a solution of non smooth optimization fast, one of the stability of the algorithm. According to the different characteristics of the beam method has developed all kinds of variable type, and is widely used in the double planning problem, chance constrained problem, minimax problem, classical optimization equilibrium problems. But in the economic, mechanical design, optimal control and other practical problems also have important applications. This paper mainly studies the nonsmooth semi infinite programming numerical algorithms, including non incremental beam smooth convex semi infinite programming Method, applied to the non smooth non convex semi infinite programming of non viable proximal bundle method, optimization method for non convex beam value maximum characteristic. The main contents of this paper can be summarized as follows: 1. the third chapter puts forward a solution of non accurate incremental beam method smooth convex semi infinite programming. The algorithm is mainly based on improved the function (improvement function), (incremental idea) and the incremental idea of imprecise data (inexact Oracle) technology. As everyone knows, the main problem lies in the difficulty of SIP has infinitely many constraints. Using the improved function of the algorithm, the semi infinite constrained problem is transformed into a non smooth unconstrained optimization problem. By using the incremental technique, only the use of a constraint function value and gradient in the structure of the cutting plane, and not all the constraint information. Then, greatly reduce the amount of data storage and computation, to speed up the calculation Speed. A new stable center, the algorithm needs to estimate the maximum function satisfy the constraint function precision value. Under the condition of EMFCQ, analysis of the convergence of the algorithm. Finally, through a large number of numerical experiments, to verify the efficiency of the algorithm and the stability of the.2. in the fourth chapter, put forward a solution non convex non smooth constrained optimization feasible beam method, this algorithm is applied to the SIP problem. By defining a maximum function, the SIP problem is transformed into a non smooth non convex optimization problem. The objective function and constraint function of the problem is a special class of non convex function, called the special nature of the lower-C2 function. Based on the lower-C2 function, use the redistribution of technology will be divided into convex parametric parameters approaching and the two part parameters. By using the improved function approximation, the constrained problem into an unconstrained problem. In order to get the iteration Point, objective function and constraint function to construct the information using convex cutting plane model. After the redistribution of approximation of the parameters and convexfication parameters are automatically updated, and will eventually be stable. Under the condition of MFCQ, the algorithm has global convergence. Under the condition of EMFCQ, between the stable point of the SIP problem the problem of stability and non smooth point are equivalent. The numerical results show that the algorithm can quickly solve some non smooth optimization problem, and can be effectively applied to semi infinite programming.3. the fifth chapter of the thesis is to investigate a special class of semi infinite optimization problems, namely the non convex optimization problem a maximum characteristic value. To solve this kind of problem backtracking proximal bundle method. Optimization can be transformed into an unconstrained semi infinite programming Max features, which is a special kind of unconstrained semi infinite programming. The special composite structure function based on the largest eigenvalue, Define a function approximation expression, called conceptual model (conceptual model) function of the linear approximation and the model is composed of inner function, thus simplifying the cutting plane model (cutting-plane model), reduce the amount of data storage in the calculation process. By using a special backtracking step (backtracking test) the degree of approximation, control concept model and the objective function effectively, and with the algorithm of structural optimization iterative process. This chapter gives the analysis of the convergence of the algorithm. The numerical results show that the proposed algorithm can quickly solve the optimization problem of the maximum value of features, and can be effectively applied to the feedback control problem.

【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O221

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