本文選題：非光滑優(yōu)化　切入點：束方法　出處：《大連理工大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：半無限規(guī)劃(Semi-Infinite Programming,簡寫為SIP)不僅在經(jīng)濟(jì)均衡、最優(yōu)控制、信息技術(shù)、工程設(shè)計等領(lǐng)域有著廣泛的應(yīng)用,而且對Chebyshev逼近理論、魯棒優(yōu)化、模糊集等理論方面的研究也起著重要作用.因此,半無限規(guī)劃的數(shù)值算法有很強(qiáng)的研究價值.束方法被公認(rèn)為求解非光滑優(yōu)化的快速的、穩(wěn)定的算法之一.針對不同問題的特性,束方法已經(jīng)發(fā)展出各類變式,并被廣泛的應(yīng)用于雙層規(guī)劃問題、機(jī)會約束問題、最小最大問題、均衡問題等經(jīng)典優(yōu)化問題,而且在經(jīng)濟(jì)、機(jī)械設(shè)計、最優(yōu)控制等實際問題中也有重要應(yīng)用.本文主要研究半無限規(guī)劃的非光滑數(shù)值算法,包括非光滑凸半無限規(guī)劃的增量束方法、應(yīng)用于非光滑非凸半無限規(guī)劃的非可行迫近束方法、非凸最大特征值優(yōu)化的束方法.本文的主要內(nèi)容可以概括如下：1.論文的第三章提出了一個求解非光滑凸半無限規(guī)劃的非精確增量束方法.該算法主要基于改進(jìn)函數(shù)(improvement function)、增量思想(incremental idea)和非精確數(shù)據(jù)(inexact oracle)技術(shù).眾所周知,SIP問題的主要難點在于具有無限多個約束.本算法使用改進(jìn)函數(shù),將半無限約束問題轉(zhuǎn)化成一個非光滑無約束優(yōu)化問題.通過使用增量技術(shù),在構(gòu)造割平面時僅使用其中一個約束的函數(shù)值和次梯度,而不是全部約束的信息.進(jìn)而,在很大程度上減小了數(shù)據(jù)存儲量和計算量,加快了計算速度.一個新的穩(wěn)定中心產(chǎn)生后,該算法需要估算出滿足一定精度的約束函數(shù)的最大函數(shù)值.在EMFCQ條件下,分析了該算法的收斂性.最后,通過大量的數(shù)值試驗,驗證了算法的效率和穩(wěn)定性.2.論文的第四章提出一個解非凸非光滑約束優(yōu)化的非可行束方法,并將該算法應(yīng)用到SIP問題.通過定義一個最大值函數(shù),可以將SIP問題轉(zhuǎn)化為一個非光滑非凸優(yōu)化問題.該問題的目標(biāo)函數(shù)和約束函數(shù)是一類特殊的非凸函數(shù),稱之為lower-C2函數(shù).基于lower-C2函數(shù)的特殊性質(zhì),使用再分配技術(shù)將迫近參數(shù)分成凸化參數(shù)和迫近參數(shù)兩部分.通過使用改進(jìn)函數(shù),將約束問題轉(zhuǎn)化為一個無約束問題.為了得到迭代點,使用凸化的目標(biāo)函數(shù)和約束函數(shù)信息來構(gòu)造割平面模型.再分配后的迫近參數(shù)和凸化參數(shù)都是自動更新,且最終都會穩(wěn)定不變的.在MFCQ條件下,本算法達(dá)到了全局收斂性.在EMFCQ條件下,SIP問題的穩(wěn)定點和非光滑問題的穩(wěn)定點之間是等價的.數(shù)值試驗結(jié)果表明：該算法即能快速地求解某些非光滑優(yōu)化問題,又能有效的應(yīng)用于半無限規(guī)劃.3.論文的第五章研究一類特殊的半無限優(yōu)化問題,即非凸最大特征值優(yōu)化問題,提出一個求解該類問題的回溯迫近束方法.最大特征值優(yōu)化可轉(zhuǎn)化為一個無約束半無限規(guī)劃,即是一類特殊的無約束半無限規(guī)劃.基于最大特征值函數(shù)的特殊復(fù)合結(jié)構(gòu),定義了目標(biāo)函數(shù)的一個近似表達(dá),稱之為概念模型(conceptual model)該模型由內(nèi)函數(shù)的線性化近似和外函數(shù)構(gòu)成,進(jìn)而簡化割平面模型(cutting-plane model),減少計算過程中的數(shù)據(jù)存儲量.通過使用一個特殊的回溯步(backtracking test),有效地控制概念模型和目標(biāo)函數(shù)的近似程度,從而隨著迭代過程優(yōu)化算法結(jié)構(gòu).本章給出了該算法的收斂性分析.數(shù)值試驗結(jié)果表明：本章提出的算法既能快速地求解各類最大特征值優(yōu)化問題,又能有效地應(yīng)用于反饋控制問題.
[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.

【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O221

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