本文選題：非線性優(yōu)化 + 等式約束優(yōu)化　； 參考：《曲阜師范大學(xué)》2017年碩士論文

【摘要】：在現(xiàn)實(shí)生活中會(huì)遇到在眾多方案中選擇一類方案使得資源使用效益最大或者目標(biāo)成本最低的問題,這樣的一類問題稱為最優(yōu)化問題.最優(yōu)化問題根據(jù)有無約束條件劃分為約束優(yōu)化問題和無約束優(yōu)化問題.在理論推理和算法設(shè)計(jì)方面,約束優(yōu)化問題和無約束優(yōu)化問題有很大的不同,但此兩類問題在某種情況下是可以相互轉(zhuǎn)化的.一般情況下,無約束優(yōu)化問題比約束優(yōu)化問題的求解相對(duì)容易.本文選擇非線性規(guī)劃中的罰函數(shù)方法將約束優(yōu)化問題轉(zhuǎn)化為無約束優(yōu)化問題,通過求解無約束的罰問題來求解帶有等式或不等式的約束優(yōu)化問題.對(duì)于傳統(tǒng)的罰函數(shù),若是簡(jiǎn)單光滑的,則一定不精確;若是簡(jiǎn)單精確的,則不光滑.因此本文的主要工作是改造傳統(tǒng)罰函數(shù),使簡(jiǎn)單罰函數(shù)既是精確的,又是光滑的.本文結(jié)構(gòu)安排如下:第一章主要介紹約束優(yōu)化問題和罰優(yōu)化問題的基本概念、基礎(chǔ)知識(shí)以及本文的主要工作.第二章針對(duì)等式約束優(yōu)化問題,通過對(duì)約束函數(shù)增加變量,提出一類簡(jiǎn)單罰函數(shù)并結(jié)合K-K-T條件和Lagrange函數(shù)證明這一類簡(jiǎn)單罰函數(shù)在有界閉集上同時(shí)具有光滑性和精確性.本章提出一種新的算法解決此類等式約束優(yōu)化問題并給出數(shù)值例子說明算法的可行性.第三章針對(duì)等式約束優(yōu)化問題,提出一類新的簡(jiǎn)單罰函數(shù)并證明它是光滑精確的.最后給出數(shù)值例子說明本章所給算法的可行性.第四章針對(duì)不等式約束優(yōu)化問題,引入目標(biāo)罰因子和約束罰因子,提出一類新的簡(jiǎn)單精確罰函數(shù).此罰函數(shù)同時(shí)懲罰目標(biāo)函數(shù)和約束函數(shù),使得約束函數(shù)的違反度減小的同時(shí)目標(biāo)函數(shù)趨近于最優(yōu)值.基于此類新的罰函數(shù)分別給出全局最優(yōu)求解算法和局部最優(yōu)求解算法,并且分別證明了算法的收斂性.最后給出數(shù)值算例,說明所給算法是可行的.
[Abstract]:In real life, there will be a problem of selecting a class of schemes in many schemes to make the use of the resources maximum or the lowest cost of the target. Such a problem is called the optimization problem. The optimization problem is divided into constrained optimization and unconstrained optimization problems based on unconstrained conditions. In theoretical reasoning and algorithm design, There are great differences between the constrained optimization problem and the unconstrained optimization problem, but the two kinds of problems can be converted to each other in some circumstances. In general, the unconstrained optimization problem is relatively easier than the constrained optimization problem. In this paper, the penalty function method in nonlinear programming is chosen to transform the constrained optimization problem into unconstrained optimization question. The problem of solving the constrained optimization problem with equality or inequality is solved by solving unconstrained penalty problems. For the traditional penalty function, if it is simple and smooth, it is not accurate; if it is simple and accurate, it is not smooth. Therefore, the main work of this paper is to transform the traditional penalty function and make the simple penalty function both accurate and smooth. The structure is arranged as follows: the first chapter mainly introduces the basic concepts, basic knowledge and the main work of the penalty optimization problem and penalty optimization problem. In the second chapter, a simple penalty function is proposed by adding variables to the constraint function, and the simple penalty is proved by combining the K-K-T condition and the Lagrange function. In this chapter, a new algorithm is proposed to solve this kind of equality constrained optimization problem and a numerical example is given to illustrate the feasibility of the algorithm. In the third chapter, a new simple penalty function is proposed for equality constrained optimization problem and it is proved to be smooth and accurate. Finally, a numerical example is given. In this chapter, the feasibility of the algorithm is explained. In the fourth chapter, a new simple exact penalty function is proposed by introducing the target penalty factor and constraint penalty factor for the inequality constrained optimization problem. This penalty function punishes both the target function and the constraint function, which makes the target function close to the optimal value at the same time, and the objective function is close to the optimal value. The new penalty function gives the global optimal solution algorithm and the local optimal solution algorithm respectively, and proves the convergence of the algorithm respectively. Finally, a numerical example is given to illustrate the feasibility of the proposed algorithm.

【學(xué)位授予單位】：曲阜師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O224

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本文編號(hào)：1803942