【摘要】：非凸規(guī)劃問(wèn)題作為一類重要的優(yōu)化問(wèn)題,能廣泛應(yīng)用于經(jīng)濟(jì)金融、信息技術(shù)、工業(yè)制造等多個(gè)重要領(lǐng)域.通常情況下,該類問(wèn)題往往存在多個(gè)非全局最優(yōu)的局部最優(yōu)解,因此尋找其全局最優(yōu)解極其困難.由于非凸優(yōu)化問(wèn)題在現(xiàn)實(shí)生活中的廣泛應(yīng)用,近年來(lái)引起越來(lái)越多研究者的關(guān)注,一些優(yōu)化方法相繼被提出,本文針對(duì)非凸優(yōu)化問(wèn)題中的兩類廣義多乘積規(guī)劃問(wèn)題分別提出了相應(yīng)的優(yōu)化算法.和已有的方法相比,本文提出的分支定界算法和迭代算法在保證最優(yōu)解的質(zhì)量的同時(shí)還很大程度上提高了其執(zhí)行效率.主要內(nèi)容如下:第一章,首先給出本文所研究的優(yōu)化問(wèn)題模型.其次對(duì)該優(yōu)化問(wèn)題的應(yīng)用背景、理論意義及當(dāng)前的研究工作做簡(jiǎn)要介紹,最后呈現(xiàn)本文的主要工作.第二章,根據(jù)廣義線性多乘積優(yōu)化問(wèn)題的特點(diǎn),提出了一個(gè)新的分支定界算法.首先通過(guò)引入變量獲得原問(wèn)題的等價(jià)問(wèn)題,接著采用凸松弛技巧將等價(jià)問(wèn)題轉(zhuǎn)化為凸規(guī)劃問(wèn)題,然后基于一個(gè)新的分支規(guī)則來(lái)求解一系列的凸規(guī)劃問(wèn)題從而獲得原問(wèn)題的全局最優(yōu)解,最后從理論上證明該算法的全局收斂性.數(shù)值實(shí)驗(yàn)結(jié)果說(shuō)明本章算法對(duì)于求解廣義線性多乘積規(guī)劃問(wèn)題具有一定的優(yōu)勢(shì).第三章.針對(duì)廣義多項(xiàng)式乘積優(yōu)化問(wèn)題,給出了一個(gè)迭代算法.首先引入變量得到與原問(wèn)題等價(jià)的廣義幾何規(guī)劃問(wèn)題,其次運(yùn)用算術(shù)-幾何平均不等式及罰函數(shù)思想將廣義幾何規(guī)劃問(wèn)題轉(zhuǎn)化成標(biāo)準(zhǔn)幾何規(guī)劃形式,然后通過(guò)求解一系列的標(biāo)準(zhǔn)幾何規(guī)劃問(wèn)題得到原問(wèn)題的最優(yōu)解,最后給出迭代算法的收斂性.數(shù)值實(shí)驗(yàn)結(jié)果表明該算法是有效可行的.
[Abstract]:As an important optimization problem, non-convex programming problem can be widely used in many important fields, such as economy and finance, information technology, industrial manufacturing and so on. In general, there are many non-global optimal local optimal solutions for this kind of problems, so it is very difficult to find their global optimal solutions. Because of the wide application of non-convex optimization problem in real life, more and more researchers pay attention to it in recent years, and some optimization methods have been proposed one after another. In this paper, two kinds of generalized multi-product programming problems in non-convex optimization problems are proposed. Compared with the existing methods, the proposed branch-and-bound algorithm and iterative algorithm not only guarantee the quality of the optimal solution, but also greatly improve its execution efficiency. The main contents are as follows: in the first chapter, the optimization model studied in this paper is given. Secondly, the application background, theoretical significance and current research work of the optimization problem are briefly introduced. Finally, the main work of this paper is presented. In chapter 2, according to the characteristics of generalized linear multiple product optimization problem, a new branch and bound algorithm is proposed. First, the equivalent problem of the original problem is obtained by introducing variables, and then the equivalent problem is transformed into a convex programming problem by using convex relaxation technique. Then a series of convex programming problems are solved based on a new branching rule to obtain the global optimal solution of the original problem. Finally, the global convergence of the algorithm is proved theoretically. Numerical results show that this algorithm has some advantages in solving generalized linear multiple product programming problems. Chapter 3 An iterative algorithm is proposed for generalized polynomial product optimization. The generalized geometric programming problem equivalent to the original problem is obtained by introducing variables, and then the generalized geometric programming problem is transformed into a standard geometric programming form by using the arithmetic-geometric mean inequality and penalty function. Then the optimal solution of the original problem is obtained by solving a series of standard geometric programming problems. Finally, the convergence of the iterative algorithm is given. Numerical results show that the algorithm is effective and feasible.
【學(xué)位授予單位】：河南師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O221

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