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Extending and Simplifying Existing Piecewise-Linear Homotopy Methods for Solving Nonlinear Systems of Equations

機(jī)譯：求解方程組非線性系統(tǒng)的現(xiàn)有分段線性同倫方法的擴(kuò)展和簡(jiǎn)化

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摘要

This dissertation research extends and simplfiies existing piecewise-linear homotopy (PL) methods to solve G(x) = 0, with G : Rn → Rm. Existing PL methods are designed to solve F(x) = 0, with F : Rn → Rn and some related point-to-set mappings. PL methods are a component of what is also known as numerical continuation methods, and they are known for being globally convergent methods. First, we present a new PL method for computing zeros of functions of the form f : R n → R by mimicking classical PL methods for computing zeros of functions of the form f : R → R. Our PL method avoids traversing subdivisions of R n x [0, 1] and instead uses an object that we refer to as triangulation-graph, which is essentially a triangulation of R x [0, 1] with hypercubes of Rn as its vertices. The hypercubes are generated randomly, and a sojourn time of an associated discrete-time Markov chain is used to show that not too many cubes are generated. Thereafter, our PL method is applied to solving G(x) = 0 for G : Rn → Rm under inequality constraints. The resultant method for solving G(x) = 0 translates into a new type of iterative method for solving systems of linear equations. Some computational illustrations are reported. A possible application to optimization problems is also indicated as a direction for further work.
機(jī)譯：本論文的研究擴(kuò)展和簡(jiǎn)化了現(xiàn)有的分段線性同倫方法（PL），用于求解G（x）= 0，其中G：Rn→Rm。現(xiàn)有的PL方法被設(shè)計(jì)為使用F：Rn→Rn和一些相關(guān)的點(diǎn)到集映射來(lái)解決F（x）= 0。 PL方法是所謂的數(shù)值連續(xù)方法的組成部分，以全局收斂方法而聞名。首先，我們通過(guò)模仿經(jīng)典的PL方法來(lái)計(jì)算形式為f的函數(shù)零：R→R，提出一種新的PL方法來(lái)計(jì)算形式為f的函數(shù)：R n→R。我們的PL方法避免了遍歷R nx的細(xì)分[0，1]，而是使用我們稱為三角剖分圖的對(duì)象，該對(duì)象本質(zhì)上是R x [0，1]的三角剖分，其中Rn的超立方體為頂點(diǎn)。超立方體是隨機(jī)生成的，并且使用關(guān)聯(lián)的離散時(shí)間馬爾可夫鏈的逗留時(shí)間來(lái)表明不會(huì)生成太多的立方體。此后，我們的PL方法應(yīng)用于在不等式約束下求解G的G（x）= 0：Rn→Rm。求解G（x）= 0的結(jié)果方法轉(zhuǎn)化為一種新型的求解線性方程組的迭代方法。報(bào)告了一些計(jì)算插圖。對(duì)于優(yōu)化問(wèn)題的可能應(yīng)用也被指示為進(jìn)一步工作的方向。

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  • 作者

    Wheaton, Ira, Jr.;

  • 作者單位

    The Florida State University.;

  • 授予單位 The Florida State University.;
  • 學(xué)科 Industrial engineering.;Operations research.;Applied mathematics.
  • 學(xué)位 Ph.D.
  • 年度 2017
  • 頁(yè)碼 78 p.
  • 總頁(yè)數(shù) 78
  • 原文格式 PDF
  • 正文語(yǔ)種 eng
  • 中圖分類
  • 關(guān)鍵詞

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