本文選題：非線性方程組 + 有限記憶BFGS��； 參考：《廣西大學(xué)》2016年碩士論文

【摘要】：非線性優(yōu)化是運籌學(xué)的一個重要分支,非線性優(yōu)化問題來源于科學(xué)與工程計算中的許多領(lǐng)域,如氣象預(yù)測問題、非線性有限元問題、非線性斷裂問題、石油地質(zhì)勘探問題等都存在著大量的大規(guī)模的非線性優(yōu)化問題.非線性方程組的求解又和非線性優(yōu)化問題的求解緊密相關(guān),因而基于非線性優(yōu)化問題的算法,研究非線性方程組的數(shù)值解,具有理論意義和實用價值.常見的求解非線性優(yōu)化問題的算法有牛頓法、擬牛頓法、共軛梯度法、信賴域算法等.本文對有限記憶BFGS算法、非精確回溯下降算法和投影共軛梯度算法進(jìn)行研究,取得的主要成果如下：(1)對有限記憶BFGS算法求解非線性方程組問題作了研究,提出了一個新的有限記憶BFGS算法,然后重點分析了它的下降性和全局收斂性等理論性質(zhì),最后設(shè)計了一些算例進(jìn)行數(shù)值試驗.數(shù)值結(jié)果表明新算法能夠較好的求解大規(guī)模非線性方程組問題,且比BFGS算法的數(shù)值表現(xiàn)要好.(2)給出了一個修正的搜索方向,利用回溯線搜索技術(shù)提出了新的算法,然后證明了該算法的下降性和全局收斂性等理論性質(zhì),最后給出了新算法和BFGS算法的數(shù)值表現(xiàn).數(shù)值結(jié)果表明新算法比BFGS算法的數(shù)值表現(xiàn)要好.(3)提出了一個新的投影共軛梯度算法,接著分析了該算法的充分下降性、信賴域性質(zhì)和全局收斂性等性質(zhì),最后設(shè)計了一些算例進(jìn)行數(shù)值試驗.數(shù)值結(jié)果表明新算法能夠較好的求解大規(guī)模非線性方程組,且比通常情況下數(shù)值結(jié)果較好的PRP算法的數(shù)值表現(xiàn)要好.
[Abstract]:Nonlinear optimization is an important branch of operational research. Nonlinear optimization problems are derived from many fields in science and engineering, such as weather prediction problem, nonlinear finite element problem, nonlinear fracture problem. There are a large number of nonlinear optimization problems in petroleum geological exploration. The solution of nonlinear equations is closely related to the solution of nonlinear optimization problems. Therefore, it is of theoretical significance and practical value to study the numerical solutions of nonlinear equations based on the algorithm of nonlinear optimization problems. Common algorithms for solving nonlinear optimization problems include Newton method, quasi-Newton method, conjugate gradient method, trust region algorithm and so on. In this paper, the finite memory BFGs algorithm, the inexact backtracking descent algorithm and the projection conjugate gradient algorithm are studied. The main results are as follows: (1) the finite memory BFGs algorithm is studied for solving nonlinear equations. In this paper, a new finite memory BFGS algorithm is proposed, and its theoretical properties such as descent and global convergence are analyzed. Finally, some numerical examples are designed for numerical experiments. Numerical results show that the new algorithm can solve large scale nonlinear equations better than BFGs algorithm. (2) A modified search direction is given, and a new algorithm is proposed by using backtracking search technique. Then the theoretical properties of the algorithm such as descent and global convergence are proved. Finally, the numerical representations of the new algorithm and the BFGS algorithm are given. Numerical results show that the new algorithm performs better than the BFGS algorithm. (3) A new projection conjugate gradient algorithm is proposed, and the sufficient descent, trust region and global convergence of the algorithm are analyzed. Finally, some numerical examples are designed to carry out numerical tests. Numerical results show that the new algorithm can solve large scale nonlinear equations, and the numerical performance of PRP algorithm is better than that of PRP algorithm.
【學(xué)位授予單位】：廣西大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：O224

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 ;TESTING DIFFERENT CONJUGATE GRADIENT METHODS FOR LARGE-SCALE UNCONSTRAINED OPTIMIZATION[J];Journal of Computational Mathematics;2003年03期

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本文編號：2101142