Armijo搜索下求解无约束优化问题的扰动BFGS方法
The Perturbed BFGS Method for the Unconstrained Optimization with the Armijo Line Search
DOI: 10.12677/AAM.2019.82029, PDF,   
作者: 严娇娇：长沙理工大学数学与统计学院，湖南 长沙
关键词: BGFS方法；Armijo线性搜索；全局收敛性；BGFS Method； Armijo Linear Search； Global Convergence
摘要: 文献[1]提出了一种求解无约束优化问题的扰动BFGS方法，并在Wolfe搜索下证明了其全局收敛性。本文证明了该扰动BFGS方法在较弱的Armijo线性搜索下求解非凸问题也具有全局收敛性。数值结果表明在Armijo搜索下该方法也具有较好的数值效果。
Abstract: In [1], a perturbed BFGS method was proposed to solve the unconstrained optimization and was proved to be globally convergent when the Wolfe line search was used. In this paper, we show that the perturbed BFGS method also possesses global convergence for nonconvex problems with the relatively weaker Armijo line search. Numerical results show that this method with the Armijo search is also promising.
文章引用：严娇娇. Armijo搜索下求解无约束优化问题的扰动BFGS方法[J]. 应用数学进展, 2019, 8(2): 250-257. https://doi.org/10.12677/AAM.2019.82029

参考文献

[1] 刘陶文. BFGS方法及其在求解约束优化问题中的应用[D]: [博士学位论文]. 长沙: 湖南大学, 2006.
[2] 李董辉, 童小娇, 万中. 数值最优化算法与理论[M]. 北京: 科学出版社, 2010: 51-63.
[3] 袁亚湘, 孙文瑜. 最优化理论与方法[M]. 北京: 中国科学出版社, 2001: 1-627.
[4] Powell, M.J.D., Cottle, R.W. and Lemke, C.E. (1976) Some Convergence Properties of a Variable Metric Algorithm for Minimization without Exact Line Search. SIAM Publications, 53-72.
[5] Dai, Y.H. (2002) Convergence Properties of the BFGS Algorithm. SIAM Journal on Optimization, 13, 693-701. [Google Scholar] [CrossRef] 
[6] Mascarenhas, W.F. (2004) The BFGS Method with Exact Line Searches Fails for Non-Convex Objective Functions. Mathematical Programming, 99, 49-61. [Google Scholar] [CrossRef] 
[7] Li, D.H. and Fukushima, M. (2001) On the Global Conver-gence of the BFGS Method for Nonconvex Unconstrained Problems. SIAM Journal on Optimization, 11, 1054-1064. [Google Scholar] [CrossRef] 
[8] Armijo, L. (1966) Minimization of Functions Having Lipschitz Continuous Partial Derivatives. Pacific Journal of Mathematics, 16, 1-3. [Google Scholar] [CrossRef] 
[9] Li, D.H. and Fukushima, M. (2001) A Modified BFGS Method and Its Global Convergence in Nonconvex Minimization. Journal of Computational and Applied Mathematics, 192, 15-35. [Google Scholar] [CrossRef] 
[10] More, J.J., Garbow, B.S. and Hillstrom, K.E. (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, 7, 17-41. [Google Scholar] [CrossRef] 

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