On averaging and representation properties of the BFGS and related secant updates

• 成果类型: Article
• 署名作者: Tapia, Richard
• 署名单位: Rice University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-014-0807-8
• 发表日期: 2015
• 页码: 363-380
• 关键词: quasi-newton methods; variable-metric methods; optimization; minimization; algorithms

摘要

In this paper we present several representation theorems and averaging theorems for members of the difference class of secant updates introduced by Brodlie et al. (J Inst Math Appl 11:73-82, 1973). Major contributions are that the integral form of the mean-value theorem leads to a proof that the BFGS update is pointwise the infinite average of all the updates on the one-dimension manifold in the Dennis class that connects the DFP secant update to the Greenstadt update, and that it can be expressed as the pointwise average of these latter two updates. Analogous results hold for all secant updates that belong to the difference class. These results contribute new understanding of the structural properties of the highly popular BFGS secant update and other updates from the difference class.