A family of variable-metric methods derived by variational means
1970/01/01 by Donald Goldfarb · 2,732 citations
Mathematics · #Advanced Optimization Algorithms Research #Applied mathematics #Combinatorics #Eigenvalues and eigenvectors #Fractional Differential Equations Solutions #Geometry #Hessian matrix #Inverse #Iterative Methods for Nonlinear Equations #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Metric (unit) #Positive definiteness #Positive-definite matrix #Rank (graph theory) #Variable (mathematics) #Weighting
paper · doi:10.1090/s0025-5718-1970-0258249-6
published in Mathematics of Computation 24(109), 23-26 (American Mathematical Society (AMS))
crossref issued 1970/01/01 · crossref published 1970/01/01 · crossref published-print 1970/01/01 · openalex publication_date 1970/01/01 · crossref created 2010/07/01 · openalex created_date 2025/10/10 · crossref deposited 2026/04/20 · openalex updated_date 2026/08/05 · crossref indexed 2026/08/07
Abstract
A new rank-two variable-metric method is derived using Greenstadt’s variational approach [ Math. Comp. , this issue]. Like the Davidon-Fletcher-Powell (DFP) variable-metric method, the new method preserves the positive-definiteness of the approximating matrix. Together with Greenstadt’s method, the new method gives rise to a one-parameter family of variable-metric methods that includes the DFP and rank-one methods as special cases. It is equivalent to Broyden’s one-parameter family [ Math. Comp. , v. 21, 1967, pp. 368–381]. Choices for the inverse of the weighting matrix in the variational approach are given that lead to the derivation of the DFP and rank-one methods directly.
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