Many iterative algorithms for optimization calculations form positive definite second derivative approximations, B say, automatically, but B is not stored explicitly because of the need to solve equations of the form \(Bd=-g\). We consider working with matrices Z, whose columns satisfy the conjugacy conditions \(Z^ TBZ=1\). Particular attention is given to updating Z in a way that corresponds to revising B by the BFGS formula. A procedure is proposed that seems to be much more stable than the direct use of a product formula [seeK. W. Brodlie, A. R. Gourlay andJ. Greenstadt, J. Inst. Math. Appl. 11, 73-82 (1973;Zbl 0252.15007)]. An extension to this procedure provides some automatic rescaling of the columns of Z, which avoids some inefficiencies due to a poor choice of the initial second derivative approximation. Our work is also relevant to active set methods for linear inequality constraints, to updating the Cholesky factorization of B, and to explaining some properties of the BFGS algorithm.

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.