Sizing and Least Change Secant Methods
RICE UNIV HOUSTON TX DEPT OF MATHEMATICAL SCIENCES
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The function omegaA traceAn detAto the 1n power introduced as a measure of deviation of a positive definite matrix from the identity. This appears to be a more uniform measure than the standard l sub 2 condition number since it takes all the eigenvalues of A into account. Optimal quasi-Newton updates are given with respect to various applications of this measure. This yields the inverse-sized BFGS and sized DFP updates suggested by Oren and Luenberger, and it gives rise to a new one-parameter class of updates based on these two updates just as the Broyden class is based on the BFGS and DFP updates. Also considered are alternatives to sizing after the first step. This leads to some interesting weighted Frobenius norm problems for weak forms of the secant condition and a particular Fletcher dual pair in the Broyden class.
- Operations Research