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Quasi-Newton algorithms with updates from the preconvex part of Broyden’s family. (English) Zbl 0661.65061
The authors study the so-called preconvex part of Broyden’s family, corresponding to values $$\phi \in (\phi^*,O]$$ (where $$\phi^*$$ is the degenerate value, $$\phi =O$$ corresponds to BFGS and $$\phi =1$$ to DFP update). By refining Powell’s arguments, they extend his global convergence theorem to the preconvex class and formulate a varying- parameter algorithm which at each iteration takes $$\phi_ k$$ so as to obtain the steepest descent direction from a given set of quasi-Newton directions. Extensive computer experiments show that the algorithm, in the authors’ own words, “outperforms BFGS by a considerable margin on the selected test problems for the given line-search algorithm”.
Reviewer: J.Rohn

##### MSC:
 65K05 Numerical mathematical programming methods 90C30 Nonlinear programming
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