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Duality in quasi-newton methods and new variational characterizations of the DFP and BFGS updates

Osman Guler(guler***at***math.umbc.edu)
Filiz Gurtuna(gurtuna1***at***math.umbc.edu)
Olena Shevchenko(olenshe1***at***math.umbc.edu)

Abstract: It is known that quasi-Newton updates can be characterized by variational means, sometimes in more than one way. This paper has two main goals. We first formulate variational problems appearing in quasi-Newton methods within the space of symmetric matrices. This simplies both their formulations and their subsequent solutions. We then construct, for the first time, duals of the variational problems for the DFP and BFGS updates and discover that the solution to a dual problem is either the same as the corresponding primal problem or the solutions are inverses of each other. Consequently, we obtain six new variational characterizations for the DFP and BFGS updates, three for each one.

Keywords: quasi-Newton methods, DFP, BFGS, variational problems, duality

Category 1: Nonlinear Optimization (Unconstrained Optimization )


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Entry Submitted: 09/28/2007
Entry Accepted: 09/28/2007
Entry Last Modified: 09/28/2007

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