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

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 vector space of symmetric matrices. This simplifies 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 solution 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.