On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating

In previous work, the authors provided a foundation for the theory of varia ble metric proximal point algorithms in Hilbert space. In that work conditi ons are developed for global, linear, and super-linear convergence. This pa per focuses attention on two matrix secant updating strategies for the fini te dimensional case. These are the Broyden and BFGS updates. The BFGS updat e is considered for application in the symmetric case, e.g., convex program ming applications, while the Broyden update can be applied to general monot one operators. Subject to the linear convergence of the iterates and a quad ratic growth condition on the inverse of the operator at the solution, supe r-linear convergence of the iterates is established for both updates. These results are applied to show that the Chen-Fukushima variable metric proxim al point algorithm is super-linearly convergent when implemented with the B FGS update.