The global convergence of partitioned BFGS on problems with convex decompositions and Lipschitzian gradients

The main purpose of this paper is the extension of Powell’s global convergence result to the partitioned BFGS method introduced by Greiwank and Toint. Even in the unpartitioned case the original result is strengthened because the search directions need not be computed exactly and the gradient is only required to be Lipschitzian rather than differentiable. Using the ℝ-functional of Byrd and Nocedal, a strong form of R-superlinear convergence is obtained if the element functions are uniformly convex and their gradients are strictly differentiable at the minimizer x