Rationale
The search direction pk at stage k is given by the solution of the analogue of the Newton equation
where is an approximation to the Hessian matrix which is updated iteratively at each stage, and is the gradient of the function evaluated at xk. A line search in the direction pk is then used to find the next point xk+1. Instead of requiring the full Hessian matrix at the point xk+1 to be computed as Bk+1, the approximate Hessian at stage k is updated by the addition of two matrices.
Both Uk and Vk are symmetric rank-one matrices but have different (matrix) bases. The symmetric rank one assumption here means that we may write
So equivalently, Uk and Vk construct a rank-two update matrix which is robust against the scale problem often suffered in the gradient descent searching (e.g., in Broyden's method).
The quasi-Newton condition imposed on this update is
Read more about this topic: BFGS Method