Broyden restricted class of variable metric methods and oblique projections

In the paper the new formulation of the Broyden restricted convex class of updates involving oblique projections is introduced. It is a sum of two terms: the first one containing special oblique projection and the second standard term ensuring verification of the quasi-Newton condition (it is also an oblique projection multiplied by appropriate scalar). The applied oblique projection involves vector defined as the convex, linear combination of the difference between consecutive iterative points and the image of the previous inverse hessian approximation on the corresponding difference of derivatives, i.e. gradients. Formula relating coefficient in the convex combination of vectors in the oblique projection with its counterpart in the standard representation of the Broyden convex class is presented. Some preliminary numerical experiments results on two twice continuously differentiable strictly convex functions with increasing dimension are included.