  Quasi-Newton methods form an important class of methods for solving nonlinear
optimization problems. In such methods, first order information is used to
approximate the second derivative. The aim is to mimic the fast convergence
that can be guaranteed by Newton-based methods. In the best case, quasi-Newton
methods will far outperform steepest descent and other first order methods,
without the computational cost of calculating the exact second derivative.
These convergence guarantees hold locally, which follows closely from the fact
that if the objective function is strongly convex it can be approximated well
by a quadratic function close to the solution. Understanding the performance of
quasi-Newton methods on quadratic problems with a symmetric positive definite
Hessian is therefore of vital importance. In the classic case, an approximation
of the Hessian is updated at every iteration and exact line search is used. It
is well known that the algorithm terminates finitely, even when the Hessian
approximation is memoryless, i.e. requires only the most recent information. In
this paper, we explore the possibilities in which reliance on exact line search
and dependence on conjugate search directions can be relaxed, while preserving
finite termination properties of quasi-Newton methods on quadratic problems.
