The main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give two classes of limited-memory quasi-Newton Hessian approximations that generate search directions parallel to those of the method of preconditioned conjugate gradients, and hence give finite termination on quadratic optimization problems. The Hessian approximations are described by a novel compact representation which provides a dynamical framework. We also discuss possible extensions of these classes and show their behavior on randomly generated quadratic optimization problems. The methods behave numerically similar to L-BFGS. Inclusion of information from the first iteration in the limited-memory Hessian approximation and L-BFGS significantly reduces the effects of round-off errors on the considered problems. In addition, we give our compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.
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