  This paper studies the convergence properties of a family of Relaxed
$\ell$-Minimal Gradient Descent methods for quadratic optimization; the family
includes the omnipresent Steepest Descent method, as well as the Minimal
Gradient method. Simple proofs are provided that show, in an appropriately
chosen norm, the gradient and the distance of the iterates from optimality
converge linearly, for all members of the family. Moreover, the function values
decrease linearly, and iteration complexity results are provided. All
theoretical results hold when (fixed) relaxation is employed. It is also shown
that, given a fixed overhead and storage budget, every Relaxed $\ell$-Minimal
Gradient Descent method can be implemented using exactly one matrix vector
product. Numerical experiments are presented that illustrate the benefits of
relaxation across the family.
