  Many convex optimization methods are conceived of and analyzed in a largely
separate fashion. In contrast to this traditional separation, this manuscript
points out and demonstrates the utility of an important but largely unremarked
common thread running through many prominent optimization methods. In
particular, we show that methods such as successive orthogonal projection,
gradient descent, projected gradient descent, the proximal-point method,
forward-backward splitting, the alternating direction method of multipliers,
and under- or over-relaxed variants of the preceding all involve updates that
are of a common type --- namely, the updates satisfy a property known as
pseudocontractivity. Moreover, since the property of pseudocontractivity is
preserved under both composition and convex combination, updates constructed
via these operations from pseudocontractive updates are themselves
pseudocontractive. Having demonstrated that pseudocontractive updates are to be
found in many optimization methods, we then provide a unified basic analysis of
methods with pseudocontractive updates. Specifically, we prove a novel bound
satisfied by the norm of the difference in iterates of pseudocontractive
updates and we then use this bound to establish that the error criterion
$\left\Vert x^{N}-Tx^{N}\right\Vert ^{2}$ is $o(1/N)$ for any method involving
pseudocontractive updates (where $N$ is the number of iterations and $T$ is the
iteration operator).
