  We consider a class of iterative numerical methods and introduce the notion
of semiglobally, practically, strictly pseudogradient (SPSP) search directions.
We demonstrate the relevance of the SPSP property in modelling a variety of
optimization algorithms, including those subject to absolute and relative
errors. We show that the attractors of iterative methods with SPSP search
directions exhibit semiglobal, practical, asymptotic stability. Moreover, the
SPSP property is robust in the sense that perturbations of SPSP search
directions also have the SPSP property.
