A Semiglobal, Practical, Strict Pseudogradient Property for Iterative Methods

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.