This works aims at understanding further convergence properties of first order local search methods with complex geometries. We focus on the composite optimization model which unifies within a simple formalism many problems of this type. We provide a general convergence analysis of the composite Gauss–Newton method as introduced in Burke and Ferris (1995) (studied further in Chong and Wang, 2002; Chong and Ng, 2007; Lewis and Wright, 2015) under tameness assumptions (an extension of semi-algebraicity). Tameness is a very general condition satisfied by virtually all problems solved in practice. The analysis is based on recent progresses in understanding convergence properties of sequential convex programming methods through the value function as introduced in Bolte and Pauwels (2016).