n this chapter we are concerned with the convex composite optimizations problem. This work is mainly motivated by the work in [17,23].We present a convergence analysis of Gauss-Newton method (defined by Algorithm (GNA) in Sec. 20.2). The convergence of GNA is based on the majorant function in [17] (to be precised in Sec. 20.2). They follow the same formulation using the majorant function provided in [23] (see [21,23,28,29]). In [3,5,8], a convergence analysis in a Banach space setting was given for (GNM) defined (formula presented).