Majorizing Functions and Convergence of the Gauss--Newton Method for Convex Composite Optimization

We introduce a notion of quasi regularity for points with respect to the inclusion F (x) ∈ C, where F is a nonlinear Fréchet differentiable function from Rv to Rm. When C is the set of minimum points of a convex real-valued function h on Rm and F ′ satisfies the L-average Lipschitz condition of Wang, we use the majorizing function technique to establish the semilocal linear/quadratic convergence of sequences generated by the Gauss–Newton method (with quasiregular initial points) for the convex composite function h◦F . Results are new even when the initial point is regular and F ′ is Lipschitz.