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

Abstract

We introduce a notion of quasi regularity for points with respect to the inclusion $F(x)\in C$, where F is a nonlinear Fréchet differentiable function from ${\mathbb{R}}^v$ to ${\mathbb{R}}^m$. When C is the set of minimum points of a convex real-valued function h on ${\mathbb{R}}^m$ 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 quasi-regular initial points) for the convex composite function $h\circ F$. Results are new even when the initial point is regular and $F'$ is Lipschitz.