Convergence analysis of the Gauss–Newton method for convex inclusion and convex-composite optimization problems

Using the convex process theory we study the convergence issues of the iterative sequences generated by the Gauss–Newton method for the convex inclusion problem defined by a cone C and a smooth function F (the derivative is denoted by F′). The restriction in our consideration is minimal and, even in the classical case (the initial point x0 is assumed to satisfy the following two conditions: F′ is Lipschitz around x0 and the convex process Tx0, defined by Tx0⋅=F′(x0)⋅−C, is surjective), our results are new in giving sufficient conditions (which are weaker than the known ones and have a remarkable property being affine-invariant) ensuring the convergence of the iterative sequence with initial point x0. The same study is also made for the so-called convex-composite optimization problem (with objective function given as the composite of a convex function with a smooth map).