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

Abstract

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 x 0 is assumed to satisfy the following two conditions: F ′ is Lipschitz around x 0 and the convex process T x 0 , defined by T x 0 ⋅ = F ′ ( x 0 ) ⋅ − 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 x 0 . 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).