Convergence analysis of the Gauss-Newton method for convex inclusion and convex-composite optimization problems
\textit{S. M. Robinson} [Numer. Math. 19, 341--347 (1972; Zbl 0227.65040)] proposed the so-called extended Newton method for solving the convex inclusion problem defined by a nonempty closed convex cone \(C\) in a Banach space \(Y\), and a smooth function \(F\) from a reflexive Banach space \(X\) into \(Y\). Under two important assumptions, he established the existence of solution to the convex inclusion problem, and provided the convergence criterion of this method for the convex inclusion problem. In this paper, the authors consider the convergence analysis of the Gauss-Newton method for the convex inclusion problem by using the convex process theory. Under some mild assumptions (which are weaker than Robinson's assumptions), they establish the convergence criterion of the Gauss-Newton method for solving convex-composite optimization problem. Particularly, they also obtain the convergence criterion for the extended Newton method for solving the convex inclusion problem presented by Robinson. Even in the special case when \(X,Y\) are finite-dimensional and \(T_{x_0}\) is surjective, their result is sharper than Robinson's result. Furthermore, the convergence criterions given in this paper are affine-invariant.
scientific article; zbMATH DE number 6009674
scientific article