Extending the Convexity of Nonlinear Image of a Ball Appearing in Optimization

Abstract

Let X, Y be Hilbert spaces and F : X → Y be Frechet differentiable. Suppose that F′ is center-Lipschitz on U(w, r) and F′(w) be a surjection. Then, S1 = F(U(w, ε1)) is convex where ε1 ≤ r. The set S1 contains the corresponding set given in [18] under the Lipschitz condition. Numerical examples where the old conditions are not satisfied but the new conditions are satisfied are provided in this paper.