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

Abstract

Let X, Y be Hilbert spaces and F : X -> Y be Frochet differentiable. Suppose that F' is center-Lipschitz on U(w, r ) and F'(w) be a siujection. Then, S-1 = F(U(w, epsilon(1))) is convex where epsilon(1) <= r The set S-1 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.