On a class of convex sets with convex images and its application to nonconvex optimization

In the present paper, conditions under which the images of uniformly convex sets through $C^{1,1}$ regular mappings between Banach spaces remain convex are established. These conditions are expressed by a certain quantitative relation betweeen the modulus of convexity of a given set and the global regularity behaviour of the mapping on it. Such a result enables one to extend to a wide subclass of convex sets the Polyak's convexity principle, which was originally concerned with images of small balls around points of Hilbert spaces. In particular, the crucial phenomenon of the preservation of convexity under regular $C^{1,1}$ transformations is shown to include the class of $r$-convex sets, where the value of $r$ depends on the regularity behaviour of the involved transformation. Two consequences related to nonconvex optimization are discussed: the first one is a sufficient condition for the global solution existence for infinite-dimensional constrained extremum problems; the second one provides a zero-order Lagrangian type characterization of optimality in nonlinear mathematical programming.