Abstract

Various characterizations of convexity for images of a vector mapping where some of its components are quadratic and the remaining ones are linear are established. In a certain sense, one might conclude that convexity of the full image is reduced to the convexity of an image in a lower dimension by deleting the linear components. The latter may be considered as the analogue to the reduction of the number of constraints once the dual is associated. The cases of having one or two quadratic components while the other are linear are particularly analyzed. This allows us to formulate some (geometric) sufficient and necessary conditions for convexity. As a byproduct, a result obtained in [Xia, Wang, and Sheu, Math. Program. Ser. A, 156 (2016), pp. 513--547] is corrected. Finally, as some applications, we obtain an S-lemma (with equality and on an affine subspace) and a characterization of strong duality in terms of convexity of some image set associated to the minimization problem under consideration.