Quadratic Maps with Convex Images

Seeking exactness for \convex relaxation", we deene a map f : < n ! < m to be ICON if its image f(< n) is convex. Analogously, f is termed LICON(AICON) if the image of every linear (every aane) vsubspace under f is convex. In this paper, we investigate quadratic maps of these three types. These maps are motivated by a fact shown here that they prescribe conditions under which feasibility and optimization problems involving quadratic constraints (called multiquadrtic programs) are reducibe to Semideenite Programs. We rst show that for every quadratic map f, there exists another quadratic map, whose image is the convex hull of the image of f. This is employed to derive a characterization of quadratic ICON maps, which is then used to show that ICON-map recognition is NP-Hard. Then we develop a characterization and a polynomial time algorithm for identifying quadratic LICON maps. Finally, a characterization of quadratic AICON maps, that lends itself to an easy recognition algorithm, is established.