Polarizable quasiconvex function generated metric projection

Abstract

We study when generalized metric projections generated by strongly quasiconvex functions are polarizable. The point of departure is Moreau's decomposition theorem, where the Euclidean metric projections onto two mutually polar closed convex cones form complementary retractions whose sum is the identity. This suggests the following fundamental question: when does a given retraction admit such a complementary polar retraction? We formulate this notion through polarizable retractions and then specialize it to generalized projections of the form $x\mapsto P_{\mathcal{C}}^{\phi }(x)$, defined by minimizing $\phi (x-y)$ with respect to $y$ over a closed convex set $\mathcal{C}$. After recalling basic properties of mutually polar retractions and first-order characterizations of generalized projections, we prove the main result: If $\phi$ is strongly quasiconvex, the generalized projection $P_{\mathcal{C}}^{\phi }$ is polarizable if and only if $\mathcal{C}$ is a closed convex cone. Thus, polarizability of the projection is characterized by the conic geometry of its range.