Pattern formation in associative neural networks is related to a quadratic optimization problem. Biological considerations imply that the functional is constrained in the L-infinity, norm and in the L-1 norm. We consider such optimization problems. We derive the Euler-Lagrange equations, and construct basic properties of the maximizers. We study in some detail the case where the kernel of the quadratic functional is finite-dimensional. In this case the optimization problem can be, fully characterized by the geometry of a certain convex and compact finite-dimensional set.