Appl Math Optim 49:55–80 (2004)
2003 Springer-Verlag New York Inc.
-Constrained Quadratic Optimization Problem with
Applications to Neural Networks
and Jacob Rubinstein
Mathematics Department, Indiana University,
Bloomington, IN 47405, USA
Communicated by M. Soner
Abstract. Pattern formation in associative neural networks is related to a quadratic
optimization problem. Biological considerations imply that the functional is con-
strained in the L
norm and in the L
norm. We consider such optimization prob-
lems. 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
Key Words. Quadratic optimization, Neural networks.
AMS Classiﬁcation. 49K30, 92B20.
Neural networks consist of a set of neurons and connections between them. A neuron
is deﬁned as a ﬁlter operating on some input. The input received by a given neuron is
determined by the activity of all the neurons that send axons to it, by the density of the
connections between the neuron and the other neurons, and by the synaptic strengths.
Therefore the system is fully characterized by the neurons density and synaptic strength.
Current address: Department of Mathematics, Technion, Haifa 32000, Israel. firstname.lastname@example.org.