  In many real world applications, data are characterized by a complex
structure, that can be naturally encoded as a graph. In the last years, the
popularity of deep learning techniques has renewed the interest in neural
models able to process complex patterns. In particular, inspired by the Graph
Neural Network (GNN) model, different architectures have been proposed to
extend the original GNN scheme. GNNs exploit a set of state variables, each
assigned to a graph node, and a diffusion mechanism of the states among
neighbor nodes, to implement an iterative procedure to compute the fixed point
of the (learnable) state transition function. In this paper, we propose a novel
approach to the state computation and the learning algorithm for GNNs, based on
a constraint optimisation task solved in the Lagrangian framework. The state
convergence procedure is implicitly expressed by the constraint satisfaction
mechanism and does not require a separate iterative phase for each epoch of the
learning procedure. In fact, the computational structure is based on the search
for saddle points of the Lagrangian in the adjoint space composed of weights,
neural outputs (node states), and Lagrange multipliers. The proposed approach
is compared experimentally with other popular models for processing graphs.
