**Forti, P. Nistri & M. Quincampoix **

**Titre:**
"*M. Generalized Neural Network for Non-Smooth Programming
Problems. *"

**Abstract:**
In 1988 Kennedy and Chua introduced the Dynamical Canonical
Nonlinear Programming Circuit (NPC) to solve in real time
nonlinear programming problems where the objective function and
the constraints are smooth (twice continuously
differentiable) functions. In this paper, a generalized circuit is
introduced (G-NPC), which is aimed at solving in real time a much
wider class of nonsmooth nonlinear programming problems
where the objective function and the constraints are assumed to
satisfy only the weak condition of being regular functions. G-NPC,
which derives from a natural extension of NPC, has a neural-like
architecture and also features the presence of constraint neurons
modeled by ideal diodes with infinite slope in the
conducting region. By using the Clarke's generalized gradient of
the involved functions, G-NPC is shown to obey a gradient system
of differential inclusions, and its dynamical behavior and
optimization capabilities, both for convex and non-convex
problems, are rigorously analyzed in the framework of nonsmooth
analysis and the theory of differential inclusions. In the special
important case of linear and quadratic programming problems,
salient dynamical features of G-NPC, namely the presence of
sliding modes, trajectory convergence in finite time, and
the ability to compute the exact optimal solution of the
problem being modeled, are uncovered and explained in the
developed analytical framework.