Approximating Networks for Functional Optimization Problems
by
Marcello Sanguineti
Department of Communication, Computer and Systems Sciences - DIST - University of Genoa
Coauthors: T. Parisini (Politecnico di Milano), R.Zoppoli (University of Genoa)

The need to solve functional optimization problems arises very often in control area, in operations research, and generally in applied mathematics. In a typical context, one has to minimize a given cost functional with respect to functions belonging to a set of feasible solutions. However, functional optimization problems can be solved analytically only if special assumptions are verified (typically, if we handle dynamic systems, they have to be linear; cost functions have to be quadratic; random variables, if present, have to be Gaussian). Otherwise, approximations are needed. The proposed approximation procedure is based on the use of neural networks and other nonlinear approximators; the solution of the original functional optimization problem is reduced to the solution of a sequence of nonlinear programming problems. The use of powerful approximators enables to face strongly nonlinear optimization problems in high-dimensional settings, without incurring into the so-called ``curse of dimensionality'' (i.e., the exponential increase of the number of parameters necessary to obtain a given approximation accuracy, with increasing dimension of the argument of the function to be approximated). As an example, the proposed procedure is applied to a highly nonlinear, high-dimensional optimal control problem, whose solution is traditionally regarded as a difficult task.

Date received: January 14, 2000