The Extended Ritz Method in Stochastic Functional Optimization: An example of Dynamic Routing in Traffic Networks

Abstract

The classical Ritz method constrains the admissible solutions of functional optimization problems to take on the structure of linear combinations of fixed basis functions. Under general assumptions, the coefficients of such linear combinations become the unknowns of a finite-dimensional nonlinear programming problem. We propose to insert "free" parameters to be optimized in the basis functions, too. This justifies the term "Extended Ritz Method." If the optimal solutions of functional optimization problems belong to classes of d-variable functions characterized by suitable regularity properties, the Extended Ritz Method may outperform the Ritz method in that the number of free parameters increases moderately (e.g., polynomially) with d, whereas the latter method may be ruled out by the curse of dimensionality. Once the functional optimization problem has been approximated by a nonlinear programming one, the solution of the latter problem is obtained by stochastic approximation techniques. The overall procedure turns out to be effective in high-dimensional settings, possibly in the presence of several decision makers. In such a context, we focus our attention on large-scale traffic networks such as communication networks, freeway systems, etc. Traffic flows may vary over time. Then the nodes of the networks (i.e., the decision makers acting at the nodes) may be requested to modify the traffic flows to be sent to their neighboring nodes. Consequently, a dynamic routing problem arises that cannot be solved analytically. In particular, we address store-and-forward packet switching networks, which exhibit the essential difficulties of traffic networks. Simulations performed on complex communication networks show the effectiveness of the proposed method.

Error Estimates for Approximate Optimization by the Extended Ritz Method

“…Also, a new branch of nonlinear approximation theory investigating approximation capabilities of neural networks was developed [11,12,21,28,38,45,50,51,52,53,54,55,56,57,58]. In a series of papers [3,5,8,9,10,64,65,66,80,81], a new method of approximate optimization was developed, called in [81] the extended Ritz method. In these papers, approximate solutions were used that were obtained over restrictions of sets of admissible solutions to linear combinations of all n-tuples of functions with varying "free" parameters, instead of linear combinations of first n functions from a basis with fixed ordering as in the classical Ritz method.…”

“…The extended Ritz method with such bases was successfully tested on a variety of problems with admissible solutions dependent on a large number of variables: stochastic optimal control [64,65,66,80] and optimal estimation of state variables [3] in nonlinear dynamic systems with a large number of state variables, team optimal control [8], optimal control of freeway traffic [81], routing in large-scale communication networks [9,10], optimal fault diagnosis [5], etc. In these applications, admissible sets of variable-basis functions were used, for which the degree n necessary to guarantee a fixed approximation accuracy grows only polynomially with the number of variables of admissible solutions.…”

“…For G formed by parameterized families of the form G = {g a : a ∈ A, g a ∈ G} with A ⊆ R p , this method was applied to a variety of optimization tasks considered in a series of papers [3,5,8,9,10,64,65,66,80] and in [81], where it was called the extended Ritz method. Here we use this term even more generally for an approximate optimization by the sequence of problems {M ∩ span n G}, where G is any subset of X.…”

“…Such a problem can be solved by algorithms based on gradient descent with stochastic perturbations [17, [4,16,40,76] and the references therein). In [5,9,10,65,66,81], applications of some of these algorithms to the extended Ritz method are described and illustrated by numerical results showing the algorithms' effectiveness in a variety of cases.…”

Optimization of approximating networks for optimal fault diagnosis

“…Such a situation may occur, for example, when the dynamic environment is represented by a large-scale traffic network, a communication network with many routers and links, etc. [14,18,19].…”