eprintid: 1672
rev_number: 11
eprint_status: archive
userid: 46
dir: disk0/00/00/16/72
datestamp: 2013-09-11 09:52:46
lastmod: 2013-09-16 12:03:00
status_changed: 2013-09-11 09:52:46
type: conference_item
metadata_visibility: no_search
creators_name: Gnecco, Giorgio
creators_name: Sanguineti, Marcello
creators_name: Zoppoli, Riccardo
creators_id: giorgio.gnecco@imtlucca.it
creators_id: 
creators_id: 
title: Functional optimization in OR problems with very large numbers of variables
ispublished: pub
subjects: QA75
divisions: CSA
full_text_status: none
pres_type: paper
keywords: Infinite-Dimensional Programming, Suboptimal Solutions, Approximation Schemes, Curse of Dimensionality, Extended Ritz Method (ERIM).
note: 42nd Conference of the Italian Operational Research Society
abstract: Functional optimization, or "infinite-dimensional programming", investigates the minimization (or maximization) of functionals with respect to admissible
solutions belonging to infinite-dimensional spaces of functions. In OR applications, such functions may express, e.g., 
-releasing policies in water-resources management;
-exploration strategies stochastic graphs;
-routing strategies in telecommunication networks;
-input/output mappings in learning from data, etc.
Infinite dimension makes inapplicable many tools used in mathematical programming, and variational methods provide closed-form solutions only in particular cases. Suboptimal solutions can be sought via "linear approximation
schemes",i.e., linear combinations of fixed basis functions (e.g., polynomial expansions):
the functional problem is reduced to optimization of the coefficients
of the linear combinations ("Ritz method"). Most often, admissible solutions
are functions dependent on many variables, related, e.g., to
-reservoirs in water-resources management;
-nodes of a communication network;
-items in inventory problems;
-freeway sections in traffic management.
Unfortunately, linear schemes may be computationally inefficient because
of the "curse of dimensionality": the number of basis functions, necessary to
obtain a desired accuracy, may grow "very fast" with the number of variables.
This motivates the "Extended Ritz Method"(ERIM), based on nonlinear approximation schemes formed by linear combinations of computational units
containing "inner" parameters which make the schemes nonlinear to be optimized (together with the coefficients of the combinations) via nonlinear programming algorithms. Experimental results show that this approach obtains surprisingly good performances. We present recent theoretical results that give insights into the possibility to cope with the curse of dimensionality in functional optimization via the ERIM, when admissible solutions contain very large numbers of variables.
date: 2011
date_type: published
pagerange: 91
event_title: AIRO 2011
event_location: Brescia, Italy
event_dates: September 6th-9th, 2011
event_type: conference
refereed: TRUE
related_url_url: http://airo2011.eco.unibs.it/
related_url_type: org
citation:   Gnecco, Giorgio and Sanguineti, Marcello and Zoppoli, Riccardo  Functional optimization in OR problems with very large numbers of variables.  In: AIRO 2011, September 6th-9th, 2011, Brescia, Italy p. 91.        (2011)