Gnecco, Giorgio and Sanguineti, Marcello and Zoppoli, Riccardo
Functional optimization in OR problems with very large numbers of variables.
In: AIRO 2011, September 6th9th, 2011, Brescia, Italy
p. 91.
(2011)
Full text not available from this repository.
Abstract
Functional optimization, or "infinitedimensional programming", investigates the minimization (or maximization) of functionals with respect to admissible
solutions belonging to infinitedimensional spaces of functions. In OR applications, such functions may express, e.g.,
releasing policies in waterresources 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 closedform 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 waterresources 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.
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