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Functional Optimization in OR Problems with Very Large Numbers of Variables

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)

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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.

Item Type: Conference or Workshop Item (Paper)
Additional Information: 42nd Conference of the Italian Operational Research Society
Uncontrolled Keywords: Infinite-Dimensional Programming, Suboptimal Solutions, Approximation Schemes, Curse of Dimensionality, Extended Ritz Method (ERIM).
Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Research Area: Computer Science and Applications
Depositing User: Giorgio Gnecco
Date Deposited: 17 Sep 2013 13:06
Last Modified: 17 Sep 2013 13:06
URI: http://eprints.imtlucca.it/id/eprint/1767

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