2005
DOI: 10.1137/s1052623403426507
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Error Estimates for Approximate Optimization by the Extended Ritz Method

Abstract: Abstract. An alternative to the classical Ritz method for approximate optimization is investigated. In the extended Ritz method, sets of admissible solutions are approximated by their intersections with sets of linear combinations of all n-tuples of functions from a given basis. This alternative scheme, called variable-basis approximation, includes functions computable by trigonometric polynomials with free frequencies, free-node splines, neural networks, and other nonlinear approximating families. Estimates o… Show more

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Cited by 65 publications
(52 citation statements)
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“…For recent investigations of these classes of approximators, the reader is referred to refs. [14,16,17,19,[22][23][24][25]30]. Now we apply some result from ref.…”
Section: Proposition 2 Let the Assumptions A1-a4 Be Verified And Denomentioning
confidence: 94%
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“…For recent investigations of these classes of approximators, the reader is referred to refs. [14,16,17,19,[22][23][24][25]30]. Now we apply some result from ref.…”
Section: Proposition 2 Let the Assumptions A1-a4 Be Verified And Denomentioning
confidence: 94%
“…[14] as a general methodology of approximate optimization, whose theoretical properties have been investigated in refs. [l4, 16,17].…”
Section: Reduction Of the Optimal Fault-diagnosis Problem To A Nonlinmentioning
confidence: 99%
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“…Therefore, a nonlinear programming problem is addressed that can be solved by means of a stochastic gradient technique. The resulting approximate control functions are sub-optimal solutions, but (thanks to the well-established approximation properties of the neural networks) one can achieve any desired degree of accuracy [5]. Once solved the off-line finite-horizon problem, only the first control function is retained in the on line phase: at any sample time t, given the system's state and the target's position and velocity, the control action is generated with a very small computational effort.…”
mentioning
confidence: 99%