Error Estimates for Approximate Optimization by the Extended Ritz Method

Kůrková, Véra; Sanguineti, Marcello

doi:10.1137/s1052623403426507

Cited by 65 publications

(52 citation statements)

References 50 publications

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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%

“…More specifically, with reference to the lth component γ The possibility of extending the property expressed by Proposition 3 to other classes of nonlinear OHL networks has been investigated in refs. [14,16,25,30]. Regarding the assumption A5, various classes of functions verifying it and exhibiting a polynomial growth of c l,i with n i have been presented in ref.…”

Section: M I )mentioning

confidence: 99%

“…It has been so successful as to be formulated in ref. [14] as a general methodology of approximate optimization, whose theoretical properties have been investigated in refs: [14,16,17]. Approximate optimization by approximating networks allows one to reduce the solution of the optimal fault-diagnosis problem to a finite-dimensional nonlinear programming problem in which the 'free' parameters of the approximating networks have to be optimized.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimization of approximating networks for optimal fault diagnosis

Alessandri

Sanguineti

2005

Optimization Methods and Software

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An optimization-based approach to fault diagnosis for nonlinear stochastic dynamic models is developed. An optimal diagnosis problem is formulated according to a receding-horizon strategy. This approach leads to a functional optimization problem (also called 'infinite optimization problem'), whose admissible solutions belong to a function space. As in such a context, the tools from mathematical programing are either inapplicable or inefficient, a methodology of approximate solution is proposed that exploits diagnosis strategies made up of combinations of a certain number of simple basis functions, easy to implement and dependent on some parameters to be optimized. The optimization of the parameters is performed in two phases. In the first, 'a-priori' knowledge of the statistics of the stochastic variables is used to initialize (off-line) the parameter values. In the second phase, the optimization continues on-line. Both off-line and on-line phases rely upon stochastic approximation algorithms. The overall procedure turns out to be effective in high-dimensional settings such as those characterized by a large dimension of the state space and a large diagnosis window. This favorable behavior results from certain properties of the proposed methodology of approximate optimization, such as polynomial bounds on the rate of growth of the number of parameterized basis functions, which guarantees the desired accuracy of approximate optimization. The effectiveness of the approach is confirmed by simulations in the context of a complex instance of the fault-diagnosis problem. The advantages over classical approaches to fault diagnosis are discussed and pointed out by numerical results.

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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%

Section: M I )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimization of approximating networks for optimal fault diagnosis

Alessandri

Sanguineti

2005

Optimization Methods and Software

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show abstract

“…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%

An Application of Receding-Horizon Neural Control in Humanoid Robotics

Ivaldi

Baglietto

Metta

et al. 2009

Nonlinear Model Predictive Control

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: Optimal trajectory planning of a humanoid arm is addressed. The goal is to make the end effector reach a desired target or track it when it moves in the arm's workspace unpredictably. As a reference setup, we considered a 7 degrees of freedom humanoid robot arm. Physical constraints require the online computations to be very quick. Following previous studies [1], a recedinghorizon method is proposed that consists in assigning the control function a fixed structure (e.g., a feedforward neural network) where a fixed number of parameters have to be tuned. More specifically, in the off-line phase, a set of neural networks (corresponding to the control functions over a finite horizon) is optimized using the Extended Ritz Method. The training set corresponds to a sampling of the arm and target position and velocity configuration space. The expected value of a suitable cost is minimized with respect to the free parameters in the neural networks. 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.

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

Baglietto

Sanguineti

Zoppoli

2003

Applied Optimization

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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.

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Error Estimates for Approximate Optimization by the Extended Ritz Method

Cited by 65 publications

References 50 publications

Optimization of approximating networks for optimal fault diagnosis

Optimization of approximating networks for optimal fault diagnosis

An Application of Receding-Horizon Neural Control in Humanoid Robotics

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

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