Intractable problems in control theory
SIAM Journal on Control and Optimization
Regularization theory and neural networks architectures
Neural Computation
Geometry and topology of continuous best and near best approximations
Journal of Approximation Theory
Approximating networks and extended Ritz method for the solution of functional optimization problems
Journal of Optimization Theory and Applications
Dynamic Programming
Minimization of Error Functionals over Variable-Basis Functions
SIAM Journal on Optimization
Error Estimates for Approximate Optimization by the Extended Ritz Method
SIAM Journal on Optimization
Learning with generalization capability by kernal methods of bounded complexity
Journal of Complexity
A recursive algorithm for nonlinear least-squares problems
Computational Optimization and Applications
Complexity of Gaussian-radial-basis networks approximating smooth functions
Journal of Complexity
Accuracy of suboptimal solutions to kernel principal component analysis
Computational Optimization and Applications
Approximate Minimization of the Regularized Expected Error over Kernel Models
Mathematics of Operations Research
Bounds on rates of variable-basis and neural-network approximation
IEEE Transactions on Information Theory
Comparison of worst case errors in linear and neural network approximation
IEEE Transactions on Information Theory
Neural approximations for infinite-horizon optimal control of nonlinear stochastic systems
IEEE Transactions on Neural Networks
Convergent on-line algorithms for supervised learning in neural networks
IEEE Transactions on Neural Networks
Distributed-information neural control: the case of dynamic routing in traffic networks
IEEE Transactions on Neural Networks
Optimization-based learning with bounded error for feedforward neural networks
IEEE Transactions on Neural Networks
Smooth Optimal Decision Strategies for Static Team Optimization Problems and Their Approximations
SOFSEM '10 Proceedings of the 36th Conference on Current Trends in Theory and Practice of Computer Science
| Hi-index | 0.00 |
Functional optimization problems entail minimization of functionals with respect to admissible solutions belonging to infinite-dimensional spaces of functions. The extended Ritz method (ERIM) searches for suboptimal solutions expressed as linear combinations of basis functions with 'free' parameters that are optimized, together with the coefficients of the linear combinations, by solving finite-dimensional nonlinear programming problems and exploiting stochastic approximation techniques. The first part of the paper gives an overview of the ERIM and introduces the concept of polynomially complex optimizing sequences. Their elements are given by the solutions of the nonlinear programming problems obtained using an increasing number of basis functions. Under some conditions, the optimizing sequences epi-converge to the optimal solution and for a given approximation accuracy, the number of free parameters increases moderately (e.g. polynomially) with the number of variables in the admissible solutions. Therefore, the ERIM may avoid the curse of dimensionality, i.e. an exponential growth of the number of parameters. In the second and third parts of the paper, the ERIM and the stochastic approximation algorithms are specialized to the solution of stochastic T-stage single-person and stochastic team decision problems, respectively. Numerical results are given for the optimal management of systems of water reservoirs and for the dynamic routing problem in communication networks. In the reservoirs management application, the ERIM is compared with dynamic programming (DP) using random sampling of the state space. The dynamic routing application considers the presence of several destinations and classes of traffic, which are useful to address quality of service requirements. Here, the ERIM is compared with an algorithm based on the open shortest path first protocol.