Numerical approximation of the solution in infinite dimensional global optimization using a representation formula

A non convex optimization problem, involving a regular functional J , on a closed and bounded subset S of a separable Hilbert space V is here considered. No convexity assumption is introduced. The solutions are represented by using a closed formula involving means of convenient random variables, ana...

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Bibliographic Details
Published in:Journal of global optimization 2016-06, Vol.65 (2), p.261-281
Main Authors: Zidani, H., De Cursi, J. E. Souza, Ellaia, R.
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Banach spaces
Calculus
Calculus of variations
Computer Science
Convexity
Hilbert space
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Random variables
Real Functions
Representations
Strategy
Studies
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Summary:A non convex optimization problem, involving a regular functional J , on a closed and bounded subset S of a separable Hilbert space V is here considered. No convexity assumption is introduced. The solutions are represented by using a closed formula involving means of convenient random variables, analogous to Pincus (Oper Res 16(3):690–694, 1968 ). The representation suggests a numerical method based on the generation of samples in order to estimate the means. Three strategies for the implementation are examined, with the originality that they do not involve a priori finite dimensional approximation of the solution and consider a hilbertian basis or enumerable dense family of V . The results may be improved on a finite-dimensional subspace by an optimization procedure, in order to get higher-quality solutions. Numerical examples involving both classical situation and an engineering application issued from calculus of variations are presented and establish that the method is effective to calculate.
ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-015-0357-5