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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Vydáno v:Journal of global optimization Ročník 65; číslo 2; s. 261 - 281
Hlavní autoři: Zidani, H., De Cursi, J. E. Souza, Ellaia, R.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.06.2016
Springer
Springer Nature B.V
Témata:
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
Representation formula
Variational calculus
49M30
Nonconvex global optimization
ISSN:0925-5001, 1573-2916
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Shrnutí: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.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-015-0357-5