Abstract
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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Zidani, H., De Cursi, J.E.S. & Ellaia, R. Numerical approximation of the solution in infinite dimensional global optimization using a representation formula. J Glob Optim 65, 261–281 (2016). https://doi.org/10.1007/s10898-015-0357-5
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DOI: https://doi.org/10.1007/s10898-015-0357-5