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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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| Published in: | Journal of global optimization 2016-06, Vol.65 (2), p.261-281 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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| container_end_page | 281 |
| container_issue | 2 |
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| container_title | Journal of global optimization |
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| creator | Zidani, H. De Cursi, J. E. Souza Ellaia, R. |
| description | 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. |
| doi_str_mv | 10.1007/s10898-015-0357-5 |
| format | article |
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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
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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.</description><subject>Algorithms</subject><subject>Approximation</subject><subject>Banach spaces</subject><subject>Calculus</subject><subject>Calculus of variations</subject><subject>Computer Science</subject><subject>Convexity</subject><subject>Hilbert space</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operations Research/Decision Theory</subject><subject>Optimization</subject><subject>Random variables</subject><subject>Real Functions</subject><subject>Representations</subject><subject>Strategy</subject><subject>Studies</subject><issn>0925-5001</issn><issn>1573-2916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><sourceid>M0C</sourceid><recordid>eNp1kcFq3TAQRUVpIK9JPyA7QzfdOJVkjyQvQ0jTQEg27VrI0uhVwbZcyYYmX1-5ziIUigSDLveImbmEXDB6ySiVXzKjqlM1ZVDTBmQN78iBgWxq3jHxnhxox6EGStkp-ZDzE6W0U8APJD2sI6ZgzVCZeU7xdxjNEuJURV8tP7HKcVj_vsN2fZjCgpULI065qIU6DrEvJc5LGMPLzq45TMfKVAnnhBmnZZd9TOM6mHNy4s2Q8eNrPSM_vt58v_5W3z_e3l1f3de2UWqpHW9917a0906AdBZBdMZJ3sqGcedRObDGALW9c0VRTEiBDqAXoucWoDkjn_d_y1i_VsyLHkO2OAxmwrhmzQpCW9oIWayf_rE-xTWV8YpLKgnApdxcl7vraAbUZRtxScaW43AMNk7oQ9GvJFMtaxiIArAdsCnmnNDrOZX9pmfNqN5i03tsusSmt9j01jXfmVy80xHTm1b-C_0B_Macsw</recordid><startdate>20160601</startdate><enddate>20160601</enddate><creator>Zidani, H.</creator><creator>De Cursi, J. 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Souza ; Ellaia, R.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c388t-d24f9440bfd657dce569ad7247312dfe8d5caa50cbdd31281676ed55b66b2c553</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Algorithms</topic><topic>Approximation</topic><topic>Banach spaces</topic><topic>Calculus</topic><topic>Calculus of variations</topic><topic>Computer Science</topic><topic>Convexity</topic><topic>Hilbert space</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operations Research/Decision Theory</topic><topic>Optimization</topic><topic>Random variables</topic><topic>Real Functions</topic><topic>Representations</topic><topic>Strategy</topic><topic>Studies</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Zidani, H.</creatorcontrib><creatorcontrib>De Cursi, J. 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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.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10898-015-0357-5</doi><tpages>21</tpages></addata></record> |
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| source | ABI/INFORM global; Springer Nature |
| 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 |
| title | Numerical approximation of the solution in infinite dimensional global optimization using a representation formula |
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