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 Zh. Vychisl. Mat. Mat. Fiz., 2009, Volume 49, Number 12, Pages 2103–2113 (Mi zvmmf4792)

Study of an adaptive single-phase method for approximating the multidimensional Pareto frontier in nonlinear systems

G. K. Kamenev

Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

Abstract: The problem of approximating the Pareto frontier (nondominated frontier) of the feasible set of criteria vectors in nonlinear multicriteria optimization problems is considered. The problem is solved by approximating the Edgeworth–Pareto hull (EPH), i.e., the maximum set with the same Pareto frontier as the original feasible set of criteria vectors. An EPH approximation method is studied that is based on the statistical accuracy estimation of the current approximation and on adaptive supplement of a metric net whose EPH approximates the desired set. The convergence of the method is proved, estimates for the convergence rate are obtained, and the efficiency of the method is studied in the case of a compact feasible set and continuous criteria functions. It is shown that the convergence rate of the method with respect to the number $k$ of iterations is no lower than $o(k^{1/\overline{\mathrm{dm}}}Y)$, where $\overline{\mathrm{dm}}Y$ is the upper metric dimension of the feasible set of criteria vectors.

Key words: multicriteria optimization, Pareto frontier, Edgeworth–Pareto hull, approximation method, statistical estimates, adaptive methods, convergence rate, metric dimension.

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English version:
Computational Mathematics and Mathematical Physics, 2009, 49:12, 2006–2016

Bibliographic databases:

UDC: 519.626

Citation: G. K. Kamenev, “Study of an adaptive single-phase method for approximating the multidimensional Pareto frontier in nonlinear systems”, Zh. Vychisl. Mat. Mat. Fiz., 49:12 (2009), 2103–2113; Comput. Math. Math. Phys., 49:12 (2009), 2006–2016

Citation in format AMSBIB
\Bibitem{Kam09} \by G.~K.~Kamenev \paper Study of an adaptive single-phase method for approximating the multidimensional Pareto frontier in nonlinear systems \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2009 \vol 49 \issue 12 \pages 2103--2113 \mathnet{http://mi.mathnet.ru/zvmmf4792} \elib{http://elibrary.ru/item.asp?id=12989093} \transl \jour Comput. Math. Math. Phys. \yr 2009 \vol 49 \issue 12 \pages 2006--2016 \crossref{https://doi.org/10.1134/S0965542509120021} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000272968700002} \elib{http://elibrary.ru/item.asp?id=15296297} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-74549209096} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. E. Berezkin, G. K. Kamenev, “Convergence analysis of two-phase methods for approximating the Edgeworth–Pareto hull in nonlinear multicriteria optimization problems”, Comput. Math. Math. Phys., 52:6 (2012), 846–854
2. Hartikainen M., Miettinen K., Wiecek M.M., “PAINT: Pareto front interpolation for nonlinear multiobjective optimization”, Comput. Optim. Appl., 52:3 (2012), 845–867
3. G. K. Kamenev, “Study of convergence rate and efficiency of two-phase methods for approximating the Edgeworth–Pareto hull”, Comput. Math. Math. Phys., 53:4 (2013), 375–385
4. A. I. Ryabikov, “Ersatz function method for minimizing a finite-valued function on a compact set”, Comput. Math. Math. Phys., 54:2 (2014), 206–218
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