Hausdorff methods for approximating the convex Edgeworth–Pareto hull in integer problems with monotone objectives

  • A. I. Pospelov


Adaptive methods for the polyhedral approximation of the convex Edgeworth–Pareto hull in multiobjective monotone integer optimization problems are proposed and studied. For these methods, theoretical convergence rate estimates with respect to the number of vertices are obtained. The estimates coincide in order with those for filling and augmentation H-methods intended for the approximation of nonsmooth convex compact bodies.


adaptive methods polyhedral approximation convergence rate multiobjective optimization Pareto frontier integer optimization 


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  1. 1.
    I. Kh. Sigal and I. I. Melamed, “An investigation of linear convolution of criteria in multicriteria discrete programming,” Comput. Math. Math. Phys. 35 (8), 1009–1017 (1995).MathSciNetzbMATHGoogle Scholar
  2. 2.
    I. Kh. Sigal and I. I. Melamed, Theory and Algorithms for Solving Multiobjective Combinatorial Optimization Problems (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 1996) [in Russian].zbMATHGoogle Scholar
  3. 3.
    I. Kh. Sigal, “Algorithms for solving the two-criterion large-scale travelling salesman problem,” Comput. Math. Math. Phys. 34 (1), 33–43 (1994).MathSciNetzbMATHGoogle Scholar
  4. 4.
    M. Ehrgott and X. Gandibleux, “An annotated bibliography of multiobjective combinatorial optimization,” Tech. Rep. No. 62 (Wirtschaftsmathematik, 2000).Google Scholar
  5. 5.
    I. I. Melamed, I. Kh. Sigal, and N. Yu. Vladimirova, “Study of the Linear parametrization of criteria in the bicriteria knapsack problem,” Comput. Math. Math. Phys. 39 (5), 721–726 (1999).MathSciNetzbMATHGoogle Scholar
  6. 6.
    X. Gandibleux and K. Klamroth, “Cardinality bounds for multiobjective knapsack problems,” Tech. Rep. (Inst. of Appl. Math., Univ. Erlangen-Nuremberg, Germany, 2006).Google Scholar
  7. 7.
    H. W. Hamacher, C. R. Pedersen, and S. Ruzika, “Finding representative systems for discrete bicriteria optimization problems by box algorithms,” Operat. Res. Lett. 35, 336–344 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D. Schweigert, Vector-weighted matchings, Combinatorics Advances, Ed. by C. Colbourn and E. Mahmoodian (Kluwer Academic, Dordrecht, 1995), pp. 267–276.Google Scholar
  9. 9.
    K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms (Wiley, Chichester, 2001).zbMATHGoogle Scholar
  10. 10.
    M. V. Evdokimov, V. G. Mednitskii, and I. Kh. Sigal, “Bicriteria problem of recomposition of a production system,” J. Computer Syst. Sci. Int. 40 (5), 764–769 (2001).MathSciNetzbMATHGoogle Scholar
  11. 11.
    A. I. Pospelov, “Approximating the convex Edgeworth–Pareto hull in integer multi-objective problems with monotone criteria,” Comput. Math. Math. Phys. 49 (10), 1686–1699 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    G. K. Kamenev and A. I. Pospelov, “Polyhedral approximation of convex compact bodies by filling methods,” Comput. Math. Math. Phys. 52 (5), 680–690 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    G. K. Kamenev, Optimal Adaptive Methods for Polyhedral Approximation of Convex Bodies (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 2007) [in Russian].zbMATHGoogle Scholar
  14. 14.
    K. Leichtweiss, Konvexe Mengen (Springer-Verlag, Berlin, 1980; Nauka, Moscow, 1985).zbMATHGoogle Scholar
  15. 15.
    P. M. Gruber, Convex and Discrete Geometry (Springer, New York, 2007).zbMATHGoogle Scholar
  16. 16.
    P. S. Aleksandrov, A. I. Markushevich, and A. Ya. Khinchin, Encyclopedia of Elementary Mathematics, Book 5: Geometry (Nauka, Moscow, 1966) [in Russian].Google Scholar
  17. 17.
    D. V. Gusev and A. V. Lotov, “Models, systems, and decisions,” Operations Research, Ed. by Yu. P. Ivanilov (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 1994) [in Russian].Google Scholar
  18. 18.
    V. A. Bushenkov, D. I. Gusev, and G. K. Kamenev, “Visualization of the Pareto set in a multidimensional decision making problem,” Dokl. Akad. Nauk 335 (5), 567–569 (1994).zbMATHGoogle Scholar
  19. 19.
    A. V. Lotov, V. A. Bushenkov, and G. K. Kamenev, Interactive Decision Maps: Approximation and Visualization of Pareto Frontier (Kluwer, Boston, 2004).CrossRefzbMATHGoogle Scholar
  20. 20.
    F. P. Preparata and M. I. Shamos, Computational Geometry: An Introduction (Springer-Verlag, New York, 1985; Mir, Moscow, 1989).zbMATHGoogle Scholar
  21. 21.
    A. V. Lotov, L. V. Bourmistrova, R. V. Efremov, V. A. Bushenkov, N. A. Brainin, and A. L. Buber, “Experience of model integration and Pareto frontier visualization in the search for preferable water quality strategies,” Environ. Model. Software 20 (2), 243–260 (2005).CrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Kharkevich Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia
  2. 2.DATADVANCE, Pokrovskii bul. 3/1BMoscowRussia

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