Skip to main content

Equivalence of convex minimization problems over base polytopes

Abstract

This paper considers convex optimization problems over base polytopes of polymatroids. We show that the decomposition algorithm for the separable convex function minimization problems helps us derive simple sufficient conditions for the rationality of optimal solutions, and leads us to some useful properties, including the equivalence of the lexicographically optimal base problem, introduced by Fujishige, and the submodular utility allocation market problem, introduced by Jain and Vazirani. In addition, we deal with a class of non-separable convex objective functions. Moreover, we describe an algorithm for the lexicographically optimal base problem, which is a variant of the Fujishige–Wolfe algorithm.

This is a preview of subscription content, access via your institution.

References

  1. Chudak, F.A., Nagano, K.: Efficient solutions to relaxations of combinatorial problems with submodular penalties via the Lovász extension and non-smooth convex optimization. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 79–88 (2007)

  2. Dutta B.: The egalitarian solution and reduced game properties in convex games. Int. J. Game Theory 19, 153–169 (1990)

    MATH  Article  Google Scholar 

  3. Dutta B., Ray D.: A Concept of egalitarianism under participation constraints. Econometrica 57, 615–635 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  4. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanai, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and Their Applications. Gordon and Breach, New York, pp. 69–87 (1970)

  5. Fleischer L., Iwata S.: A push-relabel framework for submodular function minimization and applications to parametric optimization. Discrete Appl. Math. 131, 311–322 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  6. Fujishige S.: Lexicographically optimal base of a polymatroid with respect to a weight vector. Math. Oper. Res. 5, 186–196 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  7. Fujishige S.: Submodular systems and related topics. Math. Program. Study 22, 113–131 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  8. Fujishige S.: Submodular Functions and Optimization, 2nd edn. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  9. Fujishige, S., Hayashi, T., Isotani, S.: The minimum-norm-point algorithm applied to submodular function minimization and linear programming. Research Institute for Mathematical Sciences Preprint RIMS- 1571, Kyoto University, Kyoto, Japan (2006)

  10. Gallo G., Grigoriadis M.D., Tarjan R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18, 30–55 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  11. Groenevelt H.: Two algorithms for maximizing a separable concave function over a polymatroid feasible region. Eur. J. Oper. Res. 54, 227–236 (1991)

    MATH  Article  Google Scholar 

  12. Hayrapetyan, A., Swamy, C., Tardos E.: Network design for information networks. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 933–942 (2005)

  13. Hochbaum D.S.: Lower and upper bounds for the allocation problem and other nonlinear optimization problems. Math. Oper. Res. 19, 390–409 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  14. Iwata S.: Submodular function minimization. Math. Program. 112, 45–64 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  15. Iwata S., Fleischer L., Fujishige S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  16. Iwata, S., Nagano, K.: Submodular function minimization under covering constraints. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 671–680 (2009)

  17. Jain, K., Vazirani, V.V.: Eisenberg–Gale markets: algorithms and structural properties. In: Proceedings of the 39th ACM Symposium on Theory of Computing, pp. 364–373 (2007)

  18. Lovász, L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming—The State of the Art, pp. 235–257. Springer, Berlin (1983)

  19. Maruyama, F.: A unified study on problems in information theory via polymatroids. Graduation Thesis, University of Tokyo, Japan (1978) (In Japanese.)

  20. McCormick, S.T.: Submodular function minimization. In: Aardal, K., Nemhauser, G.L., Weismantel, R. (eds.) Discrete Optimization (Handbooks in Operations Research and Management Science, vol. 12), Chapter 7, pp. 321–391. Elsevier, Amsterdam (2005)

  21. Megiddo N.: Optimal flows in networks with multiple sources and sinks. Math. Program. 7, 97–107 (1974)

    MathSciNet  MATH  Article  Google Scholar 

  22. Murota K.: Note on the universal bases of a pair of polymatroids. J. Oper. Res. Soc. Jpn. 31, 565–573 (1988)

    MathSciNet  MATH  Google Scholar 

  23. Nagano K.: A strongly polynomial algorithm for line search in submodular polyhedra. Discrete Optim. 4, 349–359 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  24. Nagano, K.: A faster parametric submodular function minimization algorithm and applications. Technical Report METR 2007-43, University of Tokyo (2007)

  25. Orlin J.B.: A faster strongly polynomial time algorithm for submodular function minimization. Math. Program. 118, 237–251 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  26. Schrijver A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory (B) 80, 346–355 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  27. Schrijver, A.: Combinatorial Optimization—Polyhedra and Efficiency. Springer, Berlin (2003)

  28. Sharma, Y., Swamy, C., Williamson, D.P.: Approximation algorithms for prize-collecting forest problems with submodular penalty functions. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1275–1284 (2007)

  29. Wolfe P.: Finding the nearest point in A polytope. Math. Program. 11, 128–149 (1976)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kiyohito Nagano.

Additional information

This research is supported by Aihara Project, the FIRST program from JSPS, and KAKENHI (22700007). A part of this work has appeared in Proceedings of the 12th IPCO Conference, Lecture Notes in Computer Science, LNCS 4513, Springer (2007), pp. 252–266.

About this article

Cite this article

Nagano, K., Aihara, K. Equivalence of convex minimization problems over base polytopes. Japan J. Indust. Appl. Math. 29, 519–534 (2012). https://doi.org/10.1007/s13160-012-0083-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-012-0083-z

Keywords

  • Convex optimization
  • Submodular function
  • Combinatorial optimization

Mathematics Subject Classification

  • 90C25
  • 52B40
  • 90C27