Skip to main content
SpringerLink
Log in

Solving a Tropical Optimization Problem via Matrix Sparsification

  • Conference paper
  • First Online:
Book cover Relational and Algebraic Methods in Computer Science (RAMICS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9348))

Abstract

An optimization problem, which arises in various applications as that of minimizing the span seminorm, is considered in the framework of tropical mathematics. The problem is to minimize a nonlinear function defined on vectors over an idempotent semifield, and calculated by means of multiplicative conjugate transposition. We find the minimum of the function, and give a partial solution which explicitly represents a subset of solution vectors. We characterize all solutions by a system of simultaneous equation and inequality, and exploit this characterization to investigate properties of the solutions. A matrix sparsification technique is developed to extend the partial solution to a wider solution subset, and then to a complete solution described as a family of subsets. We offer a backtracking procedure that generates all members of the family, and derive an explicit representation for the complete solution. Numerical examples and graphical illustrations of the results are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akian, M., Bapat, R., Gaubert, S.: Max-plus algebra. In: Hogben, L. (ed.) Handbook of Linear Algebra. Discrete Mathematics and Its Applications, pp. 25–1–25–17. Taylor and Francis, Boca Raton (2007)

    Google Scholar 

  2. Butkovič, P., Tam, K.P.: On some properties of the image set of a max-linear mapping. In: Litvinov, G.L., Sergeev, S.N. (eds.) Tropical and Idempotent Mathematics, Contemp. Math., vol. 495, pp. 115–126. AMS (2009)

    Google Scholar 

  3. Butkovič, P.: Max-linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London (2010)

    Google Scholar 

  4. Cuninghame-Green, R.A.: Describing industrial processes with interference and approximating their steady-state behaviour. Oper. Res. Quart. 13, 95–100 (1962)

    Article  Google Scholar 

  5. Cuninghame-Green, R.A., Butkovič, P.: Bases in max-algebra. Linear Algebra Appl. 389, 107–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cuninghame-Green, R.: Minimax Algebra. Lecture Notes in Economics and Mathematical Systems, vol. 166. Springer, Berlin (1979)

    MATH  Google Scholar 

  7. Giffler, B.: Scheduling general production systems using schedule algebra. Naval Res. Logist. Quart. 10, 237–255 (1963)

    Article  Google Scholar 

  8. Golan, J.S.: Semirings and Affine Equations Over Them: Theory and Applications, Mathematics and Its Applications, vol. 556. Springer, New York (2003)

    Google Scholar 

  9. Gondran, M., Minoux, M.: Graphs, Dioids and Semirings: New Models and Algorithms, Operations Research/Computer Science Interfaces, vol. 41. Springer, New York (2008)

    Google Scholar 

  10. Heidergott, B., Olsder, G.J., van der Woude, J.: Max-plus at Work: Modeling and Analysis of Synchronized Systems. Princeton Series in Applied Mathematics. Princeton Univ. Press, Princeton (2006)

    MATH  Google Scholar 

  11. Hoffman, A.J.: On abstract dual linear programs. Naval Res. Logist. Quart. 10, 369–373 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kolokoltsov, V.N., Maslov, V.P.: Idempotent Analysis and Its Applications, Mathematics and Its Applications, vol. 401. Kluwer Acad. Publ, Dordrecht (1997)

    Book  MATH  Google Scholar 

  13. Korbut, A.A.: Extremal spaces. Soviet Math. Dokl. 6, 1358–1361 (1965)

    MathSciNet  MATH  Google Scholar 

  14. Krivulin, N.K.: On solution of linear vector equations in idempotent algebra. In: Chirkov, M.K. (ed.) Mathematical Models. Theory and Applications, Issue 5, pp. 105–113. Saint Petersburg State Univ., St. Petersburg (2004) (in Russian)

    Google Scholar 

  15. Krivulin, N.K.: Methods of Idempotent Algebra for Problems in Modeling and Analysis of Complex Systems. Saint Petersburg Univ. Press, St. Petersburg (2009) (in Russian)

    Google Scholar 

  16. Krivulin, N.: A solution of a tropical linear vector equation. In: Yenuri, S. (ed.) Advances in Computer Science. Recent Advances in Computer Engineering Series, vol. 5, pp. 244–249. WSEAS Press (2012)

    Google Scholar 

  17. Krivulin, N.: Explicit solution of a tropical optimization problem with application to project scheduling. In: Biolek, D., Walter, H., Utu, I., von Lucken, C. (eds.) Mathematical Methods and Optimization Techniques in Engineering, pp. 39–45. WSEAS Press (2013)

    Google Scholar 

  18. Krivulin, N.: Complete solution of a constrained tropical optimization problem with application to location analysis. In: Höfner, P., Jipsen, P., Kahl, W., Müller, M.E. (eds.) RAMiCS 2014. LNCS, vol. 8428, pp. 362–378. Springer, Cham (2014)

    Chapter  Google Scholar 

  19. Krivulin, N.: Extremal properties of tropical eigenvalues and solutions to tropical optimization problems. Linear Algebra Appl. 468, 211–232 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Krivulin, N.: A multidimensional tropical optimization problem with nonlinear objective function and linear constraints. Optimization 64, 1107–1129 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Litvinov, G.: Maslov dequantization, idempotent and tropical mathematics: A brief introduction. J. Math. Sci (N. Y.) 140, 426–444 (2007)

    Article  MathSciNet  Google Scholar 

  22. Pandit, S.N.N.: A new matrix calculus. J. SIAM 9, 632–639 (1961)

    MathSciNet  MATH  Google Scholar 

  23. Peteanu, V.: An algebra of the optimal path in networks. Mathematica 9(2), 335–342 (1967)

    MathSciNet  MATH  Google Scholar 

  24. Romanovskiĭ, I.V.: Asymptotic behavior of dynamic programming processes with a continuous set of states. Soviet Math. Dokl. 5, 1684–1687 (1964)

    Google Scholar 

  25. Speyer, D., Sturmfels, B.: Tropical mathematics. Math. Mag. 82, 163–173 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tam, K.P.: Optimizing and Approximating Eigenvectors in Max-Algebra. PhD thesis, The University of Birmingham, Birmingham (2010)

    Google Scholar 

  27. T’kindt, V., Billaut, J.-C.: Multicriteria Scheduling: Theory, Models and Algorithms. Springer, Berlin (2006)

    MATH  Google Scholar 

  28. Vorob’ev, N.N.: The extremal matrix algebra. Soviet Math. Dokl. 4, 1220–1223 (1963)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, Saint Petersburg State University, Universitetsky Ave. 28, 198504, St. Petersburg, Russia

    Nikolai Krivulin

Authors
  1. Nikolai Krivulin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolai Krivulin .

Editor information

Editors and Affiliations

  1. McMaster University, Hamilton, Ontario, Canada

    Wolfram Kahl

  2. Brock University, St. Catharines, Ontario, Canada

    Michael Winter

  3. Universidade do Minho, Braga, Portugal

    José Oliveira

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Krivulin, N. (2015). Solving a Tropical Optimization Problem via Matrix Sparsification. In: Kahl, W., Winter, M., Oliveira, J. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2015. Lecture Notes in Computer Science(), vol 9348. Springer, Cham. https://doi.org/10.1007/978-3-319-24704-5_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-24704-5_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-24703-8

  • Online ISBN: 978-3-319-24704-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation