Advertisement

Mathematical Methods of Operations Research

, Volume 84, Issue 2, pp 359–387 | Cite as

Relationships between constrained and unconstrained multi-objective optimization and application in location theory

  • Christian Günther
  • Christiane Tammer
Article

Abstract

This article deals with constrained multi-objective optimization problems. The main purpose of the article is to investigate relationships between constrained and unconstrained multi-objective optimization problems. Under suitable assumptions (e.g., generalized convexity assumptions) we derive a characterization of the set of (strictly, weakly) efficient solutions of a constrained multi-objective optimization problem using characterizations of the sets of (strictly, weakly) efficient solutions of unconstrained multi-objective optimization problems. We demonstrate the usefulness of the results by applying it on constrained multi-objective location problems. Using our new results we show that special classes of constrained multi-objective location problems (e.g., point-objective location problems, Weber location problems, center location problems) can be completely solved with the help of algorithms for the unconstrained case.

Keywords

Multi-objective optimization Pareto efficiency Constrained optimization Unconstrained optimization Generalized convexity Location theory Gauges 

Notes

Acknowledgments

The authors wish to thank the anonymous referees for their valuable comments.

References

  1. Alzorba S, Günther C, Popovici N (2015) A special class of extended multicriteria location problems. Optimization 64(5):1305–1320MathSciNetCrossRefzbMATHGoogle Scholar
  2. Alzorba S, Günther C, Popovici N, Tammer C (2016) A new algorithm for solving planar multi-objective location problems involving the Manhattan norm. Optimization Online. http://www.optimization-online.org/DB_HTML/2016/01/5305.html (submitted)
  3. Apetrii M, Durea M, Strugariu R (2014) A new penalization tool in scalar and vector optimization. Nonlinear Anal 107:22–33MathSciNetCrossRefzbMATHGoogle Scholar
  4. Benson HP (1998) An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J Global Optim 13:1–24MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cambini A, Martein L (2009) Generalized convexity and optimization: theory and applications. Springer, BerlinzbMATHGoogle Scholar
  6. Carrizosa E, Conde E, Fernandez FR, Puerto J (1993) Efficiency in Euclidean constrained location problems. Oper Res Lett 14(5):291–295MathSciNetCrossRefzbMATHGoogle Scholar
  7. Carrizosa E, Conde E, Fernandez FR, Puerto J (1995) Planar point-objective location problems with nonconvex constraints: a geometrical construction. J Global Optim 6:77–86MathSciNetCrossRefzbMATHGoogle Scholar
  8. Carrizosa E, Plastria F (1996) A characterization of efficient points in constrained location problems with regional demand. Oper Res Lett 19(3):129–134MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chalmet L, Francis RL, Kolen A (1981) Finding efficient solutions for rectilinear distance location problems efficiently. Eur J Oper Res 6:117–124MathSciNetCrossRefzbMATHGoogle Scholar
  10. Durier R (1990) On Pareto optima and the Fermat–Weber problem. Math Program 47:65–79MathSciNetCrossRefzbMATHGoogle Scholar
  11. Durier R, Michelot C (1986) Sets of efficient points in a normed space. J Math Anal Appl 117:506–528MathSciNetCrossRefzbMATHGoogle Scholar
  12. Ehrgott M (2005) Multicriteria optimization, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  13. Fliege J (2007) The effects of adding objectives to an optimisation problem on the solution set. Oper Res Lett 35(6):782–790MathSciNetCrossRefzbMATHGoogle Scholar
  14. Gerth C, Pöhler K (1988) Dualität und algorithmische Anwendung beim vektoriellen Standortproblem. Optimization 19:491–512MathSciNetCrossRefzbMATHGoogle Scholar
  15. Giorgi G, Guerraggio A, Thierfelder J (2004) Mathematics of optimization: smooth and nonsmooth case. Elsevier, AmsterdamzbMATHGoogle Scholar
  16. Günther C, Hillmann M, Tammer C, Winkler B (2015) Facility location optimizer (FLO)—a tool for solving location problems. http://www.project-flo.de
  17. Hiriart-Urruty JB, Lemaréchal C (2001) Fundamentals of convex analysis. Springer, BerlinCrossRefzbMATHGoogle Scholar
  18. Jahn J (2011) Vector optimization: theory, applications, and extensions, 2nd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  19. Kaiser M (2015) Spatial uncertainties in continuous location problems. Dissertation, Bergische Universität WuppertalGoogle Scholar
  20. Klamroth K, Tind J (2007) Constrained optimization using multiple objective programming. J Glob Optim 37(3):325–355MathSciNetCrossRefzbMATHGoogle Scholar
  21. Ndiayea M, Michelot C (1998) Efficiency in constrained continuous location. Eur J Oper Res 104(2):288–298CrossRefzbMATHGoogle Scholar
  22. Nickel S (1995) Discretization of planar location problems. Shaker, AachenGoogle Scholar
  23. Nickel S, Puerto J, Rodríguez-Chía AM, Weissler A (2005) Multicriteria planar ordered median problems. J Optim Theory Appl 126(3):657–683MathSciNetCrossRefzbMATHGoogle Scholar
  24. Nickel S, Puerto J, Rodríguez-Chía AM (2015) Location problems with multiple criteria. In: Laporte G, Nickel S, Saldanha da Gama F (eds) Location science. Springer, Berlin, pp 205–247Google Scholar
  25. Pelegrín B, Fernández FR (1988) Determination of efficient points in multiple-objective location problems. Nav Res Logist 35:697–705MathSciNetCrossRefzbMATHGoogle Scholar
  26. Popovici N (2005) Pareto reducible multicriteria optimization problems. Optimization 54:253–263MathSciNetCrossRefzbMATHGoogle Scholar
  27. Puerto J, Fernández FR (1999) Multi-criteria minisum facility location problems. J Multicrit Decis Anal 18:268–280CrossRefzbMATHGoogle Scholar
  28. Puerto J, Rodríguez-Chía AM (2002) Geometrical description of the weakly efficient solution set for multicriteria location problems. Ann Oper Res 111:181–196MathSciNetCrossRefzbMATHGoogle Scholar
  29. Puerto J, Rodríguez-Chía AM (2008) Quasiconvex constrained multicriteria continuous location problems: structure of nondominated solution sets. Comput Oper Res 35(3):750–765MathSciNetCrossRefzbMATHGoogle Scholar
  30. Thisse JF, Ward JE, Wendell RE (1984) Some properties of location problems with block and round norms. Oper Res 32(6):1309–1327MathSciNetCrossRefzbMATHGoogle Scholar
  31. Wendell RE, Hurter AP, Lowe TJ (1977) Efficient points in location problems. AIIE Trans 9(3):238–246MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Faculty of Natural Sciences II, Institute for MathematicsMartin Luther University Halle-WittenbergHalle (Saale)Germany

Personalised recommendations