A Newton-Like Method for Variable Order Vector Optimization Problems

  • Glaydston de Carvalho Bento
  • Gemayqzel Bouza Allende
  • Yuri Rafael Leite Pereira


A Newton approach is proposed for solving variable order smooth constrained vector optimization problems. The concept of strong convexity is presented, and its properties are analyzed. It is thus obtained that the Newton direction is well defined and that the algorithm converges. Moreover, the rate of convergence is obtained under ordering structures satisfying a mild hypothesis.


Descent direction Efficient points K-strong convexity Newton method Variable order vector optimization 

Mathematics Subject Classification

90C29 90C30 26B25 65K05 



The first author was partially supported by CNPq Grants 458479/2014-4 and 3122077/2014-9, the second author was partially supported by CAPES/FAPEG 10/2013 and 08/2014, and the third author was partially supported by CAPES, CAPES/MES/Cuba Project 226/2012 Optimization and Applications. The second author was also partially supported by Alexander von Humboldt Foundation during his stay at Martin Luther University, where part of this research was carried out.


  1. 1.
    Bergstresser, K., Yu, P.L.: Domination structures and multicriteria problems in \(N\)-person games. Theory Decis. 8(1), 5–48 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Yu, P.L.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14, 319–377 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Jahn, J.: Vector optimization. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    Ehrgott, M.: Multicriteria optimization, 2nd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  5. 5.
    Baatar, D., Wiecek, M.M.: Advancing equitability in multiobjective programming. Comput. Math. Appl. 52(1–2), 225–234 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Engau, A.: Variable preference modeling with ideal-symmetric convex cones. J. Global Optim. 42(2), 295–311 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wiecek, M.M.: Advances in cone-based preference modeling for decision making with multiple criteria. Decis. Mak. Manuf. Serv. 1(1–2), 153–173 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Eichfelder, G.: Optimal elements in vector optimization with a variable ordering structure. J. Optim. Theory Appl. 151(2), 217–240 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eichfelder, G., Ha, T.X.D.: Optimality conditions for vector optimization problems with variable ordering structures. Optimization 62(5), 597–627 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eichfelder, G.: Variable ordering structures in vector optimization. Vector Optimization. Springer, Heidelberg (2014)CrossRefzbMATHGoogle Scholar
  11. 11.
    Soleimani, B.: Characterization of approximate solutions of vector optimization problems with a variable order structure. J. Optim. Theory Appl. 162(2), 605–632 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Soleimani, B., Tammer, C.: Concepts for approximate solutions of vector optimization problems with variable order structures. Vietnam J. Math. 42(4), 543–566 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bao, T.Q., Mordukhovich, B.S., Soubeyran, A.: Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality. Set Valued Var. Anal. 23(2), 375–398 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bao, T.Q., Mordukhovich, B.S., Soubeyran, A.: Variational analysis in psychological modeling. J. Optim. Theory Appl. 164(1), 290–315 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Li, X.B., Lin, Z., Peng, Z.Y.: Convergence for vector optimization problems with variable ordering structure. Optimization 65(8), 1615–1627 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bello Cruz, J.Y., Bouza Allende, G.: A steepest descent-like method for variable order vector optimization problems. J. Optim. Theory Appl. 162(2), 371–391 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fliege, J., Graña Drummond, L.M., Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20(2), 602–626 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Graña Drummond, L.M., Raupp, F.M.P., Svaiter, B.F.: A quadratically convergent Newton method for vector optimization. Optimization 63(5), 661–677 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Peressini, A.L.: Ordered topological vector spaces. Harper & Row Publishers, New York, London (1967)zbMATHGoogle Scholar
  20. 20.
    Bertsekas, D.P.: Convex analysis and optimization. Athena Scientific, Belmont (2003)zbMATHGoogle Scholar
  21. 21.
    Miettinen, K.: Nonlinear multiobjective optimization. International Series in Operations Research & Management Science, vol. 12. Kluwer Academic Publishers, Boston (1999)zbMATHGoogle Scholar
  22. 22.
    Carrizo, G.A., Lotito, P.A., Maciel, M.C.: Trust region globalization strategy for the nonconvex unconstrained multiobjective optimization problem. Math. Program. 159(1–2, Ser A), 339–369 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Villacorta, K.D.V., Oliveira, P.R., Soubeyran, A.: A trust-region method for unconstrained multiobjective problems with applications in satisficing processes. J. Optim. Theory Appl. 160(3), 865–889 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Glaydston de Carvalho Bento
    • 1
  • Gemayqzel Bouza Allende
    • 2
  • Yuri Rafael Leite Pereira
    • 3
  1. 1.IMEUniversidade Federal de GoiásGoiâniaBrazil
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of HabanaHabanaCuba
  3. 3.DMUniversidade Federal do PiauíTeresinaBrazil

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