On the existence of Pareto solutions for polynomial vector optimization problems

  • Do Sang Kim
  • Tiến-Sơn Phạm
  • Nguyen Van Tuyen
Full Length Paper Series A


We are interested in the existence of Pareto solutions to the vector optimization problem
$$\begin{aligned} \mathrm{Min}_{\,{\mathbb {R}}^m_+} \{f(x) \,|\, x\in {\mathbb {R}}^n\}, \end{aligned}$$
where \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^m\) is a polynomial map. By using the tangency variety of f we first construct a semi-algebraic set of dimension at most \(m - 1\) containing the set of Pareto values of the problem. Then we establish connections between the Palais–Smale conditions, M-tameness, and properness for the map f. Based on these results, we provide some sufficient conditions for the existence of Pareto solutions of the problem. We also introduce a generic class of polynomial vector optimization problems having at least one Pareto solution.


Existence theorems Pareto solutions M-tameness Palais–Smale conditions Properness Polynomial 

Mathematics Subject Classification

90C29 90C30 49J30 



The authors would like to thank the three referees for careful reading and constructive comments. A part of this work was done while the second and third authors were visiting Department of Applied Mathematics, Pukyong National University, Busan, Korea in September 2016. These authors would like to thank the department for hospitality and support during their stay.


  1. 1.
    Bajbar, T., Stein, O.: Coercive polynomials and their Newton polytopes. SIAM J. Optim. 25, 1542–1570 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bao, T.Q., Mordukhovich, B.S.: Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybern. 36, 531–562 (2007)MathSciNetMATHGoogle Scholar
  3. 3.
    Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122, 101–138 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  5. 5.
    Borwein, J.M.: On the existence of Pareto efficient points. Math. Oper. Res. 8, 64–73 (1983)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dias, L.R.G., Ruas, M.A.S., Tibăr, M.: Regularity at infinity of real mappings and a Morse–Sard theorem. J. Topol. 5, 323–340 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dias, L.R.G., Tibăr, M.: Detecting bifurcation values at infinity of real polynomials. Math. Z. 279, 311–319 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dias, L.R.G., Tanabé, S., Tibăr, M.: Toward effective detection of the bifurcation locus of real polynomial maps. Found. Comput. Math. 17, 837–849 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Đinh, S.T., Hà, H.V., Phạm, T.S.: A Frank–Wolfe type theorem and Hölder-type global error bound for generic polynomial systems. Preprint 2012, VIASM. Available online from http://viasm.edu.vn/wp-content/uploads/2012/11/Preprint_1227.pdf
  10. 10.
    Đinh, S.T., Hà, H.V., Phạm, T.S.: A Frank–Wolfe type theorem for nondegenerate polynomial programs. Math. Program. Ser. A. 147, 519–538 (2014)Google Scholar
  11. 11.
    Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)MATHGoogle Scholar
  12. 12.
    Fernando, J.F., Gamboa, J.M.: Polynomial images of \({\mathbb{R}}^{n}\). J. Pure Appl. Algebra 179, 241–254 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fernando, J.F., Gamboa, J.M., Ueno, C.: The open quadrant problem: a topological proof. In: A Mathematical Tribute to Professor José María Montesinos Amilibia, pp. 337–350. Dep. Geom. Topol. Fac. Cien. Mat. UCM, Madrid (2016)Google Scholar
  14. 14.
    Fernando, J.F., Ueno, C.: A short proof for the open quadrant problem. J. Symb. Comput. 79, 57–64 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gutiérrez, C., López, R., Novo, V.: Existence and boundedness of solutions in infinite-dimensional vector optimization problems. J. Optim. Theory Appl. 162, 515–547 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hà, T.X.D.: Variants of the Ekeland variational principle for a set-valued map involving the Clarke normal cone. J. Math. Anal. Appl. 316, 346–356 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hà, T.X.D.: Private communication (2017)Google Scholar
  18. 18.
    Hà, H.V., Phạm, T.S.: An estimation of the number of bifurcation values for real polynomials. Acta Math. Vietnam. 32, 141–153 (2007)MathSciNetMATHGoogle Scholar
  19. 19.
    Hà, H.V., Phạm, T.S.: Genericity in Polynomial Optimization. World Scientific Publishing, Singapore (2017)CrossRefMATHGoogle Scholar
  20. 20.
    Huong, N.T.T., Yao, J.-C., Yen, N.D.: Polynomial vector variational inequalities under polynomial constraints and applications. SIAM J. Optim. 26, 1060–1071 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ioffe, A.: A Sard theorem for tame set-valued mappings. J. Math. Anal. Appl. 335, 882–901 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jahn, J.: Vector Optimization. Theory Applications, and Extensions. Springer, Berlin (2004)MATHGoogle Scholar
  23. 23.
    Jelonek, Z.: Geometry of real polynomial mappings. Math. Z. 239, 321–333 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Jelonek, Z., Kurdyka, K.: Reaching generalized critical values of a polynomial. Math. Z. 276, 557–570 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Jeyakumar, V., Lasserre, J.B., Li, G.: On polynomial optimization over non-compact semi-algebraic sets. J. Optim. Theory Appl. 163, 707–718 (2014)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Khovanskii, A.G.: Newton polyhedra and toroidal varieties. Funct. Anal. Appl. 11, 289–296 (1978)CrossRefMATHGoogle Scholar
  27. 27.
    Kouchnirenko, A.G.: Polyhedres de Newton et nombre de Milnor. Invent. Math. 32, 1–31 (1976)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kurdyka, K., Orro, P., Simon, S.: Semialgebraic Sard theorem for generalized critical values. J. Differ. Geom. 56, 67–92 (2000)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Loi, T.L., Zaharia, A.: Bifurcation sets of functions definable in o-minimal structures. Ill. J. Math. 42, 449–457 (1998)MathSciNetMATHGoogle Scholar
  30. 30.
    Magron, V., Henrion, D., Lasserre, J.B.: Approximating Pareto curves using semidefinite relaxations. Oper. Res. Lett. 42, 432–437 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Magron, V., Henrion, D., Lasserre, J.B.: Semidefinite approximations of projections and polynomial images of semialgebraic sets. SIAM J. Optim. 25, 2143–2164 (2015)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Mawhin, J., Willem, M.: Origin and evolution of the Palais–Smale condition in critical point theory. J. Fixed Point Theory Appl. 7, 265–290 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Motzkin, T.: The arithmetic-geometric inequalities. In: Shisha O. (ed.) Inequalities. Proceeding Symposium on Wright-Patterson Air Force Base, OH, 19–27 Aug 1965, pp. 205–224. Academic Press (1967)Google Scholar
  34. 34.
    Némethi, A., Zaharia, A.: On the bifurcation set of a polynomial function and Newton boundary. Publ. Res. Inst. Math. Sci. 26, 681–689 (1990)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Păunescu, L., Zaharia, A.: On the Łojasiewicz exponent at infinity for polynomial functions. Kodai Math. J. 20, 269–274 (1997)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Rabier, P.J.: Ehresmann fibrations and Palais–Smale conditions for morphisms of Finsler manifolds. Ann. Math. 146, 647–691 (1997)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84, 497–540 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  • Do Sang Kim
    • 1
  • Tiến-Sơn Phạm
    • 2
  • Nguyen Van Tuyen
    • 3
  1. 1.Department of Applied MathematicsPukyong National UniversityBusanKorea
  2. 2.Department of MathematicsUniversity of DalatDa LatVietnam
  3. 3.Department of MathematicsHanoi Pedagogical University 2Xuan Hoa, Phuc YenVietnam

Personalised recommendations