Journal of Global Optimization

, Volume 69, Issue 4, pp 823–845 | Cite as

Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap

  • Fabián Flores-Bazán
  • William Echegaray
  • Fernando Flores-Bazán
  • Eladio Ocaña
Article
  • 154 Downloads

Abstract

Primal or dual strong-duality (or min-sup, inf-max duality) in nonconvex optimization is revisited in view of recent literature on the subject, establishing, in particular, new characterizations for the second case. This gives rise to a new class of quasiconvex problems having zero duality gap or closedness of images of vector mappings associated to those problems. Such conditions are described for the classes of linear fractional functions and that of quadratic ones. In addition, some applications to nonconvex quadratic optimization problems under a single inequality or equality constraint, are presented, providing new results for the fulfillment of zero duality gap or dual strong-duality.

Keywords

Zero duality gap Strong duality Linear fractional programming Quasiconvex programming Quadratic programming 

Mathematics Subject Classification

90C20 90C46 49N10 49N15 52A10 

References

  1. 1.
    Auslender, A.: Existence of optimal solutions and duality results under weak conditions. Math. Program. A 88, 45–59 (2000)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York (2003)MATHGoogle Scholar
  3. 3.
    Beck, A.: On the convexity of a class of quadratic mappings and its application to the problem of finding the smallest ball enclosing a given intersection of balls. J. Global Optim. 39, 113–126 (2007)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Borwein, J.M., Burachik, R., Yao, L.: Conditions for zero duality gap in convex programming. J. Convex Nonlinear Anal. 15, 167–190 (2014)MATHMathSciNetGoogle Scholar
  5. 5.
    Bot, R.I.: Conjugate Duality in Convex Optimization. Springer, Berlin (2010)CrossRefMATHGoogle Scholar
  6. 6.
    Bot, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Optim. Theory Appl. 139, 67–84 (2008)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bot, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasi-relative interior in convex programming. SIAM J. Optim. 19, 217–233 (2008)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bot, R.I., Grad, S.M., Wanka, G.: Fenchel’s duality theorem for nearly convex functions. J. Optim. Theory Appl. 132, 509–515 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bot, R.I., Kassay, G., Wanka, G.: Duality for almost convex optimization problems via the perturbation approach. J. Global Optim. 42, 385–399 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Brecker, W.W., Kassay, G.: A systematization of convexity concepts for sets and functions. J. Convex Anal. 4, 109–127 (1997)MATHMathSciNetGoogle Scholar
  11. 11.
    Burachik, R., Majeed, S.N.: Strong duality for generalized monotropic programming in infinite dimensions. J. Math. Anal. Appl. 400, 541–557 (2013)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Calabi, E.: Linear systems of real quadratic forms. Proc. Am. Math. Soc. 15, 844–846 (1964)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Cárcamo, G.; Flores-Bazán, F.: Strong duality and KKT conditions in nonconvex optimization with a single equality constraint and geometric constraint. Math. Program. B, Published on 15 October 2016. 10.1007/s10107-016-1078-3
  14. 14.
    Champion, T.: Duality gap in convex programming. Math. Program. A 99, 487–498 (2004)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Deng, S.: Coercivity properties and well posedness in vector optimization. RAIRO Oper. Res. 37, 195–208 (2003)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Dines, L.L.: On the mapping of quadratic forms. Bull. Am. Math. Soc. 47, 494–498 (1941)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Ernest, E., Volle, M.: Zero duality gap for convex programs: a generalization of the Clark–Duffin theorem. J. Optim. Theory Appl. 158, 668–686 (2013)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Ernest, E., Volle, M.: Zero duality gap and attainment with possibly non-convex data. J. Convex Anal. 23, 615–629 (2016)MATHMathSciNetGoogle Scholar
  19. 19.
    Finsler, P.: Über das Vorkommen definiter und semi-definiter Formen in scharen quadratische Formen. Comment. Mat. Helv. 9, 188–192 (1937)CrossRefMATHGoogle Scholar
  20. 20.
    Flores-Bazán, F.: Existence theory for finite-dimensional pseudomonotone equilibrium problems. Acta Appl. Math. 77, 249–297 (2003)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Flores-Bazán, F., Flores-Bazán, F., Vera, C.: A complete characterization of strong duality in nonconvex optimization with a single constraint. J. Global Optim. 53, 185–201 (2012)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Flores-Bazán, F., Flores-Bazán, F., Vera, C.: Gordan-type alternative theorems and vector optimization revisited. In: Ansari, Q.H., Yao, J.C. (eds.) Recent Developments in Vector Optimization, pp. 29–59. Springer, Berlin (2012)CrossRefGoogle Scholar
  23. 23.
    Flores-Bazán, F., Flores-Bazán, F., Vera, C.: Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivates. J. Global Optim. 63, 99–123 (2015)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Flores-Bazán, F., Hadjisavvas, N., Lara, F., Montenegro, I.: First and second order asymptotic analysis with applications in quasiconvex optimization. J. Optim. Theory Appl. 170, 372–393 (2016)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Flores-Bazán, F., Hadjisavvas, N., Vera, C.: An optimal alternative theorem and applications to mathematical programming. J. Global Optim. 37, 229–243 (2007)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Flores-Bazán, F., Jourani, A., Mastroeni, G.: On the convexity of the value function for a class of nonconvex variational problems: existence and optimality conditions. SIAM J. Control Optim. 52, 3673–3693 (2014)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Flores-Bazán, F., Mastroeni, G.: Strong duality in cone constrained nonconvex optimization. SIAM J. Optim. 23, 153–169 (2013)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Flores-Bazán, F., Mastroeni, G., Vera, C.: Proper or weak efficiency via saddle point conditions in cone constrained nonconvex vector optimization problems. Pre-print 2017-06, Departamento de Ingenieria Matematica, Universidad de ConcepcionGoogle Scholar
  29. 29.
    Flores-Bazán, F., Opazo, F.: Characterizing the convexity of joint-range for a pair of inhomogeneous quadratic functions and strong duality. Minimax Theory Appl. 1, 257–290 (2016)MATHMathSciNetGoogle Scholar
  30. 30.
    Flores-Bazán, F., Vera, C.: Unifying efficiency and weak efficiency in generalized quasiconvex vector minimization on the real-line. Int. J. Optim. Theory Methods Appl. 1, 247–265 (2009)MATHMathSciNetGoogle Scholar
  31. 31.
    Frenk, J.B.G., Kassay, G.: On classes of generalized convex functions. Gordan–Farkas type theorems, and Lagrangian duality. J. Optim. Theory Appl. 102, 315–343 (1999)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Giannessi, F.: Constrained Optimization and Image Space Analysis. Springer, Berlin (2005)MATHGoogle Scholar
  33. 33.
    Giannessi, F., Mastroeni, G.: Separation of sets and Wolfe duality. J. Global Optim. 42, 401–412 (2008)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Goberna, M.A., López, M.A., Volle, M.: Primal attainment in convex infinite optimization duality. J. Convex Anal. 21, 1043–1064 (2014)MATHMathSciNetGoogle Scholar
  35. 35.
    Grad, A.: Quasi-relative interior-type constraint qualifications ensuring strong Lagrange duality for optimization problems with cone and affine constraints. J. Math. Anal. Appl. 361, 86–95 (2010)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.): Handbook of Generalized Convexity and Generalized Monotonicity. Springer, Berlin (2005)MATHGoogle Scholar
  37. 37.
    Jeyakumar, V.: Constraint qualifications characterizing Lagrangian duality in convex optimization. J. Optim. Theory Appl. 136, 31–41 (2008)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Jeyakumar, V., Lee, G.M.: Complete characterizations of stable Farkas lemma and cone-convex programming duality. Math. Program. A 114, 335–347 (2008)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Jeyakumar, V., Li, G.Y.: Stable zero duality gaps in convex programming: complete dual characterizations with applications to semidefinite programs. J. Math. Anal. Appl. 360, 156–167 (2009)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Jeyakumar, V., Oettli, W., Natividad, M.: A solvability theorem for a class of quasiconvex mappings with applications to optimization. J. Math. Anal. Appl. 179, 537–546 (1993)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Jeyakumar, V.: Wolkowicz: zero duality gaps in infinite-dimensional programming. J. Optim. Theory Appl. 67, 87–108 (1990)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Jeyakumar, V., Li, G.Y.: Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimzation. Math. Program. A 147, 171–206 (2014)CrossRefMATHGoogle Scholar
  43. 43.
    Kuroiwa, D.: Convexity for set-valued maps. Appl. Math. Lett. 9, 97–101 (1996)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Mastroeni, G.: Some applications of the image space analysis to the duality theory for constrained extremum problems. J. Global Optim. 46, 603–614 (2010)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Polyak, B.T.: Convexity of quadratic transformations and its use in control and optimization. J. Optim. Theory Appl. 99, 553–583 (1998)CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar
  47. 47.
    Rockafellar, R.T.: Conjugate Duality and Optimization. SIAM, New Delhi (1974)CrossRefMATHGoogle Scholar
  48. 48.
    Tanaka, T.: General quasiconvexities, cones saddle points and minimax theorem for vector-valued functions. J. Optim. Theory Appl. 81, 355–377 (1994)CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Tseng, P.: Some convex programs without a duality gap. Math. Program. B 116, 553–578 (2009)CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Xia, Y., Wang, S., Sheu, R.L.: S-lemma with equality and its applications. Math. Program. A 156, 513–547 (2016)CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    Zǎlinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Fabián Flores-Bazán
    • 1
  • William Echegaray
    • 2
  • Fernando Flores-Bazán
    • 3
  • Eladio Ocaña
    • 2
  1. 1.Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile
  2. 2.IMCA, Facultad de CienciasUniversidad Nacional de IngenieríaLimaPeru
  3. 3.Departamento de Matemáticas, Facultad de CienciasUniversidad del Bío BíoConcepciónChile

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