Optimization Letters

, Volume 13, Issue 2, pp 349–366 | Cite as

Conjugate duality for constrained optimization via image space analysis and abstract convexity

  • C. L. Yao
  • S. J. LiEmail author
Original Paper


This paper is aimed at establishing a conjugate duality for the constrained optimizations equipped with some topical structures. First, we provide a dual problem for the general constrained optimization, discussing the weak duality as well as the strong duality based on the theory of abstract convexity. Transforming the zero duality gap property into a separation of two sets in image space, we involve the approach inspired by the image space analysis to study the dual theory by the aid of some separation functions. Then, using these results, we investigate this dual frame for the problem with some topical properties.


Constrained optimization Conjugate duality Abstract convexity Topical function Image space analysis 



This research was partially supported by the National Natural Science Foundation of China (Grants: 11571055 and 11601437 ), the Fundamental Research Funds for the Central Universities (Grant Number: 106112017CD-JZRPY0020) and Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant Number: KJ1501503).


  1. 1.
    Rockafellar, R.T.: Conjugate duality and optimization. In: Society for Industrial and Applied Mathematics (1974)Google Scholar
  2. 2.
    Ekeland, I., Temam, R.: Analyse Convexe et Problemes Variationnelles. Dunod Gauthier-Villars, Paris (1974)zbMATHGoogle Scholar
  3. 3.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (2015)Google Scholar
  4. 4.
    Stoer, J., Witzgall, C.: Convexity and Optimization in Finite Dimensions I. Springer, Berlin (1970)CrossRefzbMATHGoogle Scholar
  5. 5.
    Rubinov, A.M.: Abstract Convexity and Global Optimization. Kluwer Academic Publishers, Boston (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Singer, I.: Abstract Convex Analysis. Wiley, New York (1997)zbMATHGoogle Scholar
  7. 7.
    Gunawardena, J., Keane, M.: On the existence of cycle times for some nonexpansive maps. Technical Report HPL-BRIMS-95-003, Hewlett-Packard Labs (1995)Google Scholar
  8. 8.
    Mohebi, H., Samet, M.: Abstract convexity of topical functions. J. Glob. Optim. 58, 365–375 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rubinov, A.M., Singer, I.: Topical and sub-topical functions, downward sets and abstract convexity. Optimization 50, 307–551 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Doagooei, A.R., Mohebi, H.: Optimization of the difference of topical functions. J. Glob. Optim. 57, 1349–1358 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Martinez-Legaz, J.E., Rubinov, A.M., Singer, I.: Downward sets and their separation and approximation properties. J. Glob. Optim. 23, 111–137 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mohebi, H., Rubinov, A.M.: Best approximation by downward sets with applications. Anal. Theory Appl. 22, 20–40 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Castellani, G., Giannessi, F.: Decomposition of mathematical programs by means of theorems of alternative for linear and nonlinear systems. Proc. Ninth Int. Math. Program. Symp. Bp. Surv. Math. Program. 2, 423–439 (1979)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems. Part I: image space analysis. J. Optim. Theory Appl. 161, 738–762 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems. Part II: special duality schemes. J. Optim. Theory Appl. 161, 763–782 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Giannessi, F.: Constrained Optimization and Image Space Analysis, Vol. 1: Separation of Sets and Optimality Conditions. Springer, New York (2005)zbMATHGoogle Scholar
  17. 17.
    Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Luo, H.Z., Mastroeni, G., Wu, H.X.: Separation approach for augmented Lagrangians in constrained nonconvex optimization. J. Optim. Theory Appl. 144, 275290 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rubinov, A.M., Glover, B.M.: Duality for increasing positively homogeneous functions and normal sets. RAIRO Oper. Res. 32, 105–123 (1988)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gerstewitz, C., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Tech. Hochsch. Ilmenau 31, 61–81 (1985)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Göpfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2006)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingChina

Personalised recommendations