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Characterizations of Robust and Stable Duality for Linearly Perturbed Uncertain Optimization Problems

  • Nguyen Dinh
  • Miguel A. GobernaEmail author
  • Marco A. López
  • Michel Volle
Conference paper
  • 54 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 313)

Abstract

We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail.

Notes

Acknowledgements

This research was supported by the Vietnam National University - HCM city, Vietnam, project B2019-28-02, by PGC2018-097960-B-C22 of the Ministerio de Ciencia, Innovación y Universidades (MCIU), the Agencia Estatal de Investigación (AEI), and the European Regional Development Fund (ERDF), and by the Australian Research Council, Project DP180100602.

References

  1. 1.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)CrossRefGoogle Scholar
  2. 2.
    Bertsimas, D., Sim, M.: Tractable approximations to robust conic optimization problems. Math. Program. 107B, 5–36 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Borwein, J.M.: A strong duality theorem for the minimum of a family of convex programs. J. Optim. Theory Appl. 31, 453–472 (1980)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borwein, J.M., Burachik, R.S., Yao, L.: Conditions for zero duality gap in convex programming. J. Nonlinear Convex Anal. 15, 167–190 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Borwein, J.M., Lewis, A.S.: Partially finite convex programming. I. Quasi relative interiors and duality theory. Math. Program. 57B, 15-48 (1992)Google Scholar
  6. 6.
    Borwein, J.M., Lewis, A.S.: Practical conditions for Fenchel duality in infinite dimensions. In: Fixed Point Theory and Applications, pp. 83–89, Pitman Research Notes in Mathematics Series, vol. 252, Longman Scientific and Technology, Harlow (1991)Google Scholar
  7. 7.
    Bot, R.I.: Conjugate Duality in Convex Optimization. Springer, Berlin (2010)CrossRefGoogle Scholar
  8. 8.
    Bot, R.I., Jeyakumar, V., Li, G.Y.: Robust duality in parametric convex optimization. Set-Valued Var. Anal. 21, 177–189 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Burachick, R.S., Jeyakumar, V., Wu, Z.-Y.: Necessary and sufficient condition for stable conjugate duality. Nonlinear Anal. 64, 1998–2006 (2006)Google Scholar
  10. 10.
    Chu, Y.C.: Generalization of some fundamental theorems on linear inequalities. Acta Math Sinica 16, 25–40 (1966)MathSciNetGoogle Scholar
  11. 11.
    Dinh, N., Goberna, M.A., López, M.A., Volle, M.: A unifying approach to robust convex infinite optimization duality. J. Optim. Theory Appl. 174, 650–685 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dinh, N., Mo, T.H., Vallet, G., Volle, M.: A unified approach to robust Farkas-type results with applications to robust optimization problems. SIAM J. Optim. 27, 1075–1101 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dinh, N., Long, D.H.: Complete characterizations of robust strong duality for robust optimization problems. Vietnam J. Math. 46, 293–328 (2018).  https://doi.org/10.1007/s10013-018-0283-1MathSciNetCrossRefGoogle Scholar
  14. 14.
    Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)zbMATHGoogle Scholar
  15. 15.
    Goberna, M.A., López, M.A.: Recent contributions to linear semi-infinite optimization. 4OR-Q. J. Oper. Res. 15, 221–264 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Goberna, M.A., López, M.A., Volle, M.: Primal attainment in convex infinite optimization duality. J. Convex Anal. 21, 1043–1064 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Grad, S.-M.: Closedness type regularity conditions in convex optimization and beyond. Front. Appl. Math. Stat. 16, September (2016).  https://doi.org/10.3389/fams.2016.00014
  18. 18.
    Hiriart-Urruty, J.-B., Lemarechal, C.: Convex Analysis and Minimization Algoritms I. Springer, Berlin (1993)zbMATHGoogle Scholar
  19. 19.
    Hiriart-Urruty, J.-B., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey. Nonlinear Anal. 24, 1727–1754 (1995)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Jeyakumar, V., Li, G.Y.: Stable zero duality gaps in convex programming: complete dual characterisations with applications to semidefinite programs. J. Math. Anal. Appl. 360, 156–167 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, G.Y., Jeyakuma, V., Lee, G.M.: Robust conjugate duality for convex optimization under uncertainty with application to data classification. Nonlinear Anal. 74, 2327–2341 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lindsey, M., Rubinstein, Y.A.: Optimal transport via a Monge-Ampère optimization problem. SIAM J. Math. Anal. 49, 3073–3124 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    López, M.A., Still, G.: Semi-infinite programming. European J. Oper. Res. 180, 491–518 (2007)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rockafellar, R.T.: Conjugate Duality and Optimization. CBMS Lecture Notes Series No. 162. SIAM, Philadelphia (1974)CrossRefGoogle Scholar
  25. 25.
    Vera, J.R.: Geometric measures of convex sets and bounds on problem sensitivity and robustness for conic linear optimization. Math. Program. 147A, 47–79 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Volle, M.: Caculus rules for global approximate minima and applications to approximate subdifferential calculus. J. Global Optim. 5, 131–157 (1994)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, Y., Shi, R., Shi, J.: Duality and robust duality for special nonconvex homogeneous quadratic programming under certainty and uncertainty environment. J. Global Optim. 62, 643–659 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Nguyen Dinh
    • 1
  • Miguel A. Goberna
    • 2
    Email author
  • Marco A. López
    • 2
    • 3
  • Michel Volle
    • 4
  1. 1.International University, Vietnam National University - HCMCHo Chi Minh cityVietnam
  2. 2.Department of MathematicsUniversity of AlicanteAlicanteSpain
  3. 3.CIAOFederation UniversityBallaratAustralia
  4. 4.LMAAvignon UniversityAvignonFrance

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