Sectional Convexity of Epigraphs of Conjugate Mappings with Applications to Robust Vector Duality


This paper concerns the robust vector problem

$$ \mathrm{(RVP)}\enskip \text{WMin}\left\{ F(x) : x\in C, G_{u}(x)\in -S,\forall u\in\mathcal{U}\right\}, $$

where X,Y,Z are locally convex Hausdorff topological vector spaces, K is a closed and convex cone in Y with a nonempty interior, and S is a closed, convex cone in Z, \(\mathcal {U}\) is an uncertainty set, \(F\colon X\rightarrow {Y}^{\bullet },\)\(G_{u}\colon X\rightarrow Z^{\bullet }\) are proper mappings for all \( u \in \mathcal U\), and CX. Let \( A:=C\cap \left (\bigcap _{u\in \mathcal {U}}G_{u}^{-1}(-S)\right )\) and IA : XY be the indicator map defined by IA(x) = 0Y if xA and \(I_{A}(x) = + \infty _{Y}\) if xA. It is well known that the epigraph of the conjugate mapping (F + IA), in general, is not a convex set. We show that, however, it is “k-sectionally convex” in the sense that each section formed by the intersection of epi(F + IA) and any translation of a “specific k-direction-subspace” is a convex subset, for any k taking from intK. The key results of the paper are the representations of the epigraph of the conjugate mapping (F + IA) via the closure of the k-sectionally convex hull of a union of epigraphs of conjugate mappings of mappings from a family involving the data of the problem (RVP). The results then give rise to stable robust vector/convex vector Farkas lemmas which, in turn, are used to establish new results on robust strong stable duality results for (RVP). It is shown at the end of the paper that, when specifying the result to some concrete classes of scalar robust problems (i.e., when \(Y = \mathbb R\)), our results cover and extend several corresponding known ones in the literature.

This is a preview of subscription content, access via your institution.


  1. 1.

    It is also called a proper cone.

  2. 2.

    Note that, in the first part of the proof of [11, Theorem 4.2], no assumptions on the convexity or closedness of the mappings F and G are needed.

  3. 3.

    This would be an elementary result in the study of robust optimization problems. However, to the surprise of the authors, we could not find it in the references we had in hand and so we insert a short proof here.

  4. 4.

    For the sake of simplicity, we write \((x^{*}_{\alpha },r_{\alpha })_{\alpha \in D}\) for \(((x^{*}_{\alpha }, r_{\alpha }))_{\alpha \in D}\).


  1. 1.

    Aliprantis, Ch.D., Burkinshaw, O.: Positive Operators Pure and Applied Mathematics, vol. 119. Academic Press, Orlando (1985)

    Google Scholar 

  2. 2.

    Barro, M., Ouédraogo, A., Traoré, S.: On uncertain conical convex optimization problem. Pacific J. Optim. 13(1), 29–42 (2017)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Beck, A., Ben-Tal, A.: Duality in robust optimization: primal worst equals dual best. Oper. Res. Lett. 37(1), 1–6 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)

    Google Scholar 

  5. 5.

    Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Boţ, R.I.: Conjugate Duality in Convex Optimization Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin (2010)

    Google Scholar 

  7. 7.

    Boţ, R.I., Grad, S.-M., Wanka, G.: Duality in Vector Optimization. Vector Optimization. Springer, Berlin (2009)

    Google Scholar 

  8. 8.

    Dinh, N., Goberna, M.A., López, M.A., Mo, T.H.: Farkas-type results for vector-valued functions with applications. J. Optim. Theory Appl. 173(2), 357–390 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Dinh, N., Goberna, M.A., López, M. A., Mo, T.H.: Robust optimization revisited via robust vector Farkas lemmas. Optimization 66(6), 939–963 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Dinh, N., Long, D.H.: Complete characterizations of robust strong duality for robust vector optimization problems. Vietnam J. Math. 46(2), 293–328 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Dinh, N., Goberna, M.A., Long, D.H., López, M.A.: New Farkas-type results for vector-valued functions: a non-abstract approach. J. Optim. Theory Appl. 182(1), 4–29 (2019)

    MathSciNet  MATH  Article  Google 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(2), 1075–1101 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Dinh, N., Goberna, M.A., López, M.A., Volle, M.: A unifying approach to robust convex infinite optimization duality. J. Optim. Theory Appl. 174(3), 650–685 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Goberna, M.A., Jeyakumar, V., Li, G., López, M.A.: Robust linear semi-infinite programming duality under uncertainty. Math. Program. 139(1–2), Ser. B, 185–203 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Gabrel, V., Murat, C., Thiele, A.: Recent advances in robust optimization: an overview. European J. Oper. Res. 235(3), 471–483 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Jeyakumar, V., Li, G.Y.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 20(6), 3384–3407 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Jeyakumar, V., Li, G., Wang, J.H.: Some robust convex programs without a duality gap. J. Convex Anal. 20(2), 377–394 (2013)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Li, G., Ng, K.F.: On extension of Fenchel duality and its application. SIAM J. Optim. 19(3), 1489–1509 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics. 2nd edn., McGraw-Hill, New York (1991)

  20. 20.

    Tanino, T.: Conjugate duality in vector optimization. J. Math. Anal. Appl. 167 (1), 84–97 (1992)

    MathSciNet  MATH  Article  Google Scholar 

Download references


This work is supported by the project 101.01-2018.310, NAFOSTED, Vietnam.

Author information



Corresponding author

Correspondence to Nguyen Dinh.

Additional information

Dedicated to Professor Hoang Tuy’s 90th birthday

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dinh, N., Long, D.H. Sectional Convexity of Epigraphs of Conjugate Mappings with Applications to Robust Vector Duality. Acta Math Vietnam 45, 525–553 (2020).

Download citation


  • Robust vector optimization
  • Robust convex optimization
  • Robust convex strong duality
  • Robust stable vector Farkas lemma
  • Sectionally convex sets
  • Sectionally closed sets

Mathematics Subject Classification (2010)

  • 90C25
  • 49N15
  • 90C31