## Abstract

This paper concerns the robust vector problem

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 *∅* ≠ *C* ⊂ *X*. Let \( A:=C\cap \left (\bigcap _{u\in \mathcal {U}}G_{u}^{-1}(-S)\right )\) and *I*_{A} : *X* → *Y*^{∙} be the indicator map defined by *I*_{A}(*x*) = 0_{Y} if *x* ∈ *A* and \(I_{A}(x) = + \infty _{Y}\) if *x*∉*A*. It is well known that the epigraph of the conjugate mapping (*F* + *I*_{A})^{∗}, 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* + *I*_{A})^{∗} and any translation of a “specific *k*-direction-subspace” is a convex subset, for any *k* taking from int*K*. The key results of the paper are the representations of the epigraph of the conjugate mapping (*F* + *I*_{A})^{∗} 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.

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## Notes

- 1.
It is also called a proper cone.

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

## References

- 1.
Aliprantis, Ch.D., Burkinshaw, O.: Positive Operators Pure and Applied Mathematics, vol. 119. Academic Press, Orlando (1985)

- 2.
Barro, M., Ouédraogo, A., Traoré, S.: On uncertain conical convex optimization problem. Pacific J. Optim.

**13**(1), 29–42 (2017) - 3.
Beck, A., Ben-Tal, A.: Duality in robust optimization: primal worst equals dual best. Oper. Res. Lett.

**37**(1), 1–6 (2009) - 4.
Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)

- 5.
Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev.

**53**(3), 464–501 (2011) - 6.
Boţ, R.I.: Conjugate Duality in Convex Optimization Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin (2010)

- 7.
Boţ, R.I., Grad, S.-M., Wanka, G.: Duality in Vector Optimization. Vector Optimization. Springer, Berlin (2009)

- 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) - 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) - 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) - 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) - 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) - 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) - 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) - 15.
Gabrel, V., Murat, C., Thiele, A.: Recent advances in robust optimization: an overview. European J. Oper. Res.

**235**(3), 471–483 (2014) - 16.
Jeyakumar, V., Li, G.Y.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim.

**20**(6), 3384–3407 (2010) - 17.
Jeyakumar, V., Li, G., Wang, J.H.: Some robust convex programs without a duality gap. J. Convex Anal.

**20**(2), 377–394 (2013) - 18.
Li, G., Ng, K.F.: On extension of Fenchel duality and its application. SIAM J. Optim.

**19**(3), 1489–1509 (2008) - 19.
Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics. 2nd edn., McGraw-Hill, New York (1991)

- 20.
Tanino, T.: Conjugate duality in vector optimization. J. Math. Anal. Appl.

**167**(1), 84–97 (1992)

## Funding

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

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*Dedicated to Professor Hoang Tuy’s 90th birthday*

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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). https://doi.org/10.1007/s40306-019-00349-y

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### Keywords

- 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