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Duality for the Robust Sum of Functions

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Abstract

In this paper we associate with an infinite family of real extended functions defined on a locally convex space a sum, called robust sum, which is always well-defined. We also associate with that family of functions a dual pair of problems formed by the unconstrained minimization of its robust sum and the so-called optimistic dual. For such a dual pair, we characterize weak duality, zero duality gap, and strong duality, and their corresponding stable versions, in terms of multifunctions associated with the given family of functions and a given approximation parameter ε ≥ 0 which is related to the ε-subdifferential of the robust sum of the family. We also consider the particular case when all functions of the family are convex, assumption allowing to characterize the duality properties in terms of closedness conditions.

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References

  1. Aragón, F.J., Goberna, M.A., López, M.A., Rodríguez, M.M.L.: Nonlinear Optimization. Springer, Cham (2019)

    Book  Google Scholar 

  2. Ardestani-Jaafari, A., Delage, E.: Robust optimization of sums of piecewise linear functions with application to inventory problems. Oper. Res. 64, 474–494 (2016)

    Article  MathSciNet  Google Scholar 

  3. Boţ, R.I.: Conjugate duality in convex optimization. Springer, Berlin (2010)

    Book  Google Scholar 

  4. Boţ, R.I., Jeyakumar, V., Li, G.Y.: Robust duality in parametric convex optimization. Set-Valued Var. Anal. 21, 177–189 (2013)

    Article  MathSciNet  Google Scholar 

  5. Dhara, A., Natarajanyz, K.: On the polynomial solvability of distributionally robust k-sum optimization. Optim. Methods Softw. 32(4), 1–16 (2016)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Dinh, N., Goberna, M.A., López, M.A., Volle, V.: Characterizations of robust and stable duality for linearly perturbed uncertain optimization problems. In: Burachik, R., Li, G.Y (eds.) From analysis to visualization: A celebration of the life and legacy of Jonathan M. Borwein, Callaghan. to appear. Springer, Australia (2017)

  8. Dinh, N., Goberna, M.A., López, M.A., Volle, V.: Convexity and closedness in stable robust duality. Opt. Lett. 13, 325–339 (2019)

    Article  MathSciNet  Google Scholar 

  9. Ernst, E., Volle, M.: Zero duality gap and attainment with possibly non-convex data. J. Convex Anal. 23, 615–629 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Goberna, M.A., Hiriart-Urruty, J.-B., López, M.A.: Best approximate solutions of inconsistent linear inequality systems. Vietnam J. Math. 46, 271–284 (2018)

    Article  MathSciNet  Google Scholar 

  11. Hiriart-Urruty, J.B., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey. Nonlinear Anal. 23, 1727–1754 (1995)

    Article  MathSciNet  Google Scholar 

  12. Kirkwood, C.W.: Approximating risk aversion in decision analysis applications. Decision Anal. 1, 51–67 (2004)

    Article  Google Scholar 

  13. Li, G.Y., Jeyakumar, V., Lee, G.M.: Robust conjugate duality for convex optimization under uncertainty with application to data classification. Nonlinear Anal. 74, 2327–2341 (2011)

    Article  MathSciNet  Google Scholar 

  14. Li, G.Y., Ng, K.F.: On extension of Fenchel duality and its application. SIAM J. Opt. 19, 1498–1509 (2008)

    Article  MathSciNet  Google Scholar 

  15. Markowitz, H.M.: Portfolio selection: Efficient diversification of investment. Wiley, New York (1959)

    Google Scholar 

  16. Popa, C., Şerban, C.: Han-type algorithms for inconsistent systems of linear inequalities-a unified approach. Appl. Math. Comput. 246, 247–256 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Volle, M.: Calculus rules for global approximate minima and applications to approximate subdifferential calculus. J. Global Optim 5, 131–157 (1994)

    Article  MathSciNet  Google Scholar 

  18. Zheng, X.Y.: A series of convex functions on a Banach space. Acta Mathematica Sinica. 4, 19–28 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Zheng, X.Y., Ng, K.F.: Error bound moduli for conic convex systems on Banach spaces. Math. Oper. Res. 29, 213–228 (2004)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors wish to thank two anonymous referees and the Handling Editor for their valuable comments which helped us to improve the manuscript.

This research was supported by the National Foundation for Science & Technology Development (NAFOSTED), Vietnam, Project 101.01-2018.310 Some topics on systems with uncertainty and robust optimization, and by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Project PGC2018-097960-B-C22.

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Authors and Affiliations

  1. International University, Vietnam National University - HCMC, Linh Trung ward, Thu Duc district, Ho Chi Minh city, Vietnam

    N. Dinh

  2. Department of Mathematics, University of Alicante, Alicante, Spain

    M. A. Goberna

  3. Avignon University, LMA EA 2151, Avignon, France

    M. Volle

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  1. N. Dinh
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  2. M. A. Goberna
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  3. M. Volle
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Correspondence to N. Dinh.

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Dinh, N., Goberna, M.A. & Volle, M. Duality for the Robust Sum of Functions. Set-Valued Var. Anal 28, 41–60 (2020). https://doi.org/10.1007/s11228-019-00515-2

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  • DOI: https://doi.org/10.1007/s11228-019-00515-2

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