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Optimality Conditions and Duality for Multiobjective Semi-infinite Programming on Hadamard Manifolds

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Abstract

This article is devoted to studying the problem of multiobjective semi-infinite programming on Hadamard manifolds. We first establish both Karush–Kuhn–Tucker necessary and sufficient optimality conditions for some type of efficient solutions. Then, we propose Wolfe and Mond–Weir-type dual problems and examine duality relations under geodesic convexity assumptions.

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References

  1. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)

    Book  Google Scholar 

  2. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  3. Barani, A.: Convexity of the solution set of a pseudoconvex inequality in Riemannian manifolds. Numer. Funct. Anal. Optim. 39, 588–599 (2018)

    Article  MathSciNet  Google Scholar 

  4. Barani, A., Hosseini, S.: Characterization of solution sets of convex optimization problems in Riemannian manifolds. Arch. Math. 114, 215–225 (2020)

    Article  MathSciNet  Google Scholar 

  5. Bento, G.C., Melo, J.G.: Subgradient method for convex feasibility on Riemannian manifolds. J. Optim. Theory Appl. 152, 773–785 (2012)

    Article  MathSciNet  Google Scholar 

  6. Bergmann, R., Herzog, R.: Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds. SIAM J. Optim. 29, 2423–2444 (2019)

    Article  MathSciNet  Google Scholar 

  7. Borwein, J., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer, New York (2010)

    Google Scholar 

  8. Boumal, N., Mishra, B., Absil, P.A., Sepulchre, R.: Manopt, a Matlab toolbox for optimization on manifolds. J. Mach. Learn. Res. 42, 1455–1459 (2014)

    MATH  Google Scholar 

  9. Boumal, N.: An Introduction to Optimization on Smooth Manifolds. EPFL (2020)

  10. Chen, S.: The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds. Optimization (2020). https://doi.org/10.1080/02331934.2020.1810248

  11. Chen, S., Huang, N.: Vector variational inequalities and vector optimization problems on Hadamard manifolds. Optim. Lett. 10, 753–767 (2016)

  12. Chuong, T.D., Kim, D.S.: Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160, 748–762 (2014)

    Article  MathSciNet  Google Scholar 

  13. Do Carmo, M.P.: Riemannian Geometry. Birkh\({\ddot{\rm a}}\)user, Boston (1992)

  14. Farrokhiniya, M., Barani, A.: Limiting subdifferential calculus and perturbed distance function in Riemannian manifolds. J. Glob. Optim. 77, 661–685 (2020)

    Article  MathSciNet  Google Scholar 

  15. Ferreira, O.P., Iusem, A.N., Németh, S.Z.: Concepts and techniques of optimization on the sphere. TOP 22, 1148–1170 (2014)

    Article  MathSciNet  Google Scholar 

  16. Goberna, M.A., Kanzi, N.: Optimality conditions in convex multiobjective SIP. Math. Program. 164, 67–191 (2017)

    Article  MathSciNet  Google Scholar 

  17. Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)

    MATH  Google Scholar 

  18. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1993)

    Book  Google Scholar 

  19. Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (2008)

    MATH  Google Scholar 

  20. Kabgani, A., Soleimani-damaneh, M.: Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators. Optimization 67, 217–235 (2018)

    Article  MathSciNet  Google Scholar 

  21. Kanzi, N., Nobakhtian, S.: Optimality conditions for nonsmooth semi-infinite multiobjective programming. Optim. Lett. 8, 1517–1528 (2014)

    Article  MathSciNet  Google Scholar 

  22. Kanzi, N.: On strong KKT optimality conditions for multiobjective semi-infinite programming problems with Lipschitzian data. Optim. Lett. 9, 1121–1129 (2015)

    Article  MathSciNet  Google Scholar 

  23. Karkhaneei, M.M., Mahdavi-Amiri, N.: Nonconvex weak sharp minima on Riemannian manifolds. J. Optim. Theory Appl. 183, 85–104 (2019)

    Article  MathSciNet  Google Scholar 

  24. Kostyukova, O.I., Tchemisova, T.V.: Optimality conditions for convex semi-infinite programming problems with finitely representable compact index sets. J. Optim. Theory Appl. 175, 76–103 (2017)

    Article  MathSciNet  Google Scholar 

  25. Kristály, A., Li, C., López, G., Nicolae, A.: What do ‘convexities’ imply on Hadamard manifolds? J. Optim. Theory Appl. 170, 1068–1074 (2016)

  26. Lee, J.M.: Introduction to Riemannian Manifolds, 2nd edn. Springer, New York (2018)

    Book  Google Scholar 

  27. Li, C., Mordukhovich, B.S., Wang, J., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21, 1523–1560 (2011)

    Article  MathSciNet  Google Scholar 

  28. Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)

    Book  Google Scholar 

  29. Malmir, F., Barani, A.: Subdifferentials of distance function outside of target set in Riemannian manifolds. Bull. Iran. Math. Soc. https://doi.org/10.1007/s41980-020-00522-2

  30. Mond, B., Weir, T.: Generalized concavity and duality. In: Schaible, S., Ziemba, W.T. (eds.) Generalized Concavity in Optimization and Economics, pp. 263–279. Academic Press, New York (1981)

  31. Quiroz, E.A.P., Cusihuallpa, N.B., Maculan, N.: Inexact proximal point methods for multiobjective quasiconvex minimization on Hadamard manifolds. J. Optim. Theory Appl. 186, 879–898 (2020)

    Article  MathSciNet  Google Scholar 

  32. Rahimi, M., Soleimani-damaneh, M.: Isolated efficiency in nonsmooth semi-infinite multi-objective programming. Optimization 67, 1923–1947 (2018)

    Article  MathSciNet  Google Scholar 

  33. Rapcsák, T.: Smooth Nonlinear Optimization in \({\mathbb{R}}^n\). Kluwer Academic Publishers, Dordrecht (1997)

    Book  Google Scholar 

  34. Rockafellar, R.T.: Convex Analysis. Princeton Math. Ser., vol. 28. Princeton University Press, Princeton (1970)

  35. Rudin, W.: Functional Analysis. McGraw-Hill Inc, New York (1991)

    MATH  Google Scholar 

  36. Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A., Hernández-Jiménez, B.: Optimality and duality on Riemannian manifolds. Taiwan. J. Math. 22, 1245–1259 (2018)

    Article  MathSciNet  Google Scholar 

  37. Ruiz-Garzón, G., Osuna-Gómez, R., Ruiz-Zapatero, J.: Necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds. Symmetry 11, 1037 (2019)

    Article  Google Scholar 

  38. Stein, O., Still, G.: Solving semi-infinite optimization problems with interior point techniques. SIAM J. Control Optim. 42, 769–788 (2003)

    Article  MathSciNet  Google Scholar 

  39. Tung, L.T.: Karush–Kuhn–Tucker optimality conditions and duality for semi-infinite programming with multiple interval-valued objective functions. J. Nonlinear Funct. Anal. 2019, 1–21 (2019)

    Google Scholar 

  40. Tung, L.T.: Karush–Kuhn–Tucker optimality conditions and duality for multiobjective semi-infinite programming with vanishing constraints. Ann. Oper. Res. (2020). https://doi.org/10.1007/s10479-020-03742-1

    Article  MATH  Google Scholar 

  41. Tung, L.T.: Karush–Kuhn–Tucker optimality conditions and duality for multiobjective semi-infinite programming with equilibrium constraints. Yugoslav J. Oper. Res. https://doi.org/10.2298/YJOR2001

  42. Tung, L.T., Tam, D.H.: Necessary and sufficient optimality conditions for semi-infinite programming with multiple fuzzy-valued objective functions. Stat. Optim. Inf. Comput. (accepted for publication) (2021)

  43. Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic Publishers, Dordrecht (1994)

    Book  Google Scholar 

  44. Wang, X.M., Li, C., Yao, J.C.: Projection algorithms for convex feasibility problems on Hadamard manifolds. J. Nonlinear Convex Anal. 17, 3–497 (2016)

    MathSciNet  MATH  Google Scholar 

  45. Wolfe, P.: A duality theorem for nonlinear programming. Q. Appl. Math. 19, 239–244 (1961)

    Article  Google Scholar 

  46. Yang, W.H., Zhang, L.H., Song, R.: Optimality conditions for the nonlinear programming problems on Riemannian manifolds. Pac. J. Optim. 10, 415–434 (2014)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the Editors for the help in the processing of the article. The authors are very grateful to the Anonymous Referees for the very valuable remarks, which helped to improve the paper. This work is partially supported by Can Tho University.

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Correspondence to Le Thanh Tung.

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Communicated by Behzad Djafari-Rouhani.

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Tung, L.T., Tam, D.H. Optimality Conditions and Duality for Multiobjective Semi-infinite Programming on Hadamard Manifolds. Bull. Iran. Math. Soc. 48, 2191–2219 (2022). https://doi.org/10.1007/s41980-021-00646-z

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  • DOI: https://doi.org/10.1007/s41980-021-00646-z

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