Convergence, Scalarization and Continuity of Minimal Solutions in Set Optimization

doi:10.1007/s40305-022-00440-6

Abstract

Abstract: The paper deals with the study of two different aspects of stability in the given space as well as the image space, where the solution concepts are based on a partial order relation on the family of bounded subsets of a real normed linear space. The first aspect of stability deals with the topological set convergence of families of solution sets of perturbed problems in the image space and Painlevé-Kuratowski set convergence of solution sets of the perturbed problems in the given space. The convergence in the given space is also established in terms of solution sets of scalarized perturbed problems. The second aspect of stability deals with semicontinuity of the solution set maps of parametric perturbed problems in both the spaces.

Key words: Topological convergence, Painlevé-Kuratowski convergence, Upper semicontinuity, Lower semicontinuity, Stability, Scalarization

CLC Number:

Karuna, C. S. Lalitha. Convergence, Scalarization and Continuity of Minimal Solutions in Set Optimization[J]. Journal of the Operations Research Society of China, 2024, 12(3): 773-793.

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References

[1] Xiang, S., Zhou, Y.: On essential sets and essential components of efficient solutions for vector optimization problems. J. Math. Anal. Appl. 315, 317-326(2006)
[2] Song, Q.Q., Tang, G.Q., Wang, L.S.: On essential stable sets of solutions in set optimization problems. J. Optim. Theory Appl. 156, 591-599(2013)
[3] Xu, Y.D., Li, S.J.: Continuity of the solution set mappings to a parametric set optimization problem. Optim. Lett. 8, 2315-2327(2014)
[4] Xu, Y.D., Li, S.J.: On the solution continuity of parametric set optimization problems. Math. Methods Oper. Res. 84, 223-237(2016)
[5] Han, Y., Huang, N.J.: Well-posedness and stability of solutions for set optimization problems. Optimization 66, 17-33(2017)
[6] Han, Y., Zhang, K.: Semicontinuity of the minimal solution mappings to parametric set optimization problems on Banach lattices. Optimization https://doi.org/10.1080/02331934.2022.2045985(2022)
[7] Zhang, C.L., Huang, N.J.:Well-posedness and stability in set optimization with applications. Positivity 25, 1153-1173(2021)
[8] Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Convergence of solutions of a set optimization problem in the image space. J. Optim. Theory Appl. 170, 358-371(2016)
[9] Han, Y., Zhang, K., Huang, N.J.: The stability and extended well-posedness of the solution sets for set optimization problems via the Painlevé-Kuratowski convergence. Math. Methods Oper. Res. 91, 175-196(2020)
[10] Karuna, Lalitha, C.S.: External and internal stability in set optimization. Optimization 68, 833-852(2019)
[11] Anh, L.Q., Duy, T.Q., Hien, D.V.: Stability of efficient solutions to set optimization problems. J. Glob. Optim. 78, 563-580(2020)
[12] Gupta, M., Srivastava, M.: On Levtin-Polyak well posedness and stability in set optimization. Positivity 25, 1903-1921(2021)
[13] Dhingra, M., Lalitha, C.S.: Approximate solutions and scalarization in set-valued optimization. Optimization 66, 1793-1805(2017)
[14] Karuna, Lalitha, C.S.: External and internal stability in set optimization using gamma convergence. Carpathian J. Math. 35, 393-406(2019)
[15] Geoffroy, M.H.: A topological convergence on power sets well-suited for set optimization. J. Global Optim. 73, 567-581(2019)
[16] Chen, J.,Wang, G., Ou, X., Zhang,W.: Continuity of solutionsmappings of parametric set optimization problems. J. Ind. Manag. Optim. 16, 25-36(2020)
[17] Karuna, Lalitha, C.S.: Continuity of approximate weak efficient solution set map in parametric set optimization. J. Nonlinear Convex Anal. 19, 1247-1262(2018)
[18] Zhang, C.L., Huang, N.J.: On the stability of minimal solutions for parametric set optimization problems. Appl. Anal. 100, 1533-1543(2021)
[19] Liu, P.P., Wei, H.Z., Chen, C.R., Li, S.J.: Continuity of solutions for parametric set optimization problems via scalarization methods. J. Oper. Res. Soc. China 11, 1-19(2018)
[20] Karaman, E., Soyertem, M., Güvenç, · IA., Tozkan, D., KüçüK, M., KüçüK, Y.: Partial order relations on family of sets and scalarizations for set optimization. Positivity 22, 783-802(2018)
[21] Lalitha, C.S., Chatterjee, P.: Stability and scalarization of weak efficient, efficient and Henig proper efficient sets using generalized quasiconvexities. J. Optim. Theory Appl. 155, 941-961(2012)
[22] Pallaschke, D., Urbá nski, R.: Pairs of Compact Convex Sets: Fractional arithmetic with Convex Sets. Volume 548 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht (2002)
[23] Khan, A.A., Tammer, C., Zǎlinescu, C.: Set-valued Optimization: An Introduction with Applications. Springer, Berlin (2015)
[24] Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, 1822-1833(2012)
[25] Khushboo, Lalitha, C.S.: Scalarizations for a set optimization problem using generalized oriented distance function. Positivity 23, 1195-1213(2019)
[26] Hernández, E., Rodríguez-Marín, L.: Existence theorems for set optimization problems. Nonlinear Anal. 67, 1726-1736(2007)
[27] Anh, L.Q., Duy, T.Q., Hien, D.V., Kuroiwa, D., Petrot, N.: Convergence of solutions to set optimization problems with the set less order relation. J. Optim. Theory Appl. 185, 416-432(2020)
[28] Kuroiwa, D.: On duality of set-valued optimization, Research on nonlinear analysis and convex analysis (Japanese) (Kyoto, 1998) Sūrikaisekikenkyūsho Kōkyūroku. 1071, 12-16(1998)
[29] Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis: Pure and applied mathematics. Wiley, New York (1984)
[30] Kuroiwa, D.: The natural criteria in set-valued optimization, Research on nonlinear analysis and convex analysis (Japanese) (Kyoto, 1997) Sūrikaisekikenkyūsho Kōkyūroku. 1031, 85-90(1998)