The present work is devoted to the study of stability in set optimization. In particular, a sequence of perturbed set optimization problems, with a fixed objective map, is studied under suitable continuity assumptions. A formulation of external and internal stability of the solutions is considered in the image space, in such a way that the convergence of a sequence of solutions of perturbed problems to a solution of the original problem is studied under appropriate compactness assumptions. Our results can also be seen as an extension to the set-valued framework of known stability results in vector optimization.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Chen, G.Y., Huang, X., Yang, X.: Vector optimization. Set-valued and variational analysis. Springer, Berlin (2005)
Jahn, J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Berlin (2011)
Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)
Kuroiwa, D.: Some duality theorems of set-valued optimization with natural criteria. In: Takahashi, W., Tanaka, T. (eds.) Nonlinear Analysis and Convex Analysis, pp. 221–228. World Scientific Publishing, River Edge (1999)
Kuroiwa, D.: On set-valued optimization. Nonlinear Anal. 47, 1395–1400 (2001)
Ide, J., Köbis, E., Kuroiwa, D., Schöbel, A., Tammer, C.: The relationship between multi-objective robustness concepts and set-valued optimization. Fixed Point Theory Appl. 83, 1–20 (2014)
Ehrgott, M., Ide, J., Schöbel, A.: Minmax robustness for multi-objective optimization problems. Eur. J. Oper. Res. 239, 17–31 (2014)
Ide, J., Köbis, E.: Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations. Math. Methods Oper. Res. 80, 99–127 (2014)
Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. Nonlinear Anal. 30, 1487–1496 (1997)
Gutiérrez, C., Jiménez, B., Miglierina, E., Molho, E.: Scalarization in set optimization with solid and nonsolid ordering cones. J. Global Optim. 61, 525–552 (2015)
Gutiérrez, C., Jiménez, B., Novo, V., Thibault, L.: Strict approximate solutions in set-valued optimization with applications to the approximate Ekeland variational principle. Nonlinear Anal. 73, 3842–3855 (2010)
Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, 1822–1833 (2012)
Hamel, A., Löhne, A.: Minimal element theorems and Ekeland’s principle with set relations. J. Nonlinear Convex Anal. 7, 19–37 (2006)
Hernández, E., Rodríguez-Marín, L.: Existence theorems for set optimization problems. Nonlinear Anal. 67, 1726–1736 (2007)
Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)
Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)
Bednarczuk, E.: Stability Analysis for Parametric Vector Optimization Problems. Dissertationes Math. 442, Institute of Mathematics, Polish Academy of Sciences, Warsaw (2007)
Luc, D.T., Lucchetti, R., Malivert, C.: Convergence of the efficient sets. Set-Valued Anal. 2, 207–218 (1994)
Lucchetti, R., Miglierina, E.: Stability for convex vector optimization problems. Optimization 53, 517–528 (2004)
Miglierina, E., Molho, E.: Convergence of minimal sets in convex vector optimization. SIAM J. Optim. 15, 513–526 (2005)
Penot, J.-P., Sterna-Karwat, A.: Parametrized multicriteria optimization: continuity and closedness of optimal multifunctions. J. Math. Anal. Appl. 120, 150–168 (1986)
Xu, Y.D., Li, Y.D.: Continuity of the solution set mappings to a parametric set optimization problem. Optim. Lett. 8, 2315–2327 (2014)
Giannessi, F.: Constrained optimization and image space analysis. Vol. 1: separation of sets and optimality conditions. Springer, New York (2005)
Loridan, P., Morgan, J., Raucci, R.: Convergence of minimal and approximate minimal elements of sets in partially ordered vector spaces. J. Math. Anal. Appl. 239, 427–439 (1999)
Göpfert, A., Rihai, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)
Crespi, G.P., Papalia, M., Rocca, M.: Extended well-posedness of quasiconvex vector optimization problems. J. Optim. Theory Appl. 141, 285–297 (2009)
Kutateladze, S.S.: Convex \(\varepsilon \)-programming. Soviet Math. Dokl. 20, 301–303 (1979)
Crespi, G.P., Guerraggio, A., Rocca, M.: Well posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 132, 213–226 (2007)
Li, S.J., Zhang, W.Y.: Hadamard well-posed vector optimization problems. J. Global Optim. 46, 383–393 (2010)
This research was partially supported by the Ministerio de Economía y Competitividad (Spain) under project MTM2012-30942. The third author was also partially supported by MIUR PRIN MISURA Project, 2013-2015, Italy. The authors would like to thank the Editor and the anonymous referee for their useful comments.
Communicated by Lionel Thibault.
Rights and permissions
About this article
Cite this article
Gutiérrez, C., Miglierina, E., Molho, E. et al. Convergence of Solutions of a Set Optimization Problem in the Image Space. J Optim Theory Appl 170, 358–371 (2016). https://doi.org/10.1007/s10957-016-0942-x
- Set optimization
- Set relations
- Minimal solutions
- Set convergence
Mathematics Subject Classification