Scalarization, convergence and well-posedness in set optimization

Taiyong Li; Manli Yang; Taiyong Li; Manli Yang

doi:10.3934/jimo.2026044

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 3: 1194-1213. doi: 10.3934/jimo.2026044

Previous Article Next Article

Theory article

Scalarization, convergence and well-posedness in set optimization

Taiyong Li ^{1
,
,},
Manli Yang ²

1.
Jiyang College, Zhejiang A & F University, Zhuji 311800, China
2.
Public Basic Department, Huzhou College, Huzhou 313016, China

Received: 24 September 2025 Revised: 26 December 2025 Accepted: 22 January 2026 Published: 05 February 2026
49K4, 90C48, 90C31

In this paper, we introduce the notions of weak $ m $-minimal approximate solutions and $ m $-minimal approximate solutions for constrained set optimization problems, based on a novel set order relation that involves the Minkowski difference. We derive scalarization results for the sets of weak $ m $-minimal approximate solutions and $ m $-minimal approximate solutions in the context of set optimization. Utilizing these scalarizations, we analyze the Painlevé-Kuratowski convergence properties of both classes of approximate solutions. Furthermore, new notions of well-posedness for set optimization problems are proposed, and relationships between these notions are rigorously established. Finally, we establish equivalences between the well-posedness of set optimization problems and their scalar counterparts through carefully constructed optimization frameworks.
- approximate solution,
- scalarization,
- Painlevé-Kuratowski convergence,
- set optimization,
- well-posedness
Citation: Taiyong Li, Manli Yang. Scalarization, convergence and well-posedness in set optimization[J]. Journal of Industrial and Management Optimization, 2026, 22(3): 1194-1213. doi: 10.3934/jimo.2026044

Related Papers:

Abstract

In this paper, we introduce the notions of weak $ m $-minimal approximate solutions and $ m $-minimal approximate solutions for constrained set optimization problems, based on a novel set order relation that involves the Minkowski difference. We derive scalarization results for the sets of weak $ m $-minimal approximate solutions and $ m $-minimal approximate solutions in the context of set optimization. Utilizing these scalarizations, we analyze the Painlevé-Kuratowski convergence properties of both classes of approximate solutions. Furthermore, new notions of well-posedness for set optimization problems are proposed, and relationships between these notions are rigorously established. Finally, we establish equivalences between the well-posedness of set optimization problems and their scalar counterparts through carefully constructed optimization frameworks.

References

[1]	A. L. Dontchev, T. Zolezzi, Well-posed optimization problems, Lecture Notes In Mathematics, Springer, Berlin, 1993. https://doi.org/10.1007/BFb0084195
[2]	A. A. Khan, C. Tammer, C. Zalinescu, Set-valued optimization, Vector Optimization, Springer, America, 2015. https://doi.org/10.1007/978-3-642-54265-7
[3]	D. Pallaschke, R. Urbanski, Pairs of compact convex sets, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 2002. https://doi.org/10.1007/978-94-015-9920-7
[4]	D. Kuroiwa, Some duality theorems of set-valued optimization with natural criteria, Proc. Int. Conf. Nonlinear Anal. Convex Anal., World Scientific River Edge, 1999,221–228.
[5]	E. Karaman, M. Güvenç, İ. Soyertem, D. Tozkan, M. Küçük, Y. Küçük, Partial order relations on family of sets and scalarizations for set optimization, Positivity, 22 (2018), 783–802. https://doi.org/10.1007/s11117-017-0544-3 doi: 10.1007/s11117-017-0544-3
[6]	L. Q. Anh, T. Q. Duy, D. V. Hien, D. Kuroiwa, N. Petrot, Convergence of solutions to set optimization problems with the set less order relation, J. Optim. Theory Appl., 185 (2020), 416–432. https://doi.org/10.1007/s10957-020-01657-2 doi: 10.1007/s10957-020-01657-2
[7]	M. Dhingra, C. S. Lalitha, Approximate solutions and scalarization in set-valued optimization, Optimization, 66 (2017), 1793–1805. https://doi.org/10.1080/02331934.2016.1271419 doi: 10.1080/02331934.2016.1271419
[8]	M. Gupta, M. Srivastava, Approximate solutions and Levitin-Polyak well-posedness for set optimization using weak efficiency, J. Optim. Theory Appl., 186 (2020), 191–208. https://doi.org/10.1007/s10957-020-01683-0 doi: 10.1007/s10957-020-01683-0
[9]	Y. Han, K. Zhang, N. J. Huang, The stability and extended well-posedness of the solution sets for set optimization problems via the Painlevé-Kuratowski convergence, Math. Meth. Oper. Res., 91 (2020), 175–196. https://doi.org/10.1007/s00186-019-00695-5 doi: 10.1007/s00186-019-00695-5
[10]	T. Y. Li, Y. H. Xu, $\epsilon$-strict efficient solutions of vector optimization problems with set-valued maps, Asia-Pac. J. Oper. Res., 24 (2007), 841–854. https://doi.org/10.1142/S0217595907001577 doi: 10.1142/S0217595907001577
[11]	T. Y. Li, Y. H. Xu, The strictly efficient subgradient and the optimality conditions of set-valued optimization, Bull. Austral. Math. Soc., 75 (2007), 361–371. https://doi.org/10.1017/S0004972700039290 doi: 10.1017/S0004972700039290
[12]	G. P. Crespi, M. Dhingra, C. S. Lalitha, Pointwise and global well-posedness in set optimization: A direct approach, Ann. Oper. Res., 269 (2018), 49–66. https://doi.org/10.1007/s10479-017-2709-7 doi: 10.1007/s10479-017-2709-7
[13]	M. Furi, A. Vignoli, About well-posed optimization problems for functionals in metric spaces, J. Optim. Theory Appl., 5 (1970), 225–229. https://doi.org/10.1007/BF00927717 doi: 10.1007/BF00927717
[14]	M. Durea, Scalarization for pointwise well-posedness vectorial problems, Math Meth. Oper. Res., 66 (2007), 409–418. https://doi.org/10.1007/s00186-007-0162-0 doi: 10.1007/s00186-007-0162-0
[15]	W. Y. Zhang, S. J. Li, K. L. Teo, Well-posedness for set optimization problems, Nonlinear Anal., 71 (2009), 3769–3778. https://doi.org/10.1016/j.na.2009.02.036 doi: 10.1016/j.na.2009.02.036
[16]	X. J. Long, J. W. Peng, Generalized $B$-well-posedness for set optimization problems, J. Optim. Theory Appl., 157 (2013), 612–623. https://doi.org/10.1007/s10957-012-0205-4 doi: 10.1007/s10957-012-0205-4
[17]	M. Gupta, M. Srivastava, Well-posedness and scalarization in set optimization involving ordering cones with possibly empty interior, J. Glob. Optim., 73 (2019), 447–463. https://doi.org/10.1007/s10898-018-0695-1 doi: 10.1007/s10898-018-0695-1
[18]	X. J. Long, J. W. Peng, Z. Y. Peng, Scalarization and pointwise well-posedness for set optimization problems, J. Glob. Optim., 62 (2015), 763–773. https://doi.org/10.1007/s10898-014-0265-0 doi: 10.1007/s10898-014-0265-0
[19]	T. Y. Li, G. H. Xu, Scalarization and well-posedness for set optimization problems using generalized oriented distance function, J. Ind. Manage. Optim., 21 (2025), 1584–1599. https://doi.org/10.3934/jimo.2024139 doi: 10.3934/jimo.2024139
[20]	P. Loridan, Well-posedness in vector optimization, Mathematics and Its Applications, Springer, Dordrecht, 1995.
[21]	Y. Han, Connectedness of the approximate solution sets for set optimization problems, Optimization, 71 (2022), 4819–4834. https://doi.org/10.1080/02331934.2021.1969393 doi: 10.1080/02331934.2021.1969393
[22]	L. Zhu, F. Xia, Scalarization method for Levitin-Polyak well-posedness of vectorial optimization problems, Math Meth. Oper. Res., 76 (2012), 361–375. https://doi.org/10.1007/s00186-012-0410-9 doi: 10.1007/s00186-012-0410-9
[23]	E. Hern$\acute{a}$ndez, L. Rodríguez-Marín, Nonconvex scalarization in set optimization with set-valued maps, J. Math. Anal. Appl., 325 (2007), 1–18. https://doi.org/10.2298/AADM0702325M doi: 10.2298/AADM0702325M
[24]	M. Gupta, M. Srivastava, Hadamard well-posedness and stability in set optimization, Positivity, 7 (2024), 28. https://doi.org/10.1007/s11117-023-01026-z doi: 10.1007/s11117-023-01026-z
[25]	Khushboo, C. S. Lalitha, Scalarizations for a set optimization problem using generalized oriented distance function, Positivity, 23 (2019), 1195–1213. https://doi.org/10.1007/s11117-019-00659-3 doi: 10.1007/s11117-019-00659-3
[26]	Y. Zeng, Z. Y. Peng, T. Tammer, J. C. Yao, K. Deng, Scalarization and well-posedness for set optimization problems involving general set less relations, J. Optim. Theory Appl., 207 (2025), 37. https://doi.org/10.1007/s10957-025-02784-4 doi: 10.1007/s10957-025-02784-4
[27]	T. Y. Li, G. H. Xu, On Hadamard well-posedness and convergence in set optimization, Optimization, 2026, 1–21. https://doi.org/10.1080/02331934.2026.2618641 doi: 10.1080/02331934.2026.2618641
[28]	Karuna, C. S. Lalitha, External and internal stability in set optimization, Optimization, 68 (2019), 833–852. https://doi.org/10.1080/02331934.2018.1556663 doi: 10.1080/02331934.2018.1556663
[29]	Q. H. Ansari, N. Hussain, P. K. Sharma, Convergence of the solution sets for set optimization problems, J. Nonlinear Var. Anal., 6 (2022), 165–183. https://doi.org/10.23952/jnva.6.2022.3.01 doi: 10.23952/jnva.6.2022.3.01
[30]	Khushboo, C. S. Lalitha, Scalarization and convergence in unified set optimization, RAIRO-Oper. Res., 55 (2021), 3603–3616. https://doi.org/10.1051/ro/2021169 doi: 10.1051/ro/2021169
[31]	Z. Zhou, K. Feng, Q. H. Ansari, Well-posedness of set optimization problems with set order defined by Minkowski difference, J. Optim. Theory Appl., 204 (2025), 31. https://doi.org/10.1007/s10957-025-02608-5 doi: 10.1007/s10957-025-02608-5

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)