Existence of pseudo-relative sharp minimizers in set-valued optimization. (English) Zbl 1478.90115
Appl. Math. Optim. 84, No. 3, 2969-2984 (2021); correction ibid. 90, No. 1, Paper No. 8, 2 p. (2024).
Summary: In this paper, we propose new concepts of sharp minimality in set-valued optimization problems by means of the pseudo-relative interior, namely pseudo-relative \(\phi \)-sharp minimizers. Based on this notion of minimality, we extend the existence result of a unique minimum of uniformly convex real-valued functions proved by C. Zălinescu [Convex analysis in general vector spaces. Singapore: World Scientific (2002; Zbl 1023.46003)] to vector-valued as well as set-valued maps. Additionally, we provide some existence results for weak sharp minimizers in the sense of M. Durea and R. Strugariu [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 7, 2148–2157 (2010; Zbl 1225.90115)].
MSC:
| 90C29 | Multi-objective and goal programming |
| 54C60 | Set-valued maps in general topology |
| 90C25 | Convex programming |
Keywords:
set-valued optimization; pseudo-relative interior; uniform convexity; pseudo-relative \(\phi \)-sharp minimality; existence resultsReferences:
| [1] | Amahroq, T.; Oussarhan, A., Infinite-dimensional vector optimization and a separation theorem, Optimization, 69, 12, 2611-2627 (2020) · Zbl 1517.90130 · doi:10.1080/02331934.2020.1736070 |
| [2] | Amahroq, T.; Daidai, I.; Syam, A., Optimality conditions for sharp minimality of order \(\gamma\) in set-valued optimization, Le matematiche journal, 73, 1, 99-114 (2018) · Zbl 1434.49009 |
| [3] | Aubin, J.P., Cellina, A.: Differential Inclusions. Set-Valued Maps and Viability Theory, in: Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin; vol. 264 (1984) · Zbl 0538.34007 |
| [4] | Auslender, A., Stability in mathematical programming with nondifferentiable data, SIAM J. Control Optim., 22, 239-254 (1984) · Zbl 0538.49020 · doi:10.1137/0322017 |
| [5] | Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program.122(2, Ser. A), 301-347 (2010) · Zbl 1184.90149 |
| [6] | Bednarczuk, EM, Weak sharp efficiency and growth condition for vector-valued functions with applications, Optimization, 53, 455-474 (2004) · Zbl 1153.90529 · doi:10.1080/02331930412331330478 |
| [7] | Borwein, J.M., Goebel, R.: Notions of relative interior in Banach spaces, J. Math. Sci. (N. Y.), 115, 2542-2553 (2003) · Zbl 1136.49307 |
| [8] | Cromme, L., Strong uniqueness: a far reaching criterion for the convergence of iterative procedures, Numerische Mathematik, 29, 179-193 (1978) · Zbl 0352.65012 · doi:10.1007/BF01390337 |
| [9] | Durea, M.; Strugariu, R., Necessary optimality conditions for weak sharp minima in set-valued optimization, Nonlinear Analysis, 73, 2148-2157 (2010) · Zbl 1225.90115 · doi:10.1016/j.na.2010.05.041 |
| [10] | Flores-Bazán, F.; Jiménez, B., Strict efficiency in set-valued optimization, SIAM J. Control Optim., 48, 881-908 (2009) · Zbl 1201.90179 · doi:10.1137/07070139X |
| [11] | Holmes, R.B.: Geometric Functional Analysis and Its Applications. Graduate Texts in Mathematics, vol. 24. Springer, New York (1975) · Zbl 0336.46001 |
| [12] | Huang, H., Global error bounds with exponents for multifunctions with set constraints, Commun. Contemp. Math., 12, 417-435 (2010) · Zbl 1218.90190 · doi:10.1142/S0219199710003865 |
| [13] | Huang, H., Inversion theorem for nonconvex multifunctions, Math. Inequal. Appl., 13, 841-849 (2010) · Zbl 1226.90112 |
| [14] | Jiménez, B., Strict efficiency in vector optimization, J. Math. Anal. Appl., 265, 264-284 (2002) · Zbl 1010.90075 · doi:10.1006/jmaa.2001.7588 |
| [15] | Jiménez, B., Strict minimality conditions in nondifferentiable multiobjective programming, Journal of Optimization Theory and Applications, 116, 99-116 (2003) · Zbl 1030.90116 · doi:10.1023/A:1022162203161 |
| [16] | Jiménez, B.; Novo, V., Higher-order optimality conditions for strict local minima, Ann. Oper. Res., 157, 183-192 (2008) · Zbl 1190.90265 · doi:10.1007/s10479-007-0197-x |
| [17] | Khan, AA; Tammer, C.; Zălinescu, C., Set-valued optimization: An introduction with applications (2015), Heidelberg: Springer, Heidelberg · Zbl 1308.49004 · doi:10.1007/978-3-642-54265-7 |
| [18] | Kimura, Y.; Suzuki, T.; Tanaka, T., Monotonically upper semicontinuity, Bull. Kyushu Inst. Tech. Pure Appl Math., 52, 29-34 (2005) · Zbl 1076.54010 |
| [19] | Kimura, Y., Tanaka, K., Tanaka, T.: On semicontinuity of set-valued maps and marginal functions, in Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis (W. Takahashi and T. Tanaka Eds.), pp. 181-188, World Scientific, (1999) · Zbl 1001.47029 |
| [20] | Studniarski, M., Necessary and sufficient conditions for isolated local minima of nonsmooth functions, SIAM J. Control Optim., 24, 1044-1049 (1986) · Zbl 0604.49017 · doi:10.1137/0324061 |
| [21] | Tanaka, T., Seino, T.: On a Theoretically Conformable Duality for Semicontinuity of Set-Valued Mappings,, Vol.939, pp.1-10, RIMS koukyuroku, Vol.939, 1-10 (1996) |
| [22] | Ward, DE, Characterizations of strict local minima and necessary conditions for weak sharp minima, J. Optim. Theory Appl., 80, 551-571 (1994) · Zbl 0797.90101 · doi:10.1007/BF02207780 |
| [23] | Zaffaroni, A., Degrees of efficiency and degrees of minimality, SIAM J Control Optim., 42, 1071-1086 (2003) · Zbl 1046.90084 · doi:10.1137/S0363012902411532 |
| [24] | Zălinescu, C., On uniformly convex functions, J. Math. Anal. App., 95, 344-374 (1983) · Zbl 0519.49010 · doi:10.1016/0022-247X(83)90112-9 |
| [25] | Zălinescu, C., Convex Analysis in General Vector Spaces (2002), Singapore: World Scientific, Singapore · Zbl 1023.46003 · doi:10.1142/5021 |
| [26] | Zheng, X.-Y. NG, K. F.: Hölder stable minimizers, tilt stability and Hölder metric regularity of subdifferentials, SIAM Journal. Optimization 25, 416-438 (2015) · Zbl 1378.90079 |
| [27] | Zhu, SK; Li, SJ; Xue, XW, Strong Fermat Rules for Constrained Set-Valued Optimization Problems on Banach Spaces, Set-Valued Var. Anal., 20, 637-666 (2012) · Zbl 1263.49016 · doi:10.1007/s11228-012-0214-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
