Optimality conditions for vector optimization problems with difference of convex maps. (English) Zbl 1307.90160
The authors establish new optimality conditions for the approximate Pareto minimal points of some significant vector optimization problems with difference convex maps.
Reviewer: Vasile Postolică (Piatra Neamt)
MSC:
| 90C29 | Multi-objective and goal programming |
| 90C46 | Optimality conditions and duality in mathematical programming |
| 90C48 | Programming in abstract spaces |
References:
| [1] | Taa, A.: ϵ-subdifferentials of set-valued maps and ϵ-weak Pareto optimality for multiobjective optimization. Math. Methods Oper. Res. 62, 187-209 (2005) · Zbl 1109.49021 · doi:10.1007/s00186-005-0007-7 |
| [2] | Liu, J.C., Yokoyama, K.: ϵ-optimality and duality for fractional programming. Taiwan. J. Math. 3, 311-322 (1999) · Zbl 0951.90044 |
| [3] | Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Methods Oper. Res. 48, 187-200 (1998) · Zbl 0927.90095 · doi:10.1007/s001860050021 |
| [4] | Flores-Bazan, F., Oettli, W.: Simplified optimality conditions for minimizing the difference of vector-valued functions. J. Optim. Theory Appl. 108, 571-586 (2001) · Zbl 0984.90053 · doi:10.1023/A:1017535424813 |
| [5] | Gadhi, N.: Optimality conditions for the difference of convex set-valued mappings. Positivity 9, 687-703 (2005) · Zbl 1151.90536 · doi:10.1007/s11117-005-2786-8 |
| [6] | Lin, L.J.: Optimization of set-valued functions. J. Math. Anal. Appl. 186, 30-51 (1994) · Zbl 0987.49011 · doi:10.1006/jmaa.1994.1284 |
| [7] | Li, S.J., Yang, X.Q., Chen, G.Y.: Nonconvex vector optimization of set-valued mappings. J. Optim. Theory Appl. 283, 337-350 (2003) · Zbl 1137.90642 |
| [8] | Li, S.J., Chen, C.R.: High order optimality conditions for Henig efficient solutions in set-valued optimization. J. Math. Anal. Appl. 323, 1184-1200 (2006) · Zbl 1144.90481 · doi:10.1016/j.jmaa.2005.11.035 |
| [9] | Li, S.J., Zhu, S.K., Li, X.B.: Second-order optimality conditions for strict efficiency of constrained set-valued optimization. J. Optim. Theory Appl. (2012). doi:10.1007/s10957-012-0076-8 · Zbl 1268.90078 · doi:10.1007/s10957-012-0076-8 |
| [10] | Gadhi, N., Metrane, A.: Sufficient optimality condition for vector optimization problems under D.C. data. J. Glob. Optim. 28, 55-66 (2004) · Zbl 1061.90098 · doi:10.1023/B:JOGO.0000006715.69153.8b |
| [11] | Maghri, M.E., Laghdir, M.: Pareto subdifferential calculus for convex vector mappings and applications to vector optimization. SIAM J. Optim. 19, 1970-1994 (2009) · Zbl 1176.49023 · doi:10.1137/070704046 |
| [12] | Bot, R.I., Hodrea, I.B., Wanka, G.: ϵ-optimality conditions for composed convex optimization problems. J. Approx. Theory 153, 108-121 (2008) · Zbl 1158.46029 · doi:10.1016/j.jat.2008.03.002 |
| [13] | Durea, M., Dutta, J., Tammer, C.: Lagrange multipliers for ϵ-Pareto solutions in vector optimization with nonsolid cones in Banach spaces. J. Optim. Theory Appl. 145, 196-211 (2010) · Zbl 1189.90137 · doi:10.1007/s10957-009-9609-1 |
| [14] | Hiriart-Urruty, J. B.; Clarke, F. H. (ed.); Demyanov, V. F. (ed.); Giannessi, F. (ed.), From convex optimization to nonconvex optimization, 219-239 (1989), New York · Zbl 0735.90056 · doi:10.1007/978-1-4757-6019-4_13 |
| [15] | Amahroq, T., Penot, J.P., Syam, A.: On the subdifferentiability of the difference of two functions and local minimization. Set-Valued Anal. 16(4), 413-427 (2008) · Zbl 1165.26307 · doi:10.1007/s11228-008-0085-9 |
| [16] | Bot, R.I., Nechita, D.: On the Dini-Hadamard subdifferential of the difference of two functions. J. Glob. Optim. 50, 485-502 (2011) · Zbl 1220.26010 · doi:10.1007/s10898-010-9604-y |
| [17] | Laghdir, M.: Optimality conditions in DC constrained optimization. Acta Math. Vietnam. 30(2), 169-179 (2005) · Zbl 1155.90474 |
| [18] | Penot, J.P.: The directional subdifferential of the difference of two convex functions. J. Glob. Optim. 49, 505-519 (2011) · Zbl 1223.26030 · doi:10.1007/s10898-010-9615-8 |
| [19] | Gadhi, N., Laghdir, M., Metrane, A.: Optimality conditions for D.C. vector optimization problems under reverse convex constraints. J. Glob. Optim. 33, 527-540 (2005) · Zbl 1121.90403 · doi:10.1007/s10898-004-8318-4 |
| [20] | Zowe, J.: Subdifferentiability of convex functions with values in an ordered vector space. Math. Scand. 34, 69-83 (1974) · Zbl 0288.46007 |
| [21] | Penot, J.P.: Gap continuity of multimaps. Set-Valued Anal. 16(4), 429-442 (2008) · Zbl 1173.54009 · doi:10.1007/s11228-007-0061-9 |
| [22] | Baier, J., Jahn, J.: On subdifferentials of set-valued maps. J. Optim. Theory Appl. 100, 233-240 (1999) · Zbl 0922.90118 · doi:10.1023/A:1021733402240 |
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