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Optimality and duality for non-Lipschitz multiobjective optimization problems. (English) Zbl 1211.90203
Summary: Nowadays set-valued optimization means set-valued analysis and its application to optimization, and it is an extension of continuous optimization to the set-valued case. In this research area, one investigates optimization problems with constraints and/or an objective function described by set-valued maps, or investigations in set-valued analysis are applied to standard optimization problems. In this article, we are concerned with a set-valued optimization problem $$(P)$$. Using a notion of approximation derived from Jourani and Thibault, we give necessary and sufficient optimality conditions for $$(P)$$. Based on necessary optimality conditions given by T. Amahroq and N. Gadhi [J. Glob. Optim. 21, No. 4, 433–441 (2001; Zbl 1175.90409)], our approach consists of formulating the Mond-Weir dual problem $$(D)$$ and establishing duality theorems for $$(P)$$ and $$(D)$$ without any constraint qualification.
##### MSC:
 90C29 Multi-objective and goal programming 49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
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##### References:
 [1] DOI: 10.1080/02331939708844311 · Zbl 0877.49024 · doi:10.1080/02331939708844311 [2] DOI: 10.1023/A:1012748412618 · Zbl 1175.90409 · doi:10.1023/A:1012748412618 [3] Dien P.H., Acta Math Vietnamica 1 pp 109– (1983) [4] Gadhi N., Georgian Mathematical Journal 12 pp 65– (2005) [5] Giorgi G., Generalized Convexity, Generalized Monotonicity: Recent Results pp 389– (1998) [6] DOI: 10.1112/S0025579300013541 · Zbl 0713.49022 · doi:10.1112/S0025579300013541 [7] DOI: 10.1007/BF01217690 · Zbl 0889.90123 · doi:10.1007/BF01217690 [8] DOI: 10.1023/A:1021790120780 · Zbl 0956.90033 · doi:10.1023/A:1021790120780 [9] Jeyakumar V., Journal of Convex Analysis 5 pp 119– (1998) [10] Jourani A., Math. Oper. Res. 18 pp 73– (1998) [11] DOI: 10.1023/A:1017596530143 · Zbl 0987.90072 · doi:10.1023/A:1017596530143 [12] DOI: 10.1080/01630569208816467 · Zbl 0724.49010 · doi:10.1080/01630569208816467 [13] Kuroiwa D., Natural Criteria of Set-Valued Optimization (1998) · Zbl 0939.90570 [14] DOI: 10.1023/A:1021786303883 · Zbl 0915.90253 · doi:10.1023/A:1021786303883 [15] DOI: 10.1007/BF01594928 · Zbl 0718.90080 · doi:10.1007/BF01594928 [16] DOI: 10.1017/S0004972700011679 · Zbl 0755.90072 · doi:10.1017/S0004972700011679 [17] Loewen P.D., Optimization and Nonlinear Analysis, Pitman Research Notes Math, Series pp 178– (1992) · Zbl 0766.49013 [18] DOI: 10.1023/B:JOGO.0000047911.03061.88 · Zbl 1119.90049 · doi:10.1023/B:JOGO.0000047911.03061.88 [19] Mond B., Generalized Concavity in Optimization and Economics pp 263– (1981) [20] Mordukhovich B.S., Journal of Convex Analysis 2 pp 211– (1995) [21] DOI: 10.1007/BF01442543 · Zbl 0401.90104 · doi:10.1007/BF01442543
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