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http://dx.doi.org/10.14317/jami.2012.30.1_2.111

ON SUFFICIENCY AND DUALITY IN MULTIOBJECTIVE SUBSET PROGRAMMING PROBLEMS INVOLVING GENERALIZED $d$-TYPE I UNIVEX FUNCTIONS  

Jayswal, Anurag (Department of Applied Mathematics, Indian School of Mines)
Stancu-Minasian, I.M. (Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy)
Publication Information
Journal of applied mathematics & informatics / v.30, no.1_2, 2012 , pp. 111-125 More about this Journal
Abstract
In this paper, we introduce new classes of generalized convex n-set functions called $d$-weak strictly pseudo-quasi type-I univex, $d$-strong pseudo-quasi type-I univex and $d$-weak strictly pseudo type-I univex functions and focus our study on multiobjective subset programming problem. Sufficient optimality conditions are obtained under the assumptions of aforesaid functions. Duality results are also established for Mond-Weir and general Mond-Weir type dual problems in which the involved functions satisfy appropriate generalized $d$-type-I univexity conditions.
Keywords
Multiobjective subset programming problem; efficient solution; $d$-type-I univex functions; sufficient optimality conditions; duality;
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1 B. Aghezzaf and M. Hachimi, Generalized invexity and duality in multiobjective programming problems, J. Global Optim. 18 (2000)1, 91-101.   DOI
2 I. Ahmad and S. Sharma, Suffciency in multiobjective subset programming involving generalized type-I functions, J. Global Optim. 39 (2007)3, 473-481.   DOI
3 C.R. Bector, S.K. Suneja and S. Gupta, Univex functions and univex nonlinear programming, In: Proceedings of the Administrative Sciences Association of Canada, (1992) 115-124.
4 C.R. Bector and M. Singh, Duality for multiobjective B-vex programming involving n-set functions, J. Math. Anal. Appl. 202 (1996)3, 701-726.   DOI
5 D. Begis and R. Glowinski, Application de la methode des elements finis a 1'approximation d'un probleme de domaine optimal, Methodes des resolution des problemes approches, Appl. Math. Optim. 2 (1975)2, 130-169.   DOI
6 J. Cea, A. Gioan and J. Michel, Quelques resultats sur 1'identification de domaines, Calcolo. 10 (1973)3-4, 207-232.   DOI
7 H.W. Corley and S.D. Roberts, A partitioning problem with applications in regional design, Operations Res. 20 (1972), 1010-1019.   DOI
8 H.W. Corley and S.D. Roberts, Duality relationships for a partitioning problem, SIAM J. Appl. Math. 23 (1972), 490-494.   DOI
9 J.H. Chou, W.S. Hsia and T.Y. Lee, On multiple objective programming problems with set functions, J. Math. Anal. Appl. 105 (1985)2, 383-394.   DOI
10 H.W. Corely, Optimization theory for n-set functions, J. Math. Anal. Appl. 127 (1987)1, 193-205.   DOI
11 G. Dantzig and A.Wald, On the fundamental lemma of Neyman and Pearson, Ann. Math. Statistics 22 (1951), 87-93.   DOI
12 M.A. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Programming 37 (1987)1, 51-58.   DOI
13 A. Jayswal and R. Kumar, Some nondifferentiable multiobjective programming under generalized d-V -type-I univexity, J. Comput. Appl. Math. 229 (2009)1, 175-182.   DOI
14 R.N. Kaul, S.K. Suneja and M.K. Srivastava, Optimality criteria and duality in multiple-objective optimization involving generalized invexity, J. Optim. Theory Appl. 80 (1994)3, 465-482.   DOI
15 H.C. Lai and L.J. Lin, Optimality for set functions with values in ordered vector spaces, J. Optim. Theory Appl. 63 (1989)3, 371-389.   DOI
16 L.J. Lin, Optimality of differentiable vector-valued n-set functions, J. Math. Anal. Appl. 149 (1990)1, 255-270.   DOI
17 L.J. Lin, On the optimality condition of vector-valued n-set functions, J. Math. Anal. Appl. 161 (1991)2, 367-387.   DOI
18 L.J. Lin, Duality theorems of vector-valued n-set functions, Comput. Math. Appl. 21 (1991)11-12, 165-175   DOI
19 J. Neyman and E.S. Pearson, On the problem of the most efficient tests of statistical hypotheses, Philos. Trans. Soc. Lond. Ser. 231 (1933), 289-337.   DOI
20 S.K. Mishra, S.Y. Wang, K.K. Lai and J. Shi, New generalized invexity for duality in multiobjective programming problems involving N-set functions, In Generalized Convexity, Generalized Monotonicity and Applications, pp. 321-339 Edited by Andrew Eberhard, Nicolas Hadjisawas, D. T. Luc, Nonconvex Optim. Appl., 77, Springer, New York (2005).
21 V. Preda and I.M. Stancu-Minasian, Optimality and Wolfe duality for multiobjective programming problems involving n-set functions, In: Konnov, I.V. Luc, D.T. Rubinov, A.M.(eds.) Generalized Convexity and Generalized Monotonicity (Karlovassi, 1999), Lecture Notes in Econom. and Math. Systems, vol. 502, pp. 349-361. Springer-Verlag, Berlin (2001).
22 V. Preda, I.M. Stancu-Minasian and E. Koller, Optimality and duality for multiobjective programming problems involving generalized d-type-I and related n-set functions, J. Math. Anal. Appl. 283 (2003)1, 114-128.   DOI
23 V. Preda, I. M. Stancu-Minasian, M. Beldiman and A.M. Stancu, Generalized V -univexity type-I for multiobjective programming with n-set functions, J.Global Optim. 44 (2009)1, 131-148.   DOI
24 J.T. Robert and Morris: Optimal constrained selection of a measurable subset, J. Math. Anal. Appl. 70 (1979)2, 546-562.   DOI
25 N.G. Rueda and M.A. Hanson, Optimality criteria in mathematical programming involving generalized invexity, J. Math. Anal. Appl. 130 (1988)2, 375-385.   DOI
26 Y.L. Ye, D-invexity and optimality conditions, J. Math. Anal. Appl. 162 (1991)1, 242-249.   DOI
27 G.J. Zalmai, Suffciency criteria and duality for nonlinear programs involving n-set functions, J. Math. Anal. Appl. 149 (1990)2, 322-338.   DOI
28 G.J. Zalmai, Optimality conditions and duality for multiobjective measurable subset selection problems, Optimization 22 (1990)2, 221-238.   DOI