Optimality and duality in multiobjective programming involving support functions
Gupta, Rekha1 ; Srivastava, Manjari2
1 Department of Mathematics, University of Delhi 110007 Delhi, India
2 Department of Mathematics, Miranda House, University of Delhi 110007 Delhi, India
RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 2, pp. 433-446.

In this paper a vector optimization problem (VOP) is considered where each component of objective and constraint function involves a term containing support function of a compact convex set. Weak and strong Kuhn−Tucker necessary optimality conditions for the problem are obtained under suitable constraint qualifications. Necessary and sufficient conditions are proved for a critical point to be a weak efficient or an efficient solution of the problem (VOP) assuming that the functions belong to different classes of pseudoinvex functions. Two Mond Weir type dual problems are considered for (VOP) and duality results are established.

Reçu le :
Accepté le :
DOI : 10.1051/ro/2016039
Classification : 90C26, 90C29, 90C46
Mots-clés : Generalized invexity, multiobjective programming support functions, optimality conditions, Duality
Gupta, Rekha 1 ; Srivastava, Manjari 2

1 Department of Mathematics, University of Delhi 110007 Delhi, India
2 Department of Mathematics, Miranda House, University of Delhi 110007 Delhi, India 
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Gupta, Rekha; Srivastava, Manjari. Optimality and duality in multiobjective programming involving support functions. RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 2, pp. 433-446. doi : 10.1051/ro/2016039. http://www.numdam.org/articles/10.1051/ro/2016039/

M. Arana-Jiménez, A. Rufián-Lizana, R. Osuna-Gómez and G. Ruiz-Garzón, Pseudoinvexity, optimality conditions and efficiency in multiobjective problems; duality. Nonlin. Anal. 68 (2008) 24–34. | DOI | MR | Zbl

M. Arana-Jiménez, A. Rufián-Lizana, R. Osuna-Gómez and G. Ruiz-Garzón, A characterization of pseudoinvexity in multiobjective programming. Math. Comput. Model. 48 (2008) 1719–1723. | DOI | MR | Zbl

M. Arana-Jiménez, G. Ruiz-Garzón, A. Rufián-Lizana and B. Hernández-Jiménez, A characterization of pseudoinvexity for the efficiency in non-differentiable multiobjective problems. Duality. Nonlin. Anal. 73 (2010) 1109–1117. | DOI | MR | Zbl

C.R. Bector, S. Chandra and J. Dutta, Principles of optimization theory. Narosa Publishing House, New Delhi (2005)

A. Ben-Israel and B. Mond, What is invexity? J. Aust. Math. Soc. Ser. B 28 (1986) 1–9. | DOI | MR | Zbl

F.H. Clarke, Optimization and nonsmooth analysis. Interscience Publication. Wiley, New York (1983). | MR | Zbl

B.D. Craven, Invex functions and constrained local minima. J. Aust. Math. Soc. Ser. B 24 (1983) 357–366. | MR | Zbl

B.D. Craven and B.M. Glover, Invex functions and duality. J. Aust. Math. Soc. Ser. A 39 (1985) 1–20. | DOI | MR | Zbl

M.A. Hanson, On sufficiency of the Kuhn Tucker conditions. J. Math. Anal. Appl. 80 (1981) 545–550. | DOI | MR | Zbl

I. Husain and Z. Abha, Jabeen, On nonlinear programing with support functions. J. Appl. Math. Comput. 10 (2002) 83–99. | DOI | MR | Zbl

I. Husain and Z. Jabeen, On fractional programming containing support functions. J. Appl. Math. Comput. 18 (2005) 361–376. | DOI | MR | Zbl

P. Kanniappan, Necessary conditions for optimality of nondifferentiable convex multiobjective programming. J. Optim. Theory Appl. 40 (1983) 167–174. | DOI | MR | Zbl

O.L. Mangasarian, Nonlinear Programming. McGraw Hill, New York (1969) | MR | Zbl

D.H. Martin, The essence of invexity. J. Optim. Theory Appl. 47 (1985) 65–76. | DOI | MR | Zbl

B. Mond and M. Schechter, A duality theorem for a homogeneous fractional programming problem. J. Optim. Theory Appl. 25 349–359 (1978) | DOI | MR | Zbl

R. Osuna-Gómez, A. Beato-Moreno and A. Rufián-Lizana, Generalized convexity in multiobjective programming. J. Math. Anal. Appl. 233 (1999) 205–220. | DOI | MR | Zbl

R. Osuna-Gómez, A. Rufián-Lizana and P. Ruíz-Canales, Invex functions and generalized convexity in multiobjective programming. J. Optim. Theory Appl. 98 (1998) 651–661. | DOI | MR | Zbl

M. Schechter, A subgradient duality theorem. J. Math. Anal. Appl. 61 (1977) 850–855. | DOI | MR | Zbl

W. Stadler (ed.), Multicriteria optimization in mechanics: a survey. Appl. Mech. Rev. 37 (1984) 277–286.

W. Stadler, Multicriteria Optimization in Engineering and in the Sciences. Plenum Press, New York (1988). | MR | Zbl

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