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Vector variational inequalities and their relations with vector optimization

Journal Name:

Publication Year:

DOI: 
10.11121/ijocta.01.2014.00160

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Abstract (Original Language): 
In this paper, K - c quasiconvex, K - c pseudoconvex and other related functions have been introduced in terms of their Clarke subdifferentials, where K is an arbitrary closed convex, pointed cone with nonempty interior. The (strict, weakly) K -pseudomonotonicity, (strict) K -naturally quasimonotonicity and K -quasimonotonicity of Clarke subdifferential maps have also been defined. Further, we introduce Minty weak (MVVIP) and Stampacchia weak (SVVIP) vector variational inequalities over arbitrary cones. Under regularity assumption, we have proved that a weak minimum solution of vector optimization problem (VOP) is a solution of (SVVIP) and under the condition of K - c pseudoconvexity we have obtained the converse for MVVIP (SVVIP). In the end we study the interrelations between these with the help of strict K -naturally quasimonotonicity of Clarke subdifferential map.
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REFERENCES

References: 

[1] Hartman, P., Stampacchia, G., On Some
Nonlinear Elliptic Differential Functional
Equations, Acta Mathematica, 115, pp.153-
188, (1966).
[2] Giannessi, F., Theorems of the Alternative,
Quadratic Programs and Complementarity
Problem, In: R. W. Cottle, F. Giannessi and
J. L. Lions (Eds.), Variational Inequalities
and Complementarity Problems, John Wiley
and Sons, New York, pp. 151-186 (1980).
[3] Chen, G. Y., Yang, X. Q., Vector
Complementary Problem and its
Equivalence With Weak Minimal Element
in Ordered Spaces, Journal of Mathematical
Analysis and Applications, 153, pp. 136-
158, (1990).
[4] Chen, G. Y., Existence of Solutions for A
Vector Variational Inequality, An extension
of Hartman-Stampacchia theorem, Journal
of Optimization Theory and Applications,74,
pp. 445-456, (1992).
[5] Yao, J. C., Multivalued Variational
Inequalities With
K -pseudomonotone
Operators, Journal of Optimization Theory
and Applications, 83, pp. 391-403, (1994).
[6] Giannessi, F., On Minty Variational
Principle, New Trends in Mathematical
Programming, Kluwer Academic
Publishers, Dordrecht, Netherlands, (1997).
[7] Giannessi, F., On Minty Variational
Principle. In: F. Giannessi, S. Komlosi and
T. Rapesak (Eds.), New Trends in
Mathematical Programming, Dordrecht,
Netherlands: Kluwer Academic Publishers,
pp. 93-99 (1998).
[8] Giannessi, F., Vector Variational
Inequalities and Vector Equilibria,
Dordrecht, Netherlands: Kluwer Academic
Publishers, (2000).
[9] Komlosi, S., On the Stampacchia and Minty
Variational Inequalities. In: G. Giorgi and F.
Rossi (Eds), Generalized Convexity and
Optimization for Economic and Financial
Decisions, Bologna: Pitagora Editrice, pp.
231-260, (1999).
[10] Yang, X. M., Yang, X. Q., Teo, K. L.,
Some Remarks On the Minty Vector
Variational Inequality, Journal of
Optimization Theory and Applications, 121,
pp. 193- 201, (2004).
[11] Lee, G. M., Lee, K. B., Vector Variational
Inequalities for Nondifferentiable Convex
Vector Optimization Problems, Journal of
Global Optimization, 32, pp. 597-612
(2005).
[12] Mishra, S. K., Wang, S. Y., Vector
Variational–like Inequalities and Nonsmooth
Vector Optimization Problems, Nonlinear
Analysis, 64, pp. 1939-1945, (2005).
[13] Chinaie, M., Jabarootian, T., Rezaie, M.,
Zafarani, J., Minty’s Lemma and Vector
Variational –like Inequalities, Journal of
Global Optimization, 40, pp. 463- 473,
(2008).
[14] Rezaie, M., Zafarani, J., Vector
Optimization and Variational like
Inequalities, Journal of Global
Optimization, 43, pp. 47-66, (2009).
[15] Karamardian, S., Schaible, S., Seven Kinds
of Monotone Maps, Journal of Optimization
Theory and Applications, 66, pp. 37-46,
(1990).
[16] Hadjisavvas, N., Schaible, S., On Strong
Pseudomonotonicity and (Semi) Strict
Quasimonotonicity, Journal of Optimization
Theory and Applications, 79, 1, pp. 139-155,
(1993).
[17] Schaible, S., Generalized Monotonicity, A
Survey, Generalized Convexity, Edited by S.
K. Komlosi, T. Rapesak and S. Schaible,
Springer Verlag, Berlin, Germany, pp. 229-
249, (1994).
[18] Cambini, R., Some New Classes of
Generalized Concave Vector Valued
Functions, Optimization, 36, pp.11-24,
(1996).
[19] Cambini, A., Martein, L., Generalized
Convexity and Optimality Conditions in
Scalar and Vector Optimization, Handbook
of Generalized Convexity and Generalized
Monotonicity. Edited by N. Hadjisavvas,
Komlosi and S. Schaible, Springer
Netherlands, 76, pp. 151-193(2005).
[20] Vani, The Study of Optimality and Duality in
Vector Optimization Problems, Thesis
(PhD), University of Delhi, (2007).
[21] Clarke, F. H., Optimization and Nonsmooth
Analysis, Wiley Interscience, (1983).
[22] Bector, C. R., Chandra, S., Dutta, J.,
Principles of Optimization Theory, India:
Narosa Publishing House, (2005).
[23] Swaragi, Y. H., Nakayama, H., Tanino, T.,
Theory of Multiobjective Optimization,
Academic Press, New York, (1985).
[24] Jahn, J., Vector Optimization: theory,
applications, and extensions. Springer–
Verlag Berlin Heidelberg, (2004).
[25] Aggarwal, S., Optimality and Duality in
Mathematical Programming Involving
Generalized Convex Functions, Thesis
(PhD), University of Delhi, (1998).

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