Vector variational inequalities and their relations with vector optimization
DOI:
https://doi.org/10.11121/ijocta.01.2014.00160Keywords:
Generalized nonsmooth cone convexity, generalized cone monotonicity, vector optimization problem, vector variational inequality problem.Abstract
In this paper, K- quasiconvex, K- pseudoconvex and other related functions have been introduced in terms of their Clarke subdifferentials, where is an arbitrary closed convex, pointed cone with nonempty interior. The (strict, weakly) -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- 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
Hartman, P., Stampacchia, G., On Some Nonlinear Elliptic Differential Functional Equations, Acta Mathematica, 115, pp.153-188, (1966). CrossRef
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 (New York: John wiley and sons), pp. 151-186 (1980).
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). CrossRef
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). CrossRef
Yao, J. C., Multivalued Variational Inequalities With -pseudomonotone Operators, Journal of Optimization Theory and Applications, 83, pp. 391-403, (1994). CrossRef
Giannessi, F., On Minty Variational Principle, New Trends in Mathematical Programming, Kluwer Academic Publishers, Dordrecht, Netherlands, (1997).
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). CrossRef
Giannessi, F., Vector Variational Inequalities and Vector Equilibria (Dordrecht, Netherlands: Kluwer Academic Publishers), (2000). CrossRef
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).
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). CrossRef
Lee, G. M., Lee, K. B., Vector Variational Inequalities for Nondifferentiable Convex Vector Optimization Problems, Journal of Global Optimization, 32, pp. 597-612 (2005). CrossRef
Mishra, S. K., Wang, S. Y., Vector Variational–like Inequalities and Nonsmooth Vector Optimization Problems, Nonlinear Analysis, 64, pp. 1939-1945, (2005). CrossRef
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). CrossRef
Rezaie, M., Zafarani, J., Vector Optimization and Variational –like Inequalities, Journal of Global Optimization, 43, pp. 47-66, (2009). CrossRef
Karamardian, S., Schaible, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, 66, pp. 37-46, (1990). CrossRef
Hadjisavvas, N., Schaible, S., On Strong Pseudomonotonicity and (Semi) Strict Quasimonotonicity, Journal of Optimization Theory and Applications, 79, 1, pp. 139-155, (1993). CrossRef
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). CrossRef
Cambini, R., Some New Classes of Generalized Concave Vector Valued Functions, Optimization, 36, pp.11-24, (1996). CrossRef
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).
Vani, The Study of Optimality and Duality in Vector Optimization Problems, Thesis (PhD), University of Delhi, (2007).
Clarke, F. H., Optimization and Nonsmooth Analysis, Wiley Interscience, (1983).
Bector, C. R., Chandra, S., Dutta, J., Principles of Optimization Theory, India: Narosa Publishing House, (2005).
Swaragi, Y. H., Nakayama, H., Tanino, T., Theory of Multiobjective Optimization, Academic Press, New York, (1985).
Jahn, J., Vector Optimization: theory, applications, and extensions. Springer– Verlag Berlin Heidelberg, (2004).
Aggarwal, S., Optimality and Duality in Mathematical Programming Involving Generalized Convex Functions, Thesis (PhD), University of Delhi, (1998).
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