American Institute of Mathematical Sciences

2011, 1(3): 333-339. doi: 10.3934/naco.2011.1.333

General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity

 1 Texas A&M University--Kingsville, Department of Mathematics, Kingsville, Texas 78363, United States

Received  March 2011 Revised  May 2011 Published  September 2011

Motivated by the recent investigations, first a general framework for a class of $(\rho, \eta, A)$-invex n-set functions is introduced, and then some optimality conditions for multiple objective fractional programming on the generalized $(\rho, \eta, A)$-invexity are explored. The obtained results are general in nature and application-oriented to other investigations on fractional subset programming in literature.
Citation: Ram U. Verma. General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 333-339. doi: 10.3934/naco.2011.1.333
References:
 [1] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80 (1981), 545-550. doi: 10.1016/0022-247X(81)90123-2. [2] L. Caiping and Y. Xinmin, Generalized $(\rho, \theta, \eta)-$invariant monotonicity and generalized $(\rho, \theta, \eta)-$invexity of non-differentiable functions, Journal of Inequalities and Applications, 2009 (2009), Article ID\#393940, 16 pages. doi: 10.1155/2009/393940. [3] S. K. Mishra, M. Jaiswal and Pankaj, Optimality conditions for multiple objective fractional subset programming with invex and related nonconvex functions, Communications on Applied Nonlinear Analysis, 17 (2010), 89-101. [4] S. K. Mishra, S. Y. Wang and K. K. Lai, Generalized Convexity and Vector Optimization, Nonconvex Optimization and its Applications, Vol. 19, Springer-Verlag, 2009. [5] R. U. Verma, Approximation solvability of a class of nonlinear set-valued inclusions involving $(A,\eta)-$monotone mappings, Journal of Mathematical Analysis and Applications, 337 (2008), 969-975. [6] R. U. Verma, The optimality condition for multiple objective fractional subset programming based on generalized $(\rho,\eta)-$invex functions, Advances in Nonlinear Variational Inequalities, 14 (2011), 61-72. [7] G. J. Zalmai and Q. B. Zhang, Generalized $(F, \beta, \phi, \rho, \theta)-$univex functions and parametric duality in semiinfinite discrete minmax fractional programming, Advances in Nonlinear Variational Inequalities, 10 (2007), 1-20. [8] G. J. Zalmai and Q. B. Zhang, Generalized $(F, \beta, \phi, \rho, \theta)-$univex functions and global parametric sufficient optimality conditions in semiinfinite discrete minmax fractional programming, PanAmerican Mathematical Journal, 17 (2007), 1-26.

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References:
 [1] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80 (1981), 545-550. doi: 10.1016/0022-247X(81)90123-2. [2] L. Caiping and Y. Xinmin, Generalized $(\rho, \theta, \eta)-$invariant monotonicity and generalized $(\rho, \theta, \eta)-$invexity of non-differentiable functions, Journal of Inequalities and Applications, 2009 (2009), Article ID\#393940, 16 pages. doi: 10.1155/2009/393940. [3] S. K. Mishra, M. Jaiswal and Pankaj, Optimality conditions for multiple objective fractional subset programming with invex and related nonconvex functions, Communications on Applied Nonlinear Analysis, 17 (2010), 89-101. [4] S. K. Mishra, S. Y. Wang and K. K. Lai, Generalized Convexity and Vector Optimization, Nonconvex Optimization and its Applications, Vol. 19, Springer-Verlag, 2009. [5] R. U. Verma, Approximation solvability of a class of nonlinear set-valued inclusions involving $(A,\eta)-$monotone mappings, Journal of Mathematical Analysis and Applications, 337 (2008), 969-975. [6] R. U. Verma, The optimality condition for multiple objective fractional subset programming based on generalized $(\rho,\eta)-$invex functions, Advances in Nonlinear Variational Inequalities, 14 (2011), 61-72. [7] G. J. Zalmai and Q. B. Zhang, Generalized $(F, \beta, \phi, \rho, \theta)-$univex functions and parametric duality in semiinfinite discrete minmax fractional programming, Advances in Nonlinear Variational Inequalities, 10 (2007), 1-20. [8] G. J. Zalmai and Q. B. Zhang, Generalized $(F, \beta, \phi, \rho, \theta)-$univex functions and global parametric sufficient optimality conditions in semiinfinite discrete minmax fractional programming, PanAmerican Mathematical Journal, 17 (2007), 1-26.
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