We prove that a ring R has a self dudlity induced by a left R-module M if and only if its polynomial ring R[x] has a graded self duality induced by a graded left R[x]-module M[x^-1].
In this paper, we first formulate a second-order multiobjective symmetric primal-dual pair over arbitrary cones by introducing two different functions f ： R^n × R^m → Rk and g ： R^n × R^m → R^l in each k...In this paper, we first formulate a second-order multiobjective symmetric primal-dual pair over arbitrary cones by introducing two different functions f ： R^n × R^m → Rk and g ： R^n × R^m → R^l in each k-objectives as well as l-constraints. Further, appropriate duality relations are established under second-order（F, α, ρ, d）-convexity assumptions. A nontrivial example which is second-order（F, α, ρ, d）-convex but not secondorder convex/F-convex is also illustrated. Moreover, a second-order minimax mixed integer dual programs is formulated and a duality theorem is established using second-order（F, α, ρ, d）-convexity assumptions. A self duality theorem is also obtained by assuming the functions involved to be skew-symmetric.展开更多
In this paper,a new mixed-type higher-order symmetric duality in scalar-objective programming is formulated.In the literature we have results either Wolfe or Mond–Weir-type dual or separately,while in this we have co...In this paper,a new mixed-type higher-order symmetric duality in scalar-objective programming is formulated.In the literature we have results either Wolfe or Mond–Weir-type dual or separately,while in this we have combined those results over one model.The weak,strong and converse duality theorems are proved for these programs underη-invexity/η-pseudoinvexity assumptions.Self-duality is also discussed.Our results generalize some existing dual formulations which were discussed by Agarwal et al.(Generalized second-order mixed symmetric duality in nondifferentiable mathematical programming.Abstr.Appl.Anal.2011.https://doi.org/10.1155/2011/103597),Chen(Higher-order symmetric duality in nonlinear nondifferentiable programs),Gulati and Gupta(Wolfe type second order symmetric duality in nondifferentiable programming.J.Math.Anal.Appl.310,247–253,2005,Higher order nondifferentiable symmetric duality with generalized F-convexity.J.Math.Anal.Appl.329,229–237,2007),Gulati and Verma(Nondifferentiable higher order symmetric duality under invexity/generalized invexity.Filomat 28(8),1661–1674,2014),Hou andYang(On second-order symmetric duality in nondifferentiable programming.J Math Anal Appl.255,488–491,2001),Verma and Gulati(Higher order symmetric duality using generalized invexity.In:Proceeding of 3rd International Conference on Operations Research and Statistics(ORS).2013.https://doi.org/10.5176/2251-1938_ORS13.16,Wolfe type higher order symmetric duality under invexity.J Appl Math Inform.32,153–159,2014).展开更多
基金Supported by the Natural Science Foundation of Fujian Province (1994~1997)
文摘We prove that a ring R has a self dudlity induced by a left R-module M if and only if its polynomial ring R[x] has a graded self duality induced by a graded left R[x]-module M[x^-1].
基金Department of Mathematics,Indian Institute of Technology Patna,Patna 800 013,India
文摘In this paper, we first formulate a second-order multiobjective symmetric primal-dual pair over arbitrary cones by introducing two different functions f ： R^n × R^m → Rk and g ： R^n × R^m → R^l in each k-objectives as well as l-constraints. Further, appropriate duality relations are established under second-order（F, α, ρ, d）-convexity assumptions. A nontrivial example which is second-order（F, α, ρ, d）-convex but not secondorder convex/F-convex is also illustrated. Moreover, a second-order minimax mixed integer dual programs is formulated and a duality theorem is established using second-order（F, α, ρ, d）-convexity assumptions. A self duality theorem is also obtained by assuming the functions involved to be skew-symmetric.
基金The research of Khushboo Verma was supported by the Department of Atomic Energy,Govt.of India,the NBHM Post-Doctoral Fellowship Program(No.2/40(31)/2015/RD-II/9474).
文摘In this paper,a new mixed-type higher-order symmetric duality in scalar-objective programming is formulated.In the literature we have results either Wolfe or Mond–Weir-type dual or separately,while in this we have combined those results over one model.The weak,strong and converse duality theorems are proved for these programs underη-invexity/η-pseudoinvexity assumptions.Self-duality is also discussed.Our results generalize some existing dual formulations which were discussed by Agarwal et al.(Generalized second-order mixed symmetric duality in nondifferentiable mathematical programming.Abstr.Appl.Anal.2011.https://doi.org/10.1155/2011/103597),Chen(Higher-order symmetric duality in nonlinear nondifferentiable programs),Gulati and Gupta(Wolfe type second order symmetric duality in nondifferentiable programming.J.Math.Anal.Appl.310,247–253,2005,Higher order nondifferentiable symmetric duality with generalized F-convexity.J.Math.Anal.Appl.329,229–237,2007),Gulati and Verma(Nondifferentiable higher order symmetric duality under invexity/generalized invexity.Filomat 28(8),1661–1674,2014),Hou andYang(On second-order symmetric duality in nondifferentiable programming.J Math Anal Appl.255,488–491,2001),Verma and Gulati(Higher order symmetric duality using generalized invexity.In:Proceeding of 3rd International Conference on Operations Research and Statistics(ORS).2013.https://doi.org/10.5176/2251-1938_ORS13.16,Wolfe type higher order symmetric duality under invexity.J Appl Math Inform.32,153–159,2014).
