In the present paper, we move forward in the study of multiobjective programming problems and est... more In the present paper, we move forward in the study of multiobjective programming problems and establish sucient optimality conditions under generalized (
The present paper deals with the concepts of generalized fuzzy invex monotonocities and generaliz... more The present paper deals with the concepts of generalized fuzzy invex monotonocities and generalized weakly fuzzy invex functions. Some necessary conditions for weakly fuzzy invex monotonocities are presented. Moreover, the concept of fuzzy strong invex monotonocities and fuzzy strong invex functions are also discussed. To strengthen our definitions, we provide nontrivial examples of fuzzy invex monotonocities and weakly fuzzy invex functions.
Journal of Nonlinear Sciences and Applications, 2018
In this paper, we present the notion of geodesic sub-b-s-convex function on the Riemannian manifo... more In this paper, we present the notion of geodesic sub-b-s-convex function on the Riemannian manifolds. A non-trivial example of geodesic sub-b-s-convex function but not geodesic convex function is also discussed. Some fundamental properties of geodesic sub-b-s-convex functions are investigated. Moreover, we derive the optimality conditions of unconstrained and constrained programming problems under the sub-b-s-convexity.
In this paper, some interval valued programming problems are discussed. The solution concepts are... more In this paper, some interval valued programming problems are discussed. The solution concepts are adopted from Wu [7] and Chalco-Cano et al. [34]. By considering generalized Hukuhara differentiability and generalized convexity (viz. ?-preinvexity, ?-invexity etc.) of interval valued functions, the KKT optimality conditions for obtaining (LS and LU) optimal solutions are elicited by introducing Lagrangian multipliers. Our results generalize the results of Wu [7], Zhang et al. [11] and Chalco-Cano et al. [34]. To illustrate our theorems suitable examples are also provided
In the present paper, two types of second order dual models are formulated for a minmax fractiona... more In the present paper, two types of second order dual models are formulated for a minmax fractional programming problem. The concept of η-bonvexity/ generalized η-bonvexity is adopted in order to discuss weak, strong and strict converse duality theorems.
This paper is devoted to study interval-valued optimization problems. Sufficient optimality condi... more This paper is devoted to study interval-valued optimization problems. Sufficient optimality conditions are established for LU optimal solution concept under generalized (p, r)-ρ-(η, θ)-invexity. Weak, strong and strict converse duality theorems for Wolfe and Mond-Weir type duals are derived in order to relate the LU optimal solutions of primal and dual problems.
In this article, we introduce a new class of functions called r-invexity and geodesic r-preinvexi... more In this article, we introduce a new class of functions called r-invexity and geodesic r-preinvexity functions on a Riemannian manifolds. Further, we establish the relationships between r-invexity and geodesic r-preinvexity on Riemannian manifolds. It is observed that a local minimum point for a scalar optimization problem is also a global minimum point under geodesic r-preinvexity on Riemannian manifolds. In the end, a mean value inequality is extended to a Cartan-Hadamard manifold. The results presented in this paper extend and generalize the results that have appeared in the literature. MSC: 58E17; 90C26
The present paper is framed to study weak, strong and strict converse duality relations for a sem... more The present paper is framed to study weak, strong and strict converse duality relations for a semi-infinite programming problem and its Wolfe and Mond-Weir-type dual programs under generalized (H p , r)-invexity.
A nonlinear multiobjective programming problem is considered. Weak, strong and strict converse du... more A nonlinear multiobjective programming problem is considered. Weak, strong and strict converse duality theorems are established under generalized second order (F, α, ρ, d)-convexity for second order Mangasarian type and general Mond-Weir type vector duals.
A pair of Wolfe type multiobjective second order symmetric dual programs with cone constraints is... more A pair of Wolfe type multiobjective second order symmetric dual programs with cone constraints is formulated and usual duality results are established under second order invexity assumptions. These results are then used to investigate symmetric duality for minimax version of multiobjective second order symmetric dual programs wherein some of the primal and dual variables are constrained to belong to some arbitrary sets, i.e., the sets of integers. This paper points out certain omissions and inconsistencies in the earlier work of Mishra [S.K. Mishra, Multiobjective second order symmetric duality with cone constraints,
Computers & Mathematics with Applications, 2010
A pair of multiobjective mixed symmetric dual programs is formulated over arbitrary cones. Weak, ... more A pair of multiobjective mixed symmetric dual programs is formulated over arbitrary cones. Weak, strong, converse and self-duality theorems are proved for these programs under K-preinvexity and K-pseudoinvexity assumptions. This mixed symmetric dual formulation unifies the symmetric dual formulations of Suneja et al. (2002) [14] and Khurana (2005) [15].
A Mond-Weir type dual for a class of nondifferentiable minimax fractional programming problem is ... more A Mond-Weir type dual for a class of nondifferentiable minimax fractional programming problem is considered. Appropriate duality results are proved involving (F, a, q, d)-pseudoconvex functions.
The present paper deals with the properties of geodesic -preinvex functions and their relationshi... more The present paper deals with the properties of geodesic -preinvex functions and their relationships with -invex functions and strictly geodesic -preinvex functions. The geodesic -pre-pseudo-invex and geodesic -pre-quasi-invex functions on the geodesic invex set are introduced and some of their properties are discussed.
Journal of Computational and Applied Mathematics, 2008
In this paper, we study a non-differentiable minimax fractional programming problem under the ass... more In this paper, we study a non-differentiable minimax fractional programming problem under the assumption of generalized-univex function. In this paper we extend the concept of-invexity [M.A. Noor, On generalized preinvex functions and monotonicities, J. Inequalities Pure Appl. Math. 5 (2004) 1-9] and pseudo-invexity [S.K. Mishra, M.A. Noor, On vector variational-like inequality problems, J. Math. Anal. Appl. 311 (2005) 69-75] to-univexity and pseudo-univexity from a view point of generalized convexity. We also introduce the concept of strict pseudo-univex and quasi-univex functions. We derive Karush-Kuhn-Tuckertype sufficient optimality conditions and establish weak, strong and converse duality theorems for the problem and its three different form of dual problems. The results in this paper extend a few known results in the literature.
We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and r... more We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and relationships of these functions are discussed. A characterization for the existence of a global minimum point of a mathematical programming problem is presented. Moreover, a mean value inequality under geodesic log-preinvexity is extended to Cartan-Hadamard manifolds.
A Mond–Weir type dual for a class of nondifferentiable minimax fractional programming problem is ... more A Mond–Weir type dual for a class of nondifferentiable minimax fractional programming problem is considered. Appropriate duality results are proved involving (F, α, ρ, d)-pseudoconvex functions.
A second-order dual is formulated for a nondifferentiable fractional programming problem. Using t... more A second-order dual is formulated for a nondifferentiable fractional programming problem. Using the generalized second-order (F,α,ρ,d)-convexity assumptions on the functions involved, weak, strong and converse duality theorems are established in order to relate the primal and dual problems.
In this paper, we derive necessary and sufficient optimality conditions for a general minimax pro... more In this paper, we derive necessary and sufficient optimality conditions for a general minimax programming problem involving some classes of generalized convexities with the tool-right upper-Dini-derivative. Moreover, using the concept of optimality conditions, Mond-Weir type duality theory has been developed for such a minimax programming problem.
Numerical Functional Analysis and Optimization, 2007
... Second-Order Duality for Nondifferentiable Multiobjective Programming Problems. Authors: Ahma... more ... Second-Order Duality for Nondifferentiable Multiobjective Programming Problems. Authors: Ahmad, I. 1 ; Sharma, Sarita 1. ... Related content: In this: publication; By this: publisher; By this author: Ahmad, I. ; Sharma, Sarita. You are signed in as: Google (Institutional account). ...
In the present paper, we move forward in the study of multiobjective programming problems and est... more In the present paper, we move forward in the study of multiobjective programming problems and establish sucient optimality conditions under generalized (
The present paper deals with the concepts of generalized fuzzy invex monotonocities and generaliz... more The present paper deals with the concepts of generalized fuzzy invex monotonocities and generalized weakly fuzzy invex functions. Some necessary conditions for weakly fuzzy invex monotonocities are presented. Moreover, the concept of fuzzy strong invex monotonocities and fuzzy strong invex functions are also discussed. To strengthen our definitions, we provide nontrivial examples of fuzzy invex monotonocities and weakly fuzzy invex functions.
Journal of Nonlinear Sciences and Applications, 2018
In this paper, we present the notion of geodesic sub-b-s-convex function on the Riemannian manifo... more In this paper, we present the notion of geodesic sub-b-s-convex function on the Riemannian manifolds. A non-trivial example of geodesic sub-b-s-convex function but not geodesic convex function is also discussed. Some fundamental properties of geodesic sub-b-s-convex functions are investigated. Moreover, we derive the optimality conditions of unconstrained and constrained programming problems under the sub-b-s-convexity.
In this paper, some interval valued programming problems are discussed. The solution concepts are... more In this paper, some interval valued programming problems are discussed. The solution concepts are adopted from Wu [7] and Chalco-Cano et al. [34]. By considering generalized Hukuhara differentiability and generalized convexity (viz. ?-preinvexity, ?-invexity etc.) of interval valued functions, the KKT optimality conditions for obtaining (LS and LU) optimal solutions are elicited by introducing Lagrangian multipliers. Our results generalize the results of Wu [7], Zhang et al. [11] and Chalco-Cano et al. [34]. To illustrate our theorems suitable examples are also provided
In the present paper, two types of second order dual models are formulated for a minmax fractiona... more In the present paper, two types of second order dual models are formulated for a minmax fractional programming problem. The concept of η-bonvexity/ generalized η-bonvexity is adopted in order to discuss weak, strong and strict converse duality theorems.
This paper is devoted to study interval-valued optimization problems. Sufficient optimality condi... more This paper is devoted to study interval-valued optimization problems. Sufficient optimality conditions are established for LU optimal solution concept under generalized (p, r)-ρ-(η, θ)-invexity. Weak, strong and strict converse duality theorems for Wolfe and Mond-Weir type duals are derived in order to relate the LU optimal solutions of primal and dual problems.
In this article, we introduce a new class of functions called r-invexity and geodesic r-preinvexi... more In this article, we introduce a new class of functions called r-invexity and geodesic r-preinvexity functions on a Riemannian manifolds. Further, we establish the relationships between r-invexity and geodesic r-preinvexity on Riemannian manifolds. It is observed that a local minimum point for a scalar optimization problem is also a global minimum point under geodesic r-preinvexity on Riemannian manifolds. In the end, a mean value inequality is extended to a Cartan-Hadamard manifold. The results presented in this paper extend and generalize the results that have appeared in the literature. MSC: 58E17; 90C26
The present paper is framed to study weak, strong and strict converse duality relations for a sem... more The present paper is framed to study weak, strong and strict converse duality relations for a semi-infinite programming problem and its Wolfe and Mond-Weir-type dual programs under generalized (H p , r)-invexity.
A nonlinear multiobjective programming problem is considered. Weak, strong and strict converse du... more A nonlinear multiobjective programming problem is considered. Weak, strong and strict converse duality theorems are established under generalized second order (F, α, ρ, d)-convexity for second order Mangasarian type and general Mond-Weir type vector duals.
A pair of Wolfe type multiobjective second order symmetric dual programs with cone constraints is... more A pair of Wolfe type multiobjective second order symmetric dual programs with cone constraints is formulated and usual duality results are established under second order invexity assumptions. These results are then used to investigate symmetric duality for minimax version of multiobjective second order symmetric dual programs wherein some of the primal and dual variables are constrained to belong to some arbitrary sets, i.e., the sets of integers. This paper points out certain omissions and inconsistencies in the earlier work of Mishra [S.K. Mishra, Multiobjective second order symmetric duality with cone constraints,
Computers & Mathematics with Applications, 2010
A pair of multiobjective mixed symmetric dual programs is formulated over arbitrary cones. Weak, ... more A pair of multiobjective mixed symmetric dual programs is formulated over arbitrary cones. Weak, strong, converse and self-duality theorems are proved for these programs under K-preinvexity and K-pseudoinvexity assumptions. This mixed symmetric dual formulation unifies the symmetric dual formulations of Suneja et al. (2002) [14] and Khurana (2005) [15].
A Mond-Weir type dual for a class of nondifferentiable minimax fractional programming problem is ... more A Mond-Weir type dual for a class of nondifferentiable minimax fractional programming problem is considered. Appropriate duality results are proved involving (F, a, q, d)-pseudoconvex functions.
The present paper deals with the properties of geodesic -preinvex functions and their relationshi... more The present paper deals with the properties of geodesic -preinvex functions and their relationships with -invex functions and strictly geodesic -preinvex functions. The geodesic -pre-pseudo-invex and geodesic -pre-quasi-invex functions on the geodesic invex set are introduced and some of their properties are discussed.
Journal of Computational and Applied Mathematics, 2008
In this paper, we study a non-differentiable minimax fractional programming problem under the ass... more In this paper, we study a non-differentiable minimax fractional programming problem under the assumption of generalized-univex function. In this paper we extend the concept of-invexity [M.A. Noor, On generalized preinvex functions and monotonicities, J. Inequalities Pure Appl. Math. 5 (2004) 1-9] and pseudo-invexity [S.K. Mishra, M.A. Noor, On vector variational-like inequality problems, J. Math. Anal. Appl. 311 (2005) 69-75] to-univexity and pseudo-univexity from a view point of generalized convexity. We also introduce the concept of strict pseudo-univex and quasi-univex functions. We derive Karush-Kuhn-Tuckertype sufficient optimality conditions and establish weak, strong and converse duality theorems for the problem and its three different form of dual problems. The results in this paper extend a few known results in the literature.
We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and r... more We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and relationships of these functions are discussed. A characterization for the existence of a global minimum point of a mathematical programming problem is presented. Moreover, a mean value inequality under geodesic log-preinvexity is extended to Cartan-Hadamard manifolds.
A Mond–Weir type dual for a class of nondifferentiable minimax fractional programming problem is ... more A Mond–Weir type dual for a class of nondifferentiable minimax fractional programming problem is considered. Appropriate duality results are proved involving (F, α, ρ, d)-pseudoconvex functions.
A second-order dual is formulated for a nondifferentiable fractional programming problem. Using t... more A second-order dual is formulated for a nondifferentiable fractional programming problem. Using the generalized second-order (F,α,ρ,d)-convexity assumptions on the functions involved, weak, strong and converse duality theorems are established in order to relate the primal and dual problems.
In this paper, we derive necessary and sufficient optimality conditions for a general minimax pro... more In this paper, we derive necessary and sufficient optimality conditions for a general minimax programming problem involving some classes of generalized convexities with the tool-right upper-Dini-derivative. Moreover, using the concept of optimality conditions, Mond-Weir type duality theory has been developed for such a minimax programming problem.
Numerical Functional Analysis and Optimization, 2007
... Second-Order Duality for Nondifferentiable Multiobjective Programming Problems. Authors: Ahma... more ... Second-Order Duality for Nondifferentiable Multiobjective Programming Problems. Authors: Ahmad, I. 1 ; Sharma, Sarita 1. ... Related content: In this: publication; By this: publisher; By this author: Ahmad, I. ; Sharma, Sarita. You are signed in as: Google (Institutional account). ...
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