Duality on a nondifferentiable minimax fractional programming

In this article, we focus our study on a nondifferentiable minimax fractional programming problem and establish weak, strong and strict converse duality theorems under generalized higher order (F, α, p, d)-Type I assumptions. Our results extend and unify some of the known results in the literature.

Necessary and sufficient optimality conditions of Kuhn-Tucker type for a convex programming problem with subdifferentiable operator constraints have been obtained. A duality theorem of Wolfe's type has been derived. Assuming that the objective function is strictly convex, a converse duality theorem is obtained. The results are then applied to a programming problem in which the objective function is the sum of a positively homogeneous, lower-semi-continuous, convex function and a continuous convex function.

Duality for nondifferentiable minimax programming in complex spaces

Two different pairs of higher-order symmetric dual programs such as Wolfe type and Mond–Weir type are studied. The weak, strong and converse duality theorems are established for the higher-order symmetric dual programs under higher-order \(\rho -(\eta ,\theta )-\)invexity and \(\rho -(\eta ,\theta )-\)pseudo-invexity assumptions. Many examples and counterexamples are illustrated to justify our work. We also observe that several known results are obtained as special cases.

Convexity assumptions for fractional programming of variational type are relaxed to generalized invexity. The sufficient optimality conditions are employed to construct a mixed dual programming problem. It will involve the Wolfe type dual and Mond-Weir type dual as its special situations. Several duality theorems concerning weak, strong, and strict converse duality under the framework in mixed type dual form are proved.

Multiobjective second order symmetric duality with cone constraints

This paper is devoted to study interval-valued optimization problems. Sufficient optimality conditions are established for LU optimal solution concept under generalized -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. MSC: 90C46, 90C26, 90C30.

This paper deals with non-differentiable multi-objective semi-infinite programming problem. It is a problem of simultaneous minimisation of finitely many scalar valued functions subject to an arbitrary (possibly infinite) set of constraints. Non-differentiability enters, due to the square root of a quadratic form which appears in the objective functional. Concept of efficiency of order m has been extended to the above stated problem. In order to study this new solution concept, the notion of ρ-invexity of order m is also proposed which is utilised to establish sufficient optimality conditions for the non-differentiable multi-objective semi-infinite programming problem. Mond-Weir type of dual is proposed for which weak, strong and strict converse duality theorems are established.

In this paper, a pair of non-differentiable second-order multi-objective symmetrical dual fractional variational programs over arbitrary cones is constructed. We have introduced the concept of second-order (F, α, ρ, d)-pseudoconvexity and an example of a function which is second-order (F, α, ρ, d)-pseudoconvex but not second-order F-pseudoconvex has been constructed. Further, to relate the objective function value of primal and dual, we proved weak, strong, and converse duality theorems under second-order (F, α, ρ, d)-pseudoconvexity suppositions. Moreover, to validate the results of the weak duality theorem, an example of a multi-objective variational fractional problem and its dual model has been constructed. Several known results are also discussed as special cases.

In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concept of invex of order type II for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. The new functions are used to derive the sufficient optimality condition for a class of nonsmooth multiobjective programming problems. Utilizing the sufficient optimality conditions, weak and strong duality theorems are established for Wolfe type duality model.

Generalized polyhedral convex optimization problems in locally convex Hausdorff topological vector spaces are studied systematically in this paper. We establish solution existence theorems, necessary and sufficient optimality conditions, weak and strong duality theorems. In particular, we show that the dual problem has the same structure as the primal problem, and the strong duality relation holds under three different sets of conditions.

In this paper, we propose a generalisation of convexity, namely higher order semilocally strong convexity for a nonlinear multiobjective programming problem, where the function involved are semidifferentiable. The generalised Karush-Kuhn-Tucker necessary and sufficient optimality conditions are derived. Moreover, a general Mond-Weir type dual problem is presented for nonlinear multiobjective programming problem involving higher order semilocally strong convexity and usual duality theorems are discussed.

Minimax fractional programming under nonsmooth generalized [formula omitted]-d-univexity

Both parametric and parameter-free necessary and sufficient optimality conditions are established for a class of nondiffer-entiable nonconvex optimal control problems with generalized fractional objective functions, linear dynamics, and nonlinear inequality constraints on both the state and control variables. Based on these optimality results, ten Wolfe-type parametric and parameter-free duality models are formulated and weak, strong, and strict converse duality theorems are proved. These duality results contain, as special cases, similar results for minmax fractional optimal control problems involving square roots of positive semi definite quadratic forms, and for optimal control problems with fractional, discrete max, and conventional objective functions, which are particular cases of the main problem considered in this paper. The duality models presented here contain various extensions of a number of existing duality formulations for convex control problems, and subsume continuous-time generalizations of a great variety of similar dual problems investigated previously in the area of finite-dimensional nonlinear programming.

In the research of optimization problems, optimality conditions play an important role. By using some derivatives, various types of necessary and/or sufficient optimality conditions have been introduced by many researchers. Especially, in convex programming, necessary and sufficient optimality conditions in terms of the subdifferential have been studied extensively. Recently, necessary and sufficient optimality conditions for quasiconvex programming have been investigated by the authors. However, there are not so many results concerned with Karush–Kuhn–Tucker type optimality conditions for non-differentiable quasiconvex programming. In this paper, we study a Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. We show some closedness properties for Greenberg–Pierskalla subdifferential. Under the Slater constraint qualification, we show a necessary and sufficient optimality condition for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. Additionally, we introduce a necessary and sufficient constraint qualification of the optimality condition. As a corollary, we show a necessary and sufficient optimality condition for convex programming in terms of the subdifferential.