Univex Functions Research Articles

Optimality conditions for interval-valued univex programming

We introduce concepts of interval-valued univex mappings, consider optimization conditions for interval-valued univex functions for the constrained interval-valued minimization problem, and show examples for the illustration purposes.

Multi-Objective Optimization Using Local Fractional Differential Operator

In this effort, we aim to generalize the concept of Univex functions by utilizing a local fractional differential-difference operator, based on different types of local fractional calculus (fractal calculus). This study leads to a new class of these functions in some optimal problems by illustrating conditions on the generalized functions. We call it the class of local fractional Univex functions. Strong, weak, converse, and strict converse duality theorems are given. Multi-objective optimal problem involves the new process is solved (local optimal problem). The main tool employed in the analysis is based on the local fractional derivative operators.

The Modified Objective Function Method for Univex Multiobjective Variational Problems

In this paper, we use the modified objective function method for a class of nonconvex multiobjective variational problems involving univex functions. Under univexity hypotheses, we prove the equivalence between an (weakly) efficient solution of the considered multiobjective variational problem and an (weakly) efficient solution of the associated modified multiobjective variational problem constructed in the modified objective function method.

New Class of Duality Models in Discrete Minmax Fractional Programming Based on Second-Order Univexities

The main goal of this paper is first to formulate four new general higher order parametric duality models for a discrete minmax fractional programming problem, and then prove appropriate duality theorems applying various new classes of second order univex functions.

On fuzzy generalized convex mappings and optimality conditions for fuzzy weakly univex mappings

In this paper, we first introduce the weakly invex fuzzy mappings based on weakly differentiable functions and discuss the relationships between several kinds of fuzzy generalized convex mappings. Then a kind of fuzzy weakly univex functions are introduced and some properties of them are investigated. Furthermore, optimality and duality results are derived for a class of generalized convex optimization problems with fuzzy weakly univex functions.

SECOND ORDER NONDIFFERENTIABLE MULTIOBJECTIVE FRACTIONAL PROGRAMMING UNDER GENERALIZED UNIVEX FUNCTIONS

In this paper, a new class of second order $(d,\rho,\eta,\theta)$-type 1 univex function is introduced. The Wolfe type second order dual problem (SFD) of the nondifferentiable multiobjective fractional programming problem (MFP) is considered, where the objective and constraint functions involved are directionally differentiable. Also the duality results under second order$(d,\rho,\eta,\theta)$-type 1 univex functions are established.

Higher order duality in multiobjective fractional programming with square root term under generalized higher order [formula omitted]-V-type I univex functions

In this paper, a new generalized class of higher order (F,α,β,ρ,σ,d)-V-type I univex function is introduced with some examples for a differentiable multiobjective programming (MP). Then multiobjective fractional programming problem (MFP) is considered in which the numerator and denominator of objective functions contain square root of positive semidefinite quadratic form and the necessary and sufficient conditions for efficient solution for (MFP) are established under generalized higher order (F,α,β,ρ,σ,d)-V-type I univex functions. Again, higher order dual program is proposed for (MFP) and the duality results are established under generalized of higher order (F,α,β,ρ,σ,d)-V-type I univex functions. Also, some computational work has been done to substantiate the analysis.

Sufficiency and Duality in Nonsmooth Multiobjective Programming Problem under Generalized Univex Functions

We consider a nonsmooth multiobjective programming problem where the functions involved are nondifferentiable. The class of univex functions is generalized to a far wider class of (φ,α,ρ,σ)-dI-V-type I univex functions. Then, through various nontrivial examples, we illustrate that the class introduced is new and extends several known classes existing in the literature. Based upon these generalized functions, Karush-Kuhn-Tucker type sufficient optimality conditions are established. Further, we derive weak, strong, converse, and strict converse duality theorems for Mond-Weir type multiobjective dual program.

Sufficiency and duality for nonsmooth multiobjective programming problems involving generalized univex functions

In this paper, nonsmooth univex, nonsmooth quasiunivex, and nonsmooth pseudounivex functions are introduced. By utilizing these new concepts, sufficient optimality conditions for a weakly efficient solution of the nonsmooth multiobjective programming problem are established. Weak and strong duality theorems are also derived for Mond-Weir type multiobjective dual programs.

Multiobjective programming under (φ, d)-V-type I univexity

In this paper, we have introduced a new class of (φ, d)-V-type I univex, quasi (φ, d)-V-type I univex, pseudo (φ, d)-V-type I univex, quasi-pseudo (φ, d)-V-type I univex and pseudo-quasi (φ, d)-V-type I univex functions in case of nonlinear multiobjective programming problem where functions involved are nondifferentiable and illustrated through non-trivial examples that this class extends some known classes in literature. Various Karush-Kuhn-Tucker type sufficient optimality conditions are obtained under this newly introduced class of functions. Also, for mixed type multiobjective dual program, we have established weak, strong, converse and strict converse duality results in order to relate the efficient and weak efficient solutions of primal and dual problem.

SECOND-ORDER UNIVEX FUNCTIONS AND GENERALIZED DUALITY MODELS FOR MULTIOBJECTIVE PROGRAMMING PROBLEMS CONTAINING ARBITRARY NORMS

In this paper, we introduce three new broad classes of second-order generalized convex functions, namely, (<TEX>$\mathcal{F}$</TEX>, <TEX>$b$</TEX>, <TEX>${\phi}$</TEX>, <TEX>${\rho}$</TEX>, <TEX>${\theta}$</TEX>)-sounivex functions, (<TEX>$\mathcal{F}$</TEX>, <TEX>$b$</TEX>, <TEX>${\phi}$</TEX>, <TEX>${\rho}$</TEX>, <TEX>${\theta}$</TEX>)-pseudosounivex functions, and (<TEX>$\mathcal{F}$</TEX>, <TEX>$b$</TEX>, <TEX>${\phi}$</TEX>, <TEX>${\rho}$</TEX>, <TEX>${\theta}$</TEX>)-quasisounivex functions; formulate eight general second-order duality models; and prove appropriate duality theorems under various generalized (<TEX>$\mathcal{F}$</TEX>, <TEX>$b$</TEX>, <TEX>${\phi}$</TEX>, <TEX>${\rho}$</TEX>, <TEX>${\theta}$</TEX>)-sounivexity assumptions for a multiobjective programming problem containing arbitrary norms.

Nonsmooth multiobjective optimization involving generalized univex functions

In this paper, we have considered a class of constrained nonsmooth multi objective programming problem involving semi-directionally differentiable functions from a view point of generalized convexity. A new generalized class of (dI—ρ—σ)-V-type I univex functions is introduced under which various weak, strong, converse and strict converse duality theorems are established for Mond-Weir type dual program in order to relate the efficient and weak efficient solutions of primal and dual problem. Also, we have illustrated through various non-trivial examples that this class extends many earlier studied classes in literature.

Optimality and Duality for Multiobjective Fractional Programming Involving Nonsmooth Generalized(ℱ,b,ϕ,ρ,θ)-Univex Functions

We establish properly efficient necessary and sufficient optimality conditions for multiobjective fractional programming involving nonsmooth generalized(ℱ,b,ϕ,ρ,θ)-univex functions. Utilizing the necessary optimality conditions, we formulate the parametric dual model and establish some duality results in the framework of generalized(ℱ,b,ϕ,ρ,θ)-univex functions.

Univex Interval-Valued Mapping with Differentiability and Its Application in Nonlinear Programming

Interval-valued univex functions are introduced for differentiable programming problems. Optimality and duality results are derived for a class of generalized convex optimization problems with interval-valued univex functions.

Minimax fractional programming problem involving nonsmooth generalized ?-univex functions

In this paper, we introduce a new class of generalized ?-univex functions where the involved functions are locally Lipschitz. We extend the concept of ?-type I invex [S. K. Mishra, J. S. Rautela, On nondifferentiable minimax fractional programming under generalized ?-type I invexity, J. Appl. Math. Comput. 31 (2009) 317-334] to ?-univexity and an example is provided to show that there exist functions that are ?-univex but not ?-type I invex. Furthermore, Karush-Kuhn-Tucker-type sufficient optimality conditions and duality results for three different types of dual models are obtained for nondifferentiable minimax fractional programming problem involving generalized ?-univex functions. The results in this paper extend some known results in the literature.

Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity

Using a parametric approach, we establish necessary and sufficient optimality conditions and derive some duality theorems for a class of nonsmooth minmax fractional programming problems containing generalized univex functions. The results obtained in this paper extend and improve some corresponding results in the literature.

Sufficiency and duality for nonsmooth multiobjective programming problems involving generalized [formula omitted]-univex functions

In this paper, we introduce a new class of generalized ( F , α , ρ , θ ) – d – V -univex functions for a nonsmooth multiobjective programming problem. Sufficient optimality conditions under generalized ( F , α , ρ , θ ) – d – V -univex functions are established for a feasible solution to be an efficient solution. Appropriate duality theorems for a Mond–Weir-type dual are also presented under the aforesaid assumptions.

Generalized (F,β,Φ,ρ,θ) -univex functions and optimality conditions in semiinfinite fractional programming

In this paper, we establish a set of necessary optimality conditions, and formulate and discuss a fairly large number of sets of global sufficient optimality criteria under various generalized (F,β,Φ,ρ,θ) –univexity assumptions for a semiinfinite fractional programming problem.

Generalized (F,β,Φ,ρ,θ) -univex functions and duality models in semiinfinite fractional programming

In this paper, we formulate and discuss a fairly large number of dual problems for (P) and establish a multitude of duality results under various generalized (F,β,Φ,ρ,θ) - univexity assumptions for a semiinfinite fractional programming problem.

Generalized (,α, ρ, θ)–d–V–univex functions and non-smooth alternative theorems

Under a new kind of generalized convexity and non-smooth generalized (, α, ρ, θ)–d–V–univexity, some basic alternative theorems are established and proved. These results extend some other generalized convex alternative theorems. To illustrate their applications, we present characterizations of saddle-point type necessary optimality conditions for properly efficient solutions of vector optimization problems. To conclude, the alternative theorems are applied to yield a canonical convex alternative theorem as a corollary.