## Hybrid Parametric Optimality Constraints for Discrete Rivmax Fractional Programming

### Ram U Verma*

*Department of Mathematics, University of North Texas, USA*

**Submission: ** April 29, 2017; **Published: ** June 02, 2017

***Corresponding author:** Ram U Verma, Department of Mathematics, University of North Texas, USA, Email: **[email protected]**

**How to cite this article:** Ram U V. Hybrid Parametric Optimality Constraints for Discrete Minmax Fractional Programming. Biostat Biometrics Open Acc
02 J. 2017;2(1): 555579. DOI: 10.19080/BBOAJ.2017.02.555579

**Abstract**

Several classes of generalized higher order parametric sufficient optimality constraints for a discrete minmax fractional programming problem are investigated toward establishing advanced results on higher order fractional programming. These results are established by applying advanced partitioning schemes and various types of generalized second-order (F,β,φ,π,ρ,θ,m)-univexity assumptions. The obtained results are new and generalize most of the results on (F,β,π,ρ,θ,m)-univexity in the literature.

** Keywords:** Discrete minmax fractional programming; Second-order univex functions; Generalized sufficient optimality conditions

**Introduction**

In this paper, we intend to establish several sets of generalized parametric sufficient optimality conditions for the following discrete minmax fractional programming problem:

The present investigation is aimed at establishing various second-order necessary and sufficient optimality conditions for several types of optimization problems, using the generalized concepts of second-order invexity, pseudoinvexity, and quasiinvexity originally defined by Hanson [1], and a set of second-order necessary optimality conditions by introducing the new classes of generalized second-oder invex functions. We shall apply two partitioning schemes by Mond & Weir [2] and Yang [3] , in conjunction with the new classes of generalized second- order invex functions to formulate and discuss numerous sets of generalized second-order sufficient optimality conditions for (P). To the best of our knowledge, all the second-order sufficient optimality results established in this paper are new in the area of discrete minmax fractional programming and encompass most of the investigations in the literature. The generalized optimality conditions established here can be utilized for constructing some generalized second-order parametric duality models for (P) and proving numerous weak, strong, and strict converse duality theorems. For more details on the discrete minmax fractional programming and related literature, we refer the reader [1-9].

The rest of this paper is organized as follows. In the remainder of this section, we generalize a few basic definitions and recall some auxiliary results which will be needed in the sequel. In Section 2, we state and prove various second-order parametric sufficient optimality results for (P) using a variety of generalized (F,β,φ,π,ρ,θ,m)-sounivexity assumptions. Finally, in Section 3 we summarize our main results and also point out some further research opportunities arising from certain modifications of the principal problem investigated in the present paper.

We next define some new classes of generalized second- order univex functions, called (strictly) (F,β,φ,π,ρ,θ,m)-sounivex, (strictly) (F,β,φ,π,ρ,θ,m)-pseudosounivex, and (pre-strictly) (F,β,φ,π,ρ,θ,m)-quasisounivex functions. These are further extensions of the classes of second-order (strictly) (^/p,p,0,m)- sonvex, (strictly) (^,n,p,0,m)-pseudosonvex, and (prestrictly) (^,n,p,0,m)-quasisonvex functions which were introduced recently in [4]. The second-order univex functions are also referred to as “sounivex functions” in the litera-ture. For more on the generalized F-convex and other related functions, we
refer the reader [8,9]. Now we present the new classes of (F,P/rc,9,p,0,m)-sounivex functions at x*. Let *f : X* ^ M be a twice differentiable function.

**Definition**

The function f is said to be (strictly) (F,β,φ,π,ρ,θ,m)- pseudosounivex at x* if there exist functions *ρ: χxχ*→R_{+}
≡(o,»), *φ*: ℝ→ℝ, *ρ:X x X Rπ,*, *θ: X x X ℝ ^{n},*Hybrid optimality constriants a sublinear function

*f*x,x

^{*}’.): ℝ→ℝ , and a positive integer m such that for each xϵX(x≠x

^{*}) and ,

*z ϵ*ℝ

^{n}

Where ‖. ‖ is a norm on ℝ^{n} and ≺a,b≻ is the innerproduct of the vectors a and b.The function f is said to be (strictly) (F,β,φ,π,ρ,θ,m)- pseudosounivex at X if it (strictly) (F,β,φ,π,ρ,θ,m)pseudosounivex at each x^{*}ϵX.

**Definition**

The function f is said to be (prestrictly) (F,β,φ,π,ρ,θ,m)- pseudosounivex at x* if there exist functions *ρ: χxχ*→R_{+} ≡(o,»), *φ*: ℝ→ℝ, *ρ:X x X Rπ,*, *θ: X x X ℝ*^{n}, a sublinear function *f*x,x^{*}’.): ℝ→ℝ and a positive integer m such that for each x£X

The function f is said to be (strictly) (F,β,φ,π,ρ,θ,m)- pseudosounivex on X if it is (strictly) (F,β,φ,π,ρ,θ,m)- pseudosounivex at each x*£X.

**Definition**

The function f is said to be (prestrictly) (F,β,φ,π,ρ,θ,m)- quasisounivex at x* if there exist functions *ρ: χxχ*→R_{+} ≡(o,»), *φ*: ℝ→ℝ, *ρ:X x X Rπ,*, *θ: X x X ℝ ^{n}*, a sublinear function

*f*x,x

^{*}’.): ℝ→ℝ and a positive integer m such that for each x£X

The function f is said to be (prestrictly) (F,β,φ,π,ρ,θ,m)- quasisounivex on X if it is (prestrictly) (F,β,φ,π,ρ,θ,m)- quasisounivex at each x*£X.

From the above definitions it is clear that if f is (F,β,φ,π,ρ,θ,m)- sounivex at x*, then it is both (F,β,φ,π,ρ,θ,m)-pseudosounivex and (F,β,φ,π,ρ,θ,m)-quasisounivex at x*, if f is (F,β,φ,π,ρ,θ,m)- quasisounivex at x*, then it is prestrictly (F,β,φ,π,ρ,θ,m)- quasisounivex at x*, and if f is strictly (F,β,φ,π,ρ,θ,m)- pseudosounivex at x*, then it is (F,β,φ,π,ρ,θ,m)-quasisounivex at x*.

In the proofs of the duality theorems, sometimes it may be more convenient to use certain alternative but equivalent forms of the above definitions. These are obtained by considering the contrapositive statements. For example, (F,β,φ,π,ρ,θ,m)- quasisounivexity can be defined in the following equivalent way: The function f is said to be (F,β,φ,π,ρ,θ,m)-quasisounivex at x* if there exist functions *ρ: χxχ*→R_{+} ≡(o,»), *φ*: ℝ→ℝ, *ρ:X x X Rπ,*, *θ: X x X ℝ ^{n},* a sublinear function

*f*x,x

^{*}’.): ℝ→ℝ , and a positive integer m such that for each xϵX(x≠x

^{*}) and ,

*z ϵ*ℝ

^{n}

We conclude this section by recalling a set of second-order parametric necessary optimality conditions for (P). The form and features of this result will provide clear guidelines for formulating various sets of second-order parametric sufficient optimality conditions for (P).

**Theorem**

[4] Let x* be an optimal solution of (P), let λ* = ϕ(x*) be defined by

**Second-Order Sufficient Optimality**

In this section, we discuss several families of sufficient optimality results under various generalized (F,β,φ,π,ρ,θ,m)- sounivexity hypotheses imposed on certain combinations of the problem functions. This is accomplished by employing a certain partitioning scheme which was originally proposed in [2] for the purpose of constructing generalized dual problems for nonlinear programming problems. For this we need some additional notation.

In the proofs of our sufficiency theorems, we shall make frequent use of the following auxiliary result which provides an alternative expression for the objective function of (P).

**Lemma**

Making use of the sets and functions defined above, we can now formulate our first collection of generalized second-order parametric sufficient optimality results for (P) as follows.

**Theorem**

Assume, furthermore, that any one of the following four sets of hypotheses is satisfied:

**Proof**

Let x be an arbitrary feasible solution of (P).

(a) : In view of the sub linearity of *f*(x,*x*; ), it is clear that?
(2.1) and (2.2) can be expressed as follows:

The rest of the proof is identical to that of part (a). (c) and (d): The proofs are similar to those of parts (a) and (b).

**Theorem**

Then x* is an optimal solution of (P).

**Proof**

**Theorem**

**Proof(a)**

(a) : Suppose to the contrary that x* is not an optimal solution of (P ). This implies that

**Concluding Remarks**

Based on Dinkelbach’s parametric model [1], we have established numerous sets of higher order parametric sufficient optimality criteria for a discrete minmax fractional programming problem using a variety of generalized (*F,β,φ,π,ρ,θ,m*) -sounivexity constraints. These optimality models can be applied for constructing various duality models as well as for developing new algorithms for the numerical solution of minmax fractional programming problems. Furthermore, the obtained results generalize most of the results available in the semiinfinite aspects of the mathematical programming, for example, the following semiinfinite minmax fractional programming problem:

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