New Class of duality models in discrete minmax fractional programming based on second-order univexities

The main purpose 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 utilizing various new classes of second-order (F , β, φ, ζ, ρ, θ,m)-univex functions.


Introduction and Preliminaries
In this paper, we introduce four generalized second-order parametric duality models and prove a variety of weak, strong, and strict converse duality theorems using the notion of second-order (F, β, ϕ, ζ, ρ, θ, m)-univex functions for the following discrete minmax fractional programming problem: subject to G j (x) ≤ 0, j ∈ q, H k (x) = 0, k ∈ r, x ∈ X, where X is an open convex subset of R n (n-dimensional Euclidean space), f i , g i , i ∈ p ≡ {1, 2, . . ., p}, G j , j ∈ q, and H k , k ∈ r, are real-valued functions defined on X, and for each i ∈ p, g i (x) > 0 for all x satisfying the constraints of (P ).
The problems of this nature are more frequently referred to as "generalized fractional programming problems" to the context of mathematical programming.These problems provide realistic models for some significant real-world problems (more notably encountered in multiobjective programming, approximation theory, goal programming, location planning and economics), while their mathematical tractability empowers equivalent parametric nonlinear programming problems with nonfractional objective functions.
Recently, Verma and Zalmai [10] established some second-order parametric necessary optimality conditions as well as numerous second-order sufficient optimality conditions for a discrete minmax fractional programming In the present paper, we plan to introduce new classes of second-order (F, β, ϕ, ζ, ρ, θ, m)-univex functions, which generalize most of the existing notions of second-order univex functions.The second-order univex functions are also referred to as "sounivex functions" in the literature.Here, we shall utilize two partitioning schemes due to Mond and Weir [6] and Yang [13], in conjunction with the generalized versions of the new classes of second-order (F, β, ϕ, ζ, ρ, θ, m)-univex functions to formulate four generalized duality models for (P ) and prove appropriate duality theorems.The duality models and the related duality theorems established in this paper generalize those presented in [10 -12] and others.To the best of our knowledge, all of these duality results are new in the area of discrete minmax fractional programming.In fact, it seems that results of this type, which are based on second-order necessary and sufficient optimality conditions, have not yet appeared in any shape or form for any type of mathematical programming problems.For extensive lists of publications dealing with second-order and higher-order duality results for various categories of nonlinear programming problems, which are essentially based on first-order optimality conditions and different kinds of generalized convexity concepts, we refer the reader ([2]- [5], [7], [9]- [12], [15], [16]).
All the parametric duality results established in this paper can easily be modified (and specialized) for each one of the following three classes of nonlinear programming problems to the context (P ): where F (assumed to be nonempty) is the feasible set of (P ), that is, The paper is organized as follows: In Section 1, we introduce a few basic definitions and recall some auxiliary results that will be used in the sequel.In Section 2, we utilize a partitioning scheme due to Mond and Weir [6], and formulate two general second-order parametric duality models for (P ) and prove weak, strong, and strict converse duality theorems using various generalized (F, β, ϕ, ζ, ρ, θ, m)-sounivexity assumptions.In Section 3, we shall make use of another partitioning method due to Yang [13] and construct another pair of general second-order parametric duality models with different constraint structures and discuss several second-order duality results under a variety of generalized (F, β, ϕ, ζ, ρ, θ, m)-sounivexity assumptions.Finally, in Section 4 we present some remarks on our main results aiming at some future research endeavors arising from certain modifications of the principal minmax model formulated in the present investigation.
We next introduce some generalized versions of the sounivexity (Zalmai [15]) and others in the literature.Let f : X → R be a twice differentiable function.Suppose that ∥ • ∥ denotes a norm on R n , and ⟨a, b⟩ is the inner product of the vectors a and b.
Definition 1.1.The function f is said to be (strictly Stat., Optim.Inf.Comput.Vol.The function f is said to be (strictly The function f is said to be (strictly The function f is said to be (prestrictly From the above definitions it is clear that if We observe that in some contexts during proving the duality theorems, it may be easier to apply certain alternative but equivalent forms of the above definitions by considering the contrapositive statements.For example, (F, β, ϕ, ρ, ζ, θ, m)-quasisounivexity can be defined in the following equivalent way: a sublinear function (x, x * ; •) : R n → R, and a positive integer m such that for each x ∈ X and z ∈ R n , As the generalized sounivexity of functions generalizes most of the convex functions, including generalized invexity, pseudoinvexity and quasiinvexity by identifying appropriate choices of F, β, ϕ, ρ, ζ, θ, and m where η is a given function from X × X to R n , then we obtain the definitions of (ϕ, η, ρ, ζ, θ, m)-sonvex, (ϕ, η, ρ, ζ, θ, m)-pseudosonvex, and (ϕ, η, ρ, ζ, θ, m)-quasisonvex functions introduced recently in [10].For example, let f : X → R be a twice differentiable function.
Example 1.Let f : X → R be a twice differentiable function with the norm ∥ • ∥ on R n and inner product ⟨a, b⟩ of the vectors a and b.Then the function f is said to be (strictly We conclude this section by recalling a set of second-order necessary optimality conditions for (P ).
Theorem 1.1.[15] Let x * be an optimal solution of (P), let and assume that the functions f i , g i , i ∈ p, G j , j ∈ q, and H k , k ∈ r, are twice continuously differentiable at x * , and that the second-order Guignard constraint qualification holds at x * .Then for each where C(x * ) is the set of all critical directions of (P) at x * , that is, For brevity, we shall henceforth refer to x * as a normal optimal solution of (P ) if it is an optimal solution and satisfies the second-order Guignard constraint qualification.
In the remainder of this paper, we shall assume that the functions f i , g i , i ∈ p, G j , j ∈ q, and H k , k ∈ r, are twice continuously differentiable on the open set X.Moreover, we shall assume, without loss of generality, that g i (x) > 0, i ∈ p, and φ(x) ≥ 0 for all x ∈ X.

Duality Model I
In this section, we discuss several families of duality results under various generalized (F, β, ϕ, ρ, ζ, θ, m)sounivexity hypotheses imposed on certain combinations of the functions involved in the considered optimization problem.This is accomplished by employing a certain partitioning scheme which was originally proposed in [6] for the purpose of constructing generalized dual problems for nonlinear programming problems.For this we need some additional notation.
Consider the following two problems: where F(x, y; •) is a sublinear function from R n to R.
The feasible set F (DI) (assumed to be nonempty) of (DI) is defined as: Comparing (DI) and ( DI), we see that ( DI) is relatively more general than (DI) in the sense that any feasible solution of (DI) is also feasible for ( DI), but the converse is not necessarily true.Furthermore, we observe that (2.1) is a system of n equations, whereas (2.6) is a single inequality.Clearly, from a computational point of view, (DI) is preferable to ( DI) because of the dependence of (2.6) on the feasible set of (P ).
Despite these apparent differences, it turns out that the statements and proofs of all the duality theorems for (P ) − (DI) and (P ) − ( DI) are almost identical and, therefore, we shall consider only the pair (P ) − (DI).
In the proofs of our duality theorems, we shall make frequent use of the following auxiliary result which provides an alternative expression for the objective function of (P ).Lemma 1. [10] For each x ∈ X, .
The next two theorems show that (DI) is a dual problem for (P ).
Theorem 2.1.(Weak Duality) Let x and y be arbitrary feasible solutions of (P) and (DI), respectively.Furthermore, assume that any one of the following four sets of hypotheses is satisfied: )-quasisounivex at y, φt is increasing, and φt (0) = 0; )-quasisounivex at y, φt is increasing, and φt (0) = 0; ]) ≥ 0. (2.7) ≤ 0 (by the primal feasibility of x) (by (2.4) and the dual feasibility of y) Thus, it follows from (ii) that Summing over t ∈ M and using the sublinearity of F(x, y; •), we obtain By (i), the above inequality implies that φ( Φ(x, u, v, w, λ) − Φ(y, u, v, w, λ) But φ(a) ≥ 0 ⇒ a ≥ 0, and hence we get where the second inequality follows from (2.3) and the dual feasibility of y.Since x ∈ F, the above inequality reduces to (2.11) Now using (2.11) and Lemma 2.1, we see that (b): The proof is similar to that of part (a).
(d): The proof is similar to that of part (c).

Theorem 2.3. (Strict Converse Duality) Let
x * be a normal optimal solution of (P), and let x be an optimal solution of (DI) with respect to z, ũ, ṽ, w, λ.Furthermore assume that any one of the following four sets of conditions holds:

Proof
Since x * is a normal optimal solution of (P ), by Theorem 2.1, there exist ũ, ṽ, w, and λ such that x (with respect to z, ũ, ṽ, w, λ) is a feasible solution of (DI) and φ(x )-quasisounivex at y, and for each t ∈ M , φt is increasing and φt (0) = 0, where v, w, λ) is prestrictly (F, β, φi , ρi , ζ, θ, m)-quasisounivex at y, and for each i ∈ I + , φi is strictly increasing and φi (0) = 0, where {I 1+ , I 2+ } is a partition of )-quasisounivex at y, and for t ∈ M , φt is increasing and Proof (a) : Suppose to the contrary that φ(x) < λ.This implies that ( Keeping in mind that u ≥ 0, we see that for each i ∈ I + , and so it follows from the properties of φi that Hence, by (i), for each i ∈ I + , the above inequality implies that F ( x, y; β(x, y) Since u ≥ 0, u i = 0 for each i ∈ p\I + , ∑ p i=1 u i = 1, and F(x, y; •) is sublinear, the above inequalities yield
(b) -(g) : The proofs are similar to that of part (a).
Theorem 2.5.(Strong Duality) Let x * be a normal optimal solution of (P) and assume that any one of the seven sets of conditions set forth in Theorem 2.4 is satisfied for feasible solutions of (DI).Then for each z * ∈ C(x * ), there exist u * , v * , w * , and λ * such that x * is an optimal solution of (DI) and φ(x * ) = λ * .

Proof
The proof is similar to that of Theorem 2.2.

Duality Model II
In this section we discuss two additional duality models for (P ).In these duality formulations, we utilize a partition of p in addition to those of q and r.This partitioning scheme, which is a slightly extended version of the one initially proposed by Mond and Weir [6], was used by Yang [13] for formulating a generalized duality model for a multiobjective fractional programming problem.In our duality theorems, we impose appropriate generalized (F, β, ϕ, ρ, ζ, θ, m)-sounivexity requirements on certain combinations of the problem functions.Let {I 0 , I 1 , . . ., I ℓ } be a partition of p such that L = {0, 1, 2, . . ., ℓ} ⊂ M = {0, 1, . . ., M }, and let the realvalued function ξ → Π t (ξ, u, v, w, λ) be defined, for fixed u, v, w, and λ, on X by Consider the following two problems: ( DII) Maximize λ subject to (3.2) -(3.5) and where F(x, y; •) is a sublinear function from R n to R.
The feasible set F (DII) (assumed to be nonempty) of (DII) is defined as: The comments and observations made earlier about the relationship between (DI) and ( DI) are, of course, also valid for (DII) and ( DII).
The following two theorems show that (DII) is a dual problem for (P ).
Theorem 3.1.(Weak Duality) Let x and y be arbitrary feasible solutions of (P) and (DII), respectively.Furthermore, assume that any one of the following seven sets of hypotheses is satisfied: (a) (i) for each t ∈ L, ξ → Π t (ξ, u, v, w, λ) is strictly (F, β, ϕ t , ρ t , ζ, θ, m)-pseudosounivex at y, ϕ t is increasing, and ζ, θ, m)-quasisounivex at y, ϕ t is increasing, and ϕ t (0) = 0; (iii) )-quasisounivex at y, and for each t ∈ L, ϕ t is increasing and ϕ t (0) = 0, where )-quasisounivex at y, and for each t ∈ L, ϕ t is increasing and ϕ t (0) = 0, where )-quasisounivex at y, and for each t ∈ M \ L, ϕ t is increasing and ϕ t (0) = 0, where Proof (a) : Suppose to the contrary that φ(x) < λ.This implies that Since u ≥ 0 and u ̸ = 0, we see that for each Now using this inequality, we see that for each t ∈ L, and hence Hence, by (i), for each t ∈ L, the above inequality implies that  Theorem 3.2.(Strong Duality) Let x * be a normal optimal solution of (P) and assume that any one of the seven sets of conditions set forth in Theorem 3.1 is satisfied for all feasible solutions of (DII).Then for each z * ∈ C(x * ), there exist u * , v * , w * , and λ * such that x * is an optimal solution of (DII) and φ(x * ) = λ * .

Proof
The proof is similar to that of Theorem 2.2.

Concluding Remarks
It seems that the results presented in this paper will prove useful in investigating other related classes on nonlinear programming problems and applying similar generalized convexity concepts to nonlinear fractional programming problems, including finite and semiinfinite aspects, for example, a class of semiinfinite minmax fractional programming problems.