ON DUALITY THEORY FOR MULTIOBJECTIVE SEMI-INFINITE FRACTIONAL OPTIMIZATION MODEL USING HIGHER ORDER CONVEXITY

In the article, a semi-infinite fractional optimization model having multiple objectives is first formulated. Due to the presence of support functions in each numerator and denominator with constraints, the model so constructed is also non-smooth. Further, three different types of dual models viz Mond-Weir, Wolfe and Schaible are presented and then usual duality results are proved using higherorder (K × Q) − (F , α, ρ, d)-type I convexity assumptions. To show the existence of such generalized convex functions, a nontrivial example has also been exemplified. Moreover, numerical examples have been illustrated at suitable places to justify various results presented in the paper. The formulation and duality results discussed also generalize the well known results appeared in the literature. Mathematics Subject Classification. 90C29, 90C32, 90C46. Received October 9, 2020. Accepted April 17, 2021.


Introduction
A semi-infinite model (SIM) is an optimization problem having finite number of variables with the infinite number of constraints. Initially, in 1962, SIM is named by Charnes et al. [6], in which a survey of SIM mainly about a linear model and duality results with convex property have been done. In this direction, some important theorems for the linear model have been generalized using the pairing of finite space of sequences and vector space of finite dimension. Later, the application of SIM in Euclidean space has been shown by Charnes et al. [7] and its implication in duality results for a n-dimensional convex minimization problem have been demonstrated. For these convex problems, Karney [18] proposed duality results using its Lagrangian dual. Jeyakumar [14] introduced new constraints qualifications for convex SIM and then developed a strong duality relation. SIM is important for both its results and latent applications in different mathematical fields. It is not only used in the practical problems in which constraints have time or space parameters but also in the areas related to statistics, robotics, transportation problems, game theory and engineering. For more details about significance of SIM, we refer to [11,12,20,23,28,36]. Ito et al. [13] have derived optimality conditions and duality results for the convex SIM using Slater's constraint qualification. After that, considering constraints over arbitrary cones, Shapiro [27] has developed weak and strong duality relations for convex SIM. Next, Gupta and Srivastava [10] have discussed KKT results for the nonsmooth multiobjective programming problem and then developed usual duality relations. An algorithm based on parametric dual for the quadratic semi-infinite problem have been proposed and convergence of the method is shown in Liu et al. [21]. Further, Basu et al. [5] have discussed the duality gap for SIM with the support dual.
Using generalized (η, ρ)-invexity, Zalmai and Zhang [37] have developed non-parametric duality relations for semi-infinite discrete minimax fractional problem and further, second-order parameter free duality results are established by Zalmai [36]. After that, Antczak and Zalmai [3] have established second-order relations for semiinfinite minimax type fractional optimization using (Φ, ρ) − V -invexity assumptions. These invexity conditions are later on extended to higher-order in Stancu-Minasian et al. [30]. Considering the same optimization model [3], Verma and Zalmai [35] have studied a parameter free dual model and established duality results using (φ, ρ, θ,m)-sonvexity. Later, the approximate duality relations for nonsmooth minimax fractional optimization model using higher order B − (p, r) invexity have been discussed in Sonali et al. [29].
Mishra and Jaiswal [22] have discussed SIM involving equilibrium constraints and derived optimality conditions with duality results using invexity property. For SIM, considering the concept of convexificators, Pandey and Mishra [24,25] have proposed necessary as well as sufficient optimality conditions. Further, they formulated Mond-Weir and Wolfe type duals, and proved related theorems with the help of ∂ * -convexity\generalized convexity. Slater's constraints qualification is used for a quasiconvex SIM and then optimality theorems with duality relations are established by Kanzi and Soleimani-damaneh [17]. A fractional semi-infinite problem with (H p , R)-invexity have been studied in Patel and Patel [26].
Recently, a robust approximation approach is applied in fractional semi-infinite programming and some interesting results for optimality solution with approximation have been established in Zeng et al. [38]. After that, mixed type dual models are formulated and approximate dual relations are discussed for nonlinear SIM in Sun et al. [31,32]. For a robust vector optimization problem, inspiring from the concept of Quasi -solution for SIM in Jiao et al. [15], necessary and sufficient optimality relations between feasible solution and -solution are developed in Antczak et al. [4]. Using convex decomposition, optimality conditions and extended duality results are developed for generalized SIM by Aboussoror et al. [1]. In Tung [33,34], subdifferential in terms of tangential convexity is used for developing KKT and strong KKT optimality results for multiobjective SIM. In terms of invexity and equilibrium constraints, sufficient optimality conditions and duality results for two dual models have been derived in Joshi [16]. Recently, Emam [9] has studied a nonsmooth SIM involving E-convexity and support functions, and further established duality results by constructing Mond-Weir type dual model.
Liang et al. [19] have introduced the concept of generalized (F, α, ρ, d) convexity and further for fractional optimization model, they have derived optimality relations and usual duality theorems. Using the same type of convexity, higher order dual models are formulated and optimality relations are derived for minimax type problems in Ahmad et al. [2]. Motivated by the work in [2,19,34], in this paper, we have studied a new class of semi-infinite fractional programming over arbitrary cones. The main outcomes of the paper are briefly explained below: -Problem formulation: A new class of semi-infinite fractional multiple objective problem over arbitrary cones has been formulated. Due to the presence of support functions in each numerator and denominator of the objective function and in each constraint, the problem becomes non-smooth. This not only generalizes all the existing semi-infinite models but also gives infinitely many optimization problems since it involves arbitrary cones. -Assumptions: The concept of higher order (K × Q) − (F, α, ρ, d)-type I convexity is introduced whose existence is further illustrated by citing a non-trivial example. This paper is organized as : In Section 2, some notations and preliminary results are recalled. Also, the concept of higher order (K × Q) − (F, α, ρ, d)-type I convexity is introduced and further, a non-trivial example has been demonstrated. In Sections 3-5, for a class of a non-smooth multiple objective semi-infinite fractional  programming problem, higher order Mond-Weir, Wolfe and Schaible type dual models are constructed, and  usual duality results are proved under aforesaid assumption. To validate and clarify the duality results, different numerical examples are also shown at suitable places. In the last section, the conclusion with future scope is given.

Preliminaries
Consider the following cone optimization model: denotes the feasible region of the problem (MP).
, for all γ ∈ R + and b ∈ R n . Definition 2.3. Let F : B × B × R n → R be a sublinear functional in the third variable. Then the pair (ψ, φ) is called (strictly) higher order (K × Q) − (F, α, ρ, d)-type I convex atũ ∈ R n with respect to L : B × R n → R k and S : B × R n → R m , if for all x ∈ B, p, q ∈ R n , there exist real valued function α(· , ·) : . . , k, j = 1, 2, . . . , m, such that Next, we will show a non-trivial example to illustrate the existence of such functions.

Definition 2.5 ([31]
). Let f : R n → R be a convex function. Then, the subdifferential of f atx is defined as Definition 2.6 ( [8]). The support function of a compact convex set A ⊆ R n is defined as The subdifferential of support function Ω(x|A) atx is given by Now, consider the semi-infinite multiobjective fractional programming problem as follows: . , m}, f : R n → R k , g : R n → R k , h : R n → R m are continuously differentiable functions and for compact convex sets C i , D i , E j and M j in R n , respective support functions are Ω(x|C i ), Ω(x|D i ), Ω(x|E j ) and Ω(x|M j ) for i ∈Ĩ, j ∈J. Also, assume that f i (·) + Ω((·)|C i ) ≥ 0 and g i (·) − Ω((·)|D i ) > 0 for all feasible x and T is an infinite index set.
for all t ∈ T } represents the feasible region of the problem (SIFP). Let K * and Q * be positive dual cones of K and Q, respectively.

Mond-Weir type dual
For (SIFP) model, consider the following Mond-Weir type higher order dual: Theorem 3.1 (Weak duality). Assume that x and (u, v, w 1 , w 2 , λ, µ, z, p, q) be feasible for the problems (SIFP) and (MD), respectively. Let a sublinear functional (in third variable) be F : Then Proof. By hypothesis (i) and Definition 2.3 at u with respect to L : B × R n → R k and S : B × R n → R m , we have It follows from λ ∈ int K * and (3.5) that Using the sublinearity property of F, λ ∈ int K * ⊆ int R k + and dual constraint (3.3), we get Using µ > 0, along with sublinearity of F, the above inequality gives Further, using inequality (3.2) in the addition of (3.7) and (3.8), we obtain It follows from assumption (iii), dual constraint (3.1), sublinearity of F and F x, Now, on the contrary, suppose that (3.4) is not correct. Then, which contradicts the inequality (3.9). This completes the proof.
given as:
Let the sublinear functional be F x,u (b) = bxu and α(x, y) = 1 + x 2 y 2 . Let Thus, their support functions will be Validation of Theorem 3.1: First we will show that all the hypothesis of Theorem 3.1 are satisfied.
Further, for w 1 Hence, the hypothesis (i) of Theorem 3.1 is satisfied. Moreover, R 2 Thus, hypotheses (ii) and (iii) of Theorem 3.1 also hold. Now, for β, the expression Hence, the Theorem 3.1 is verified at x = 0, −1 ∈ S 0 and the point β feasible for (MD).

Validation of Theorem 3.3:
For the points β and x = 0, it has been shown above that the assumptions (ii), (iii) and (iv) of Theorem 3.3 hold true. Also, the value of the expression Thus, the assumption (i) of Theorem 3.3 also holds. Hence verified.

Wolfe type dual
Consider the following Wolfe type higher order dual model for (SIFP):

Theorem 4.1 (Weak duality)
. Let x and (u, v, w 1 , w 2 , λ, µ, z, p, q) be feasible for the problems (SIFP) and (WD), respectively. Let a sublinear functional (in third variable) be F : B × B × R n → R. Also, assume that . . , h m (·, τ ) + (·) T w 1 m is higher order (K × Q) − (F, α, ρ, d)-type I convex at u with respect to L and S, Then Proof. It follows from hypotheses (i) and (ii) and sublinearity of functional F that (4.5) Using (4.2) in (4.5) and then adding with (4.4), we get It follows from the hypothesis (iii) and sublinearity of F that Further, applying inequality (4.1) and using F x,u (0) = 0, we get Now, if possible, suppose that Finally, using (4.8) in (4.7), we obtain which contradicts (4.6). This proves the theorem.
Also, R 2 Thus, all the hypotheses of Theorem 4.1 are satisfied. Now for the feasible point (of (WD)) β 2 , we get This validates the result of Theorem 4.1 for x = 0, −1 ∈ S 0 and β 2 feasible for (WD).

Validation of Theorem 4.3:
For the points β 2 and x = −1, it has been proved above that the assumptions (ii), (iii) and (iv) of Theorem 4.3 are satisfied. Further, Hence, the assumption (i) also holds at x = u = −1. This completes the validation of Theorem 4.3.

Without the assumption (i) of Theorem 4.1:
For any ρ and R 2 + = K, R 2 + ⊂ Q. Therefore, assumption (ii) and (iii) of Theorem 4.1 are satisfied but at x = 0, Hence, the pair On the other hand, at x = 0 and β 3 , This shows the significance of assumption (i) in Theorem 4.1, without which the result may not satisfy.
Proof. It follows on the lines of Theorem 4.2.

Conclusion
To the best of our knowledge, the class of conic non-smooth semi-infinite multiobjective fractional programming problem has not been studied so far. In this article, semi-infinite model with multiple fractional type objective function is formulated. Further, introducing the idea of higher order (K × Q) − (F, α, ρ, d)-type I convex function, the duality relations for Mond-Weir, Wolfe and Schaible type dual models have been developed. Validation of various results obtained have also been shown by demonstrating non trivial examples. Further, it has been shown by giving examples that considering the assumptions of higher order (K × Q) − (F, α, ρ, d)-type I convexity is significant since without this, the duality results obtained may not hold. Exploring optimality relations and duality theorems for (SIFP) over space of symmetric matrices by using E-convexity in objective functions be an enthralling future work in this direction.