Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) Optimality and Duality for Multiobjective Semiinfinite Programming Problems Involving Generalized (C, α, η, ρ, d)-Invexity Pooja Gupta Department of Basic Science & Humanities Maharana Pratap Group of Institution, Kanpur, India Email: pujjubhumath@gmail.com Alka Katiyar Department of Mathematics, School of Basic Sciences CSJM University, Kanpur, India Email:alkakatiyar0092@gmail.com Sandeep Kumar Porwal∗ Department of Mathematics, School of Basic Sciences CSJM University, Kanpur, India Email:skpmathsdstcims@gmail.com ∗ Corresponding author (Article History: Received: 12-01-2025; Revised: 25-02-2025; Accepted: 05-03-2025 ) Abstract In this paper, we formulate generalized (C,α, η, ρ, d)-invexity and based on these definitions, we derive several sufficient conditions for optimality in multiobjective semi-infinite programming problems. Further, under the assumptions of dual model we solve corresponding weak, strong and strict converse duality theorems for these multiobjective semiinfinite programming problem. 2020 Mathematics Subject Classification: 90C25, 90C29, 90C30, 90C34, 90C46 Keywords: Invex set, invex function, multiobjective programming, semiinfinite programming, feasible solution, efficient solution. 1 Introduction The theory surrounding semi-infinite programming involves minimizing a function with a finite number of variables while adhering to an arbitrary number of inequalities. When there are multi- ple objective functions, this is referred to as a multiobjective semi-infinite programming problem. Shapiro [7] provided an overview of the foundational theory of semi-infinite programming, exploring various methods for establishing duality, discretization, and both first and second-order optimality conditions. For further information and applications related to semi-infinite programming, please consult the cited references. [2, 3, 4, 5, 6, 8, 11, 12, 33, 34]. In [28], Preda introduced the idea of (F, ρ)-convexity, expanding upon the concepts of F -convexity [18] and ρ-convexity [16], and derived several duality results. Liang et al. in [31] later presented (F, α, ρ, d)-convexity to address nonlinear fractional programming problems, which encompasses the (F, ρ)-convex functions. Subsequently, Liang et al. in [30] broadened the findings from [31] to in- clude a specific category of multiobjective fractional programming problems. Yuan et al. defined (C,α, ρ, d)-convexity in [13], as a generalization of (F, α, ρ, d)-convexity and established optimality conditions along with duality results for nondifferentiable minimax fractional programming problems that utilize generalized convex functions. Additionally, Long in [29] and Mishra et al. in [25] derived sufficient optimality conditions and duality theorems employing (C,α, ρ, d)-convexity for nondiffer- entiable multiobjective fractional programming and nondifferentiable multiobjective semi-infinite https://internationalpubls.com 1140 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) programming problems, respectively. Mishra et al. [21] also achieved optimality and duality results for minimax fractional programming involving support functions within the framework of (C,α, ρ, d)-convexity. Invexity is crucial for establishing optimality conditions and duality results across a range of optimization problems. Invex functions represent a broader category than convex functions, retain- ing many of their key properties. Additionally, invexity and its generalizations can be viewed as alternatives to convexity. For further information on invex functions, please refer to the relevant literature [1, 9, 10, 17, 22, 23, 32]. Weir [27] examined a multiobjective programming problem that incorporates invex functions and derived duality results under the condition that the multipliers for all objective functions are strictly positive. Kuk et al. in [15] formulated generalized K-K-T necessary and sufficient optimal- ity conditions, along with duality theorems, for nonsmooth multiobjective fractional programming problems that involve V −ρ− invex functions. Caristi et al. [14] focused on multiobjective program- ming with a set of constraints defined within a compact set, obtaining Kuhn-Tucker type optimality criteria under relaxed invexity conditions. They also introduced dual problems where both weak and strong duality properties are maintained in the same framework. Additionally, Mishra et al. [24] developed Wolfe and Mond-Weir-type dual models, establishing duality theorems for the nons- mooth semi-infinite programming problem discussed in [20]. Motivated by the research of Ben-Israel and Mond [1], Caristi et al. [14] and Jaiswal and Mishra [19], we extend the definition of gener- alized (C,α, ρ, d)-convexity to generalized (C,α, η, ρ, d)-invexity. Additionally, we present sufficient conditions for optimality and duality theorems. The structure of this paper is organized as follows: In Section 2, we provide some preliminaries, our problem and some definitions. Section 3 is dedicated to establishing definitions that will be essential for our theorems. In Section 4, we derive sufficient conditions for optimality. Finally, Section 5 presents weak, strong, and strict converse duality theorems that connect the primal problem with the Mond-Weir dual problem under the framework of generalized (C,α, η, ρ, d)-invexity. 2 Preliminaries In an n-dimensional Euclidean space Rn, let Rn + is the non-negative orthant . Consider the nonlinear multiobjective semiinfinite programming problem defined as: (SP) Min f(x) Subject to gj(x) ≦ 0, j ∈ J, Let f(x) = (f1(x), ..., fp(x)), where each fi(i ∈ p ≡ {1, 2, ..., p}) and gj(j ∈ J) are differentiable functions defined on a non-empty open subset X ⊆ Rn and map to R. The set J is an index set that may be infinite. Here we take the non-empty feasible set S of (SP): S = {x ∈ X : gj(x) ≦ 0, j ∈ J}. and I = {j ∈ J : gj(x0) = 0} where I represents the index set of active constraints for x0 ∈ S. Let C : X×X×Rn → R be a function such that for any (x, x0) ∈ X×X, it satisfies C(x,x0)(0) = 0. Additionally, let α : X ×X → R+ ∖ {0}, ρ ∈ R, and d : X ×X → R+, (d(x, x0) = 0 iff x = x0). Definition 2.1. A function C : X ×X ×Rn → R is convex on Rn if and only if for any fixed point (x, x0) ∈ X ×X and for any y1, y2 ∈ Rn, the following condition holds: C(x,x0)(λy1 + (1− λ)y2) ≦ λC(x,x0)(y1) + (1− λ)C(x,x0)(y2), ∀ λ ∈ (0, 1). Definition 2.2. A feasible point x0 ∈ X is considered an efficient solution for problem (SP) if and only if there is no point x ∈ X such that: f(x) ≤ f(x0). Definition 2.3. A feasible point x0 ∈ X is a weak efficient solution for problem (SP) iff there is no point x ∈ X such that: f(x) < f(x0). https://internationalpubls.com 1141 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) The necessary optimality conditions for (SP) presented below are taken from [19] Theorem 2.4. (Necessary Optimality Conditions) Let x0 be an efficient solution for (SP) and I(x0) ̸= ϕ. If (SP) satisfies the suitable constraint qualification (see [26]) at x0 then there exist u ∈ Rp, v = (vj)j∈J , such that uT∇f(x0) + vI T∇gI(x0) = 0, (2.1) vT g(x0) = 0, u ≥ 0, v ≥ 0 and vj ̸= 0 for finitely many j ∈ I. 3 Definitions In 2006, Yuan and Liu (see [13]) gave the definition of (C,α, ρ, d) convex function. We see that this definition is not hold when α ≤ 0. In this paper, we introduce the definitions of (C,α, η, ρ, d) invex function and its generalizations, which will also hold when η ≤ 0. These definitions will be used in the sequal. Definition 3.1. The vector-valued function f : X → Rp is called (C,α, η, ρ, d)-invex at x0 ∈ X if, for each fi : X → R, there exists a vectorial function ηi : X × X → R such that for every x ∈ X and i ∈ p, the following condition holds: fi(x)− fi(x0) αi(x, x0) (>) ≧ C(x,x0)(∇fi(x0))ηi(x, x0) + ρi di(x, x0) αi(x, x0) . The function f is said to be (C,α, η, ρ, d)-invex on X if and only if it is (C,α, η, ρ, d)-invex at every point in X. By replacing η(x, x0) with 1 in the above definition, we will get the definition of (C,α, ρ, d) con- vexity. In this example, we try to illustrate the importance of invex functions and its generalization. Exmple. Let X = {x : π 4 ≤ x ≤ π 2 }, ρ = −1, α(x, x0) = 1, d(x, x0) = (x − x0), C(x,x0)(a) = a2(x− x0) for any (x, x0) ∈ X ×X and let f(x) = cos2x. Then, we see that f(x) is not (C,α, ρ, d)- convex at x = π 4 , but it is (C,α, η, ρ, d)-invex at x = π 4 with η(x, x0) = (1− cos(x− x0) (x− x0) ). As we know that the definition of (C,α, ρ, d) convexity is not hold when α ≤ 0. But on the other hand we see that, the definition of (C,α, η, ρ, d) invexity holds for all real values of η. Definition 3.2. The vector-valued function f : X → Rp is called (C,α, η, ρ, d)-pseudo-invex at x0 ∈ X if, for each fi : X → R, there exists a vectorial function ηi : X ×X → R such that for every x ∈ X and i ∈ p, the following condition holds: fi(x)(≦) < fi(x0) ⇒ C(x,x0)(∇fi(x0))ηi(x, x0) + ρi di(x, x0) αi(x, x0) < 0. The function f is said to be (C,α, η, ρ, d)-pseudo-invex on X iff it is (C,α, η, ρ, d)-pseudo-invex at each point in X. Definition 3.3. The vector-valued function f : X → Rp is called weak strictly (C,α, η, ρ, d)-pseudo- invex at x0 ∈ X if, for each fi : X → R there exists a vectorial function ηi : X ×X → R such that for every x ∈ X and i ∈ p, the following condition holds: fi(x) ≤ fi(x0) ⇒ C(x,x0)(∇fi(x0))ηi(x, x0) + ρi di(x, x0) αi(x, x0) < 0. The function f is said to be weak strictly (C,α, η, ρ, d)-pseudo-invex on X iff it is weak strictly (C,α, η, ρ, d)-pseudo-invex at each point in X. Definition 3.4. The vector-valued function f : X → Rp is said to be strong (C,α, η, ρ, d)-pseudo- invex at x0 ∈ X if, for each fi : X → R there exists a vectorial function ηi : X ×X → R such that for each x ∈ X and i ∈ p, the following condition holds: fi(x) ≤ fi(x0) ⇒ C(x,x0)(∇fi(x0))ηi(x, x0) + ρi di(x, x0) αi(x, x0) ≤ 0. https://internationalpubls.com 1142 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) The function f is said to be strong (C,α, η, ρ, d)-pseudo-invex on X iff it is strong (C,α, η, ρ, d)- pseudo-invex at each point in X. Definition 3.5. The vector-valued function f : X → Rp is said to be (C,α, η, ρ, d)-quasi-invex at x0 ∈ X if, for each fi : X → R there exists a vectorial function ηi : X ×X → R such that for each x ∈ X and i ∈ p, the following condition holds: fi(x) ≦ fi(x0) ⇒ C(x,x0)(∇fi(x0))ηi(x, x0) + ρi di(x, x0) αi(x, x0) ≦ 0. The function f is said to be (C,α, η, ρ, d)-quasi-invex on X iff it is (C,α, η, ρ, d)-quasi-invex at each point in X. Definition 3.6. The vector-valued function f : X → Rp is said to be weak (C,α, η, ρ, d)-quasi-invex at x0 ∈ X if, for each fi : X → R there exists a vectorial function ηi : X × X → R such that for each x ∈ X and i ∈ p, the following condition holds: fi(x) ≤ fi(x0) ⇒ C(x,x0)(∇fi(x0))ηi(x, x0) + ρi di(x, x0) αi(x, x0) ≦ 0. The function f is said to be weak (C,α, η, ρ, d)-quasi-invex on X if and only if it is weak (C,α, η, ρ, d)- quasi-invex at each point in X. 4 Optimality For the nonlinear multiobjective semiinfinite programming problem (SP), we state the following optimality conditions. Theorem 4.1. If there exist a feasible solution x0 for (SP) and, u ∈ Rp and v = (vj)j∈J are vectors which satisfies uT∇f(x0) + vTI ∇gI(x0) = 0, vT g(x0) = 0, (4.1) u ≥ 0, v ≥ 0 and vj ̸= 0 for finite many j ∈ I. Also, if f is strong (C,α1, η1, ρ1, d1)-pseudoinvex function at x0 and gI is (C,α2, η2, ρ2, d2)-quasi- invex function at x0 w.r.t. the same η(x, x0) (i.e., η 1 = η2 = η) with p∑ i=1 uiρ 1 i d1i (x, x0) α1 i (x, x0) + ∑ j∈I vjρ 2 j d2j (x, x0) α2 j (x, x0) ≧ 0. (4.2) Then, x0 is an efficient solution of (SP) Proof. Assume that x0 is not an efficient solution of (SP), then for a feasible solution x ∈ S, we have fi(x) ≤ fi(x0). Since gI(x0) = 0, hence gI(x) ≦ gI(x0). Since f is strong (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is (C,α2, η2, ρ2, d2)-quasi-invex at x0, therefore we have C(x,x0)(∇fi(x0))η(x, x0) + ρ1i d1i (x, x0) α1 i (x, x0) ≤ 0, and C(x,x0)(∇gI(x0))η(x, x0) + ρ2I d2I(x, x0) α2 I(x, x0) ≦ 0. Since C is convex, thus from the above inequalities, we get C(x,x0)  p∑ i=1 1 τ ui∇fi(x0) + ∑ j∈I 1 τ vj∇gj(x0)  η(x, x0) https://internationalpubls.com 1143 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) + p∑ i=1 1 τ uiρ 1 i d1i (x, x0) α1 i (x, x0) + ∑ j∈I 1 τ vjρ 2 j d2j (x, x0) α2 j (x, x0) < 0, where τ = p∑ i=1 ui + ∑ j∈I vj . Since C(x, x0)(0) = 0, thus from equation (4.1) we have p∑ i=1 uiρ 1 i d1i (x, x0) α1 i (x, x0) + ∑ j∈I vjρ 2 j d2j (x, x0) α2 j (x, x0) < 0, which go against our assumption (4.2). Thus x0 is an efficient solution of (SP). Example: Consider the semiinfinite problem as follows: (SP 1) Minimum f(x) s.t. gj(x) ≦ 0, j ∈ J, x ∈ R, where the functions F : X → R2 and gj : X → R, (X = R) are defined as: F (x) = (f1(x), f2(x)) = (x2 − 2x, x3 − x2) and g1(x) = x2(x− 2), g2(x) = x3 + x, gk(x) = x+ 1 k , k = 3, 4, ... Also the function C : R × R × R → R is defined by C(x,x0)(a) = −a2(x + x0), the function η : X ×X → R is defined by η(x, x0) = (2x − x0) and the set S of feasible solutions for (SP 1) is defined by S = {x ∈ R : gj(x) ≦ 0} = {x ∈ R : x ≦ −1 3 } We can check that fi(i = 1, 2) are strong (C,αi, ηi, ρi, di)-pseudo-invex at x0 = −1 3 , with αi(x, x0) = 1, ηi = η, ρi = 0 and di(x, x0) = 1, also gj(j ∈ I) are (C,αj , ηj , ρj , dj)-quasi-invex at x0 = −1 3 , with αj(x, x0) = 1, ηj = η, ρj = 0 and dj(x, x0) = 1. Clearly, ηi = ηj = η and x0 = −1 3 is a feasible solution for (SP 1) that meets the requirements of Theorem 4.1, in which u = ( 3 4 , 1) and v = (0, 0, 1, 0, ..., 0, ...). For x0 = −1 3 , I = {j ∈ J : gj(x0) = 0} = {3} is the index set of active constraints. We conclude that there is no point x ∈ S such that f(x) ≤ f(x0). Hence, x0 = −1 3 is an efficient solution of (SP 1). Theorem 4.2. If there exist a feasible solution x0 for (SP) and, u ∈ Rp and v = (vj)j∈J are vectors which satisfies uT∇f(x0) + vTI ∇gI(x0) = 0, vT g(x0) = 0, u ≥ 0, v ≥ 0 and vj ̸= 0 for finitely many j ∈ I. Also, if f is weak strictly (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is (C,α2, η2, ρ2, d2)-quasi-invex at x0 with respect to same η(x, x0) (i.e., η 1 = η2 = η) with p∑ i=1 uiρ 1 i d1i (x, x0) α1 i (x, x0) + ∑ j∈I vjρ 2 j d2j (x, x0) α2 j (x, x0) ≧ 0. Then, x0 is an efficient solution of (SP). https://internationalpubls.com 1144 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) Proof. Assume that x0 is not an efficient solution of (SP). Then for a feasible solution x ∈ S, we have fi(x) ≤ fi(x0). Since gI(x0) = 0, hence gI(x) ≦ gI(x0). Since f is weak strictly (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is (C,α2, η2, ρ2, d2)-quasi-invex at x0, therefore we have C(x,x0)(∇fi(x0))η(x, x0) + ρi 1 d 1 i (x, x0) α1 i (x, x0) < 0, and C(x,x0)(∇gI(x0))η(x, x0) + ρ2I d2I(x, x0) α2 I(x, x0) ≦ 0. The remaining part of the proof is similar to proof of Theorem 4.1. Theorem 4.3. If there exist a feasible solution x0 for (SP) and, u ∈ Rp and v = (vj)j∈J are vectors which satisfies uT∇f(x0) + vTI ∇gI(x0) = 0, vT g(x0) = 0, u ≥ 0, v ≥ 0 and vj ̸= 0 for finitely many j ∈ I. Also, if f is weak (C,α1, η1, ρ1, d1)-quasi-invex at x0 and gI is strictly (C,α2, η2, ρ2, d2)-pseudo-invex at x0 with respect to same η(x, x0) (i.e., η 1 = η2 = η) with p∑ i=1 uiρ 1 i d1i (x, x0) α1 i (x, x0) + ∑ j∈I vjρ 2 j d2j (x, x0) α2 j (x, x0) ≧ 0. Then, x0 is an efficient solution of (SP). Proof. Assume that x0 is not an efficient solution of (SP). Then for a feasible solution x ∈ S, we have fi(x) ≤ fi(x0). Since gI(x0) = 0, hence gI(x) ≦ gI(x0). Since f is weak (C,α1, η1, ρ1, d1)-quasi-invex at x0 and gI is strictly (C,α2, η2, ρ2, d2)-pseudo-invex at x0, therefore we have C(x,x0)(∇fi(x0))η(x, x0) + ρ1i d1i (x, x0) α1 i (x, x0) ≦ 0, and C(x,x0)(∇gI(x0))η(x, x0) + ρ2I d2I(x, x0) α2 I(x, x0) < 0. The remaining part of the proof is similar to proof of Theorem 4.1. Theorem 4.4. If there exist a feasible solution x0 for (SP) and, u ∈ Rp and v = (vj)j∈J are vectors which satisfies uT∇f(x0) + vTI ∇gI(x0) = 0, vT g(x0) = 0, u ≥ 0, v ≥ 0 and vj ̸= 0 for finitely many j ∈ I. Also, if f is weak strictly (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is strictly (C,α2, η2, ρ2, d2)- pseudo-invex at x0 with respect to same η(x, x0) (i.e., η 1 = η2 = η) with p∑ i=1 uiρ 1 i d1i (x, x0) α1 i (x, x0) + ∑ j∈I vjρ 2 j d2j (x, x0) α2 j (x, x0) ≧ 0. Then, x0 is an efficient solution of (SP). https://internationalpubls.com 1145 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) Proof. Assume that x0 is not an efficient solution of (SP). Then for a feasible solution x ∈ S, we have fi(x) ≤ fi(x0). Since gI(x0) = 0, hence gI(x) ≦ gI(x0). Since f is weak strictly (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is strictly (C,α2, η2, ρ2, d2)- pseudo-invex at x0, therefore we have C(x,x0)(∇fi(x0))η(x, x0) + ρi 1 d 1 i (x, x0) α1 i (x, x0) < 0, and C(x,x0)(∇gI(x0))η(x, x0) + ρ2I d2I(x, x0) α2 I(x, x0) < 0. The remaining part of the proof is similar to proof of Theorem 4.1. Theorem 4.5. If there exist a feasible solution x0 for (SP) and, u ∈ Rp and v = (vj)j∈J are vectors which satisfies uT∇f(x0) + vTI ∇gI(x0) = 0, vT g(x0) = 0, u ≥ 0, v ≥ 0 and vj ̸= 0 for finitely many j ∈ I. Also, if f is strong (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is strictly (C,α2, η2, ρ2, d2)-pseudo- invex at x0 with respect to same η(x, x0) (i.e., η 1 = η2 = η) with p∑ i=1 uiρ 1 i d1i (x, x0) α1 i (x, x0) + ∑ j∈I vjρ 2 j d2j (x, x0) α2 j (x, x0) ≧ 0. Then, x0 is an efficient solution of (SP). Proof. Assume that x0 is not an efficient solution of (SP). Then for a feasible solution x ∈ S, we have fi(x) ≤ fi(x0). Since gI(x0) = 0, hence gI(x) ≦ gI(x0). Since f is strong (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is strictly (C,α2, η2, ρ2, d2)-pseudo- invex at x0, therefore we have C(x,x0)(∇fi(x0))η(x, x0) + ρ1i d1i (x, x0) α1 i (x, x0) ≤ 0, and C(x,x0)(∇gI(x0))η(x, x0) + ρ2I d2I(x, x0) α2 I(x, x0) < 0. The remaining part of the proof is similar to proof of Theorem 4.1. Since it is clear from the definitions that, an efficient solution is also a weak efficient solution for (SP) but the converse need not be true, therefore Theorem 4.1 - Theorem 4.5 are still valid for weak efficiency. Theorem 4.6. If there exist a feasible solution x0 for (SP) and, u ∈ Rp and v = (vj)j∈J are vectors which satisfies uT∇f(x0) + vTI ∇gI(x0) = 0, vT g(x0) = 0, u ≥ 0, v ≥ 0 and vj ̸= 0 for finitely many j ∈ I. Also, if f is (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is strictly (C,α2, η2, ρ2, d2)-pseudo-invex at x0 with respect to same η(x, x0) (i.e., η 1 = η2 = η) with p∑ i=1 uiρ 1 i d1i (x, x0) α1 i (x, x0) + ∑ j∈I vjρ 2 j d2j (x, x0) α2 j (x, x0) ≧ 0. Then, x0 is a weak efficient solution of (SP). https://internationalpubls.com 1146 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) Proof. Assume that x0 is not a weak efficient solution of (SP). Then for a feasible solution x ∈ S, we have fi(x) < fi(x0). Since gI(x0) = 0, hence gI(x) ≦ gI(x0). Since f is (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is strictly (C,α2, η2, ρ2, d2)-pseudo-invex at x0, therefore we have C(x,x0)(∇fi(x0))η(x, x0) + ρi 1 d 1 i (x, x0) α1 i (x, x0) < 0, and C(x,x0)(∇gI(x0))η(x, x0) + ρ2I d2I(x, x0) α2 I(x, x0) < 0. The remaining part of the proof is similar to proof of Theorem 4.1. Theorem 4.7. If there exist a feasible solution x0 for (SP) and, u ∈ Rp and v = (vj)j∈J are vectors which satisfies uT∇f(x0) + vTI ∇gI(x0) = 0, vT g(x0) = 0, u ≥ 0, v ≥ 0 and vj ̸= 0 for finitely many j ∈ I. Also, if f is (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is (C,α2, η2, ρ2, d2)-quasi-invex at x0 with respect to same η(x, x0) (i.e., η 1 = η2 = η) with p∑ i=1 uiρ 1 i d1i (x, x0) α1 i (x, x0) + ∑ j∈I vjρ 2 j d2j (x, x0) α2 j (x, x0) ≧ 0. Then, x0 is a weak efficient solution of (SP). Proof. Assume that x0 is not a weak efficient solution of (SP). Then for a feasible solution x ∈ S, we have fi(x) < fi(x0). Since gI(x0) = 0, hence gI(x) ≦ gI(x0). Since f is (C,α1, η1, ρ1, d1)-pseudo-invex at x0 and gI is (C,α 2, η2, ρ2, d2)-quasi-invex at x0, therefore we have C(x,x0)(∇fi(x0))η(x, x0) + ρ1i d1i (x, x0) α1 i (x, x0) < 0, and C(x,x0)(∇gI(x0))η(x, x0) + ρ2I d2I(x, x0) α2 I(x, x0) ≦ 0. The remaining part of the proof is similar to proof of Theorem 4.1. 5 Duality We now prove duality relations between nonlinear multiobjective semiinfinite programming problem (SP) and Mond-Weir-type dual problem (MWD): (MWD) Max f(y) = (f1(y), ..., fp(y)), Subject to p∑ i=1 ui∇fi(y) + ∑ j∈J vj∇gj(y) = 0, (5.1) vT g(y) ≧ 0, v = (vj)j∈J , vj ∈ R+ and vj ̸= 0 for finitely many j ∈ J, p∑ i=1 ui = 1, ui > 0 (i = 1, ..., p), y ∈ X ⊆ Rn, where fi and gi are differnetiable functions from X to R, and X is nonempty open subset of Rn. https://internationalpubls.com 1147 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) Theorem 5.1. (Weak Duality) Let x0 and (y0, u, v) be feasible solution for (SP) and (MWD) respectively, and p∑ i=1 uiρ 1 i d1i (x0, y0) α1 i (x0, y0) + ∑ j∈J vjρ 2 j d2j (x0, y0) α2 j (x0, y0) ≧ 0. (5.2) (a) Let f be strong (C,α1, η1, ρ1, d1)-pseudo-invex at y0 and vT g be (C,α2, η2, ρ2, d2)-quasi-invex at y0 with respect to a common kernal η(x, x0) (i.e., η 1 = η2 = η). (b) Let f be weak strictly (C,α1, η1, ρ1, d1)-pseudo-invex at y0 and vT g be (C,α2, η2, ρ2, d2)-quasi- invex at y0 with respect to a common kernal η(x, x0) (i.e., η 1 = η2 = η). (c) Let f be weak (C,α1, η1, ρ1, d1)-quasi-invex at y0 and vT g be strictly (C,α2, η2, ρ2, d2)-pseudo- invex at y0 with respect to a common kernal η(x, x0) (i.e., η 1 = η2 = η). (d) Let f be weak strictly (C,α1, η1, ρ1, d1)-pseudo-invex at y0 and vT g be strictly (C,α2, η2, ρ2, d2)- pseudo-invex at y0 with respect to a common kernal η(x, x0) (i.e., η 1 = η2 = η). (e) Let f be strong (C,α1, η1, ρ1, d1)-pseudo-invex at y0 and vT g be strictly (C,α2, η2, ρ2, d2) - pseudo-invex at y0 with respect to a common kernal η(x, x0) (i.e., η 1 = η2 = η). If any one of the following assumptions will hold, then the following inequality will not hold f(x0) ≤ f(y0). (5.3) Proof. (a) Suppose that the assumption (a) hold. Since x0 and (y0, u, v) are the feasible solution for (SP) and (MWD). Therefore, vjgj(x0) ≦ 0 ≦ vjgj(y0). Also, vT g is (C,α2, η2, ρ2, d2)-quasi-invex at y0, which implies that C(x0,y0)(vj∇gj(y0))η(x0, y0) + ρ2j d2j (x0, y0) α2 j (x0, y0) ≦ 0. (5.4) Let (5.3) holds, then by the definition of strong (C,α1, η1, ρ1, d1)-pseudo-invexity, we have C(x0,y0)(∇fi(y0))η(x0, y0) + ρ1i d1i (x0, y0) α1 i (x0, y0) ≤ 0. (5.5) Let τ = p∑ i=1 ui + ∑ j∈J vj . Then, by the convexity of C(x0,y0)(·) and equations (5.1),(5.2),(5.4),(5.5), we get 0 > p∑ i=1 ui τ C(x0,y0)(∇fi(y0))η(x0, y0) + ∑ j∈J vj τ C(x0,y0)(∇gj(y0))η(x0, y0) + p∑ i=1 ui τ ρ1i d1i (x0, y0) α1 i (x0, y0) + ∑ j∈J vj τ ρ2j d2j (x0, y0) α2 j (x0, y0) ≧ C(x0,y0) 1 τ  p∑ i=1 ui∇fi(y0) + ∑ j∈J vj∇gj(y0)  η(x0, y0) + 1 τ  p∑ i=1 uiρ 1 i d1i (x0, y0) α1 i (x0, y0) + ∑ j∈J vjρ 2 j d2j (x0, y0) α2 j (x0, y0)  ≧ 0. Which gives a contradiction. Thus, the proof of part (a) is complete. Similarly we can prove part (b) - (e). Theorem 5.2. (Strong Duality) Let f and vT g be satisfy any of the five assumptions specified in Theorem 5.1. If x0 ∈ S is an efficient solution for (SP) and (SP) satisfies a suitable constraint qualification (see[26]). Then, ∃ u ∈ Rp + u > 0, v = (vj)(j∈J), vj ∈ R+, such that (x0, u, v) is an efficient solution of (MWD) and the respective objective values of (SP) and (MWD) are equal. https://internationalpubls.com 1148 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) Proof. Since x0 ∈ S is an efficient solution for (SP ) and a suitable constraint qualification is satisfied (see[26]). Therefore by Theorem 4.1, there exist u ∈ R+ p u > 0, v = (vj)(j∈J), vj ∈ R+, such that (x0, u, v) is a feasible solution for (MWD). Suppose that (x0, u, v) is not an efficient solution for (MWD), this implies there exist a feasible solution (y0, u0, v0) for (MWD), s.t. (f1(x0), ..., fp(x0)) ≤ (f1(y0), ..., fp(y0)), which contradicts Theorem 5.1. Hence, the proof is complete. Theorem 5.3. (Strict Converse Duality) Let the assumptions of Theorem 5.2 be satisfied and f be strictly (C,α1, η1, ρ1, d1)-pseudo-invex at y0. If x0 and (y0, u0, v0) are feasible solutions for (SP) and (MWD) respectively, then x0 = y0. Proof. Suppose that x0 ̸= y0. By strong duality theorem ∃ u ∈ Rp +, u > 0, v = (vj)(j∈J), vj ∈ R+, such that (x0, u, v) is an efficient solution for (MWD). Hence, f(x0) = f(y0). (5.6) Since x0 and (y0, u0, v0) are the feasible solution for (SP) and (MWD), this implies that v0jgj(x0) ≦ 0 ≦ v0jgj(y0). (5.7) By definition of (C,α2, η2, ρ2, d2)-quasi-invexity, we have C(x0,y0)(v0j∇gj(y0))η(x0, y0) + ρ2j d2j (x0, y0) α2 j (x0, y0) ≦ 0, (η2 = η). (5.8) Again, by the assumption on fi(i = 1, 2, ..., p), we have C(x0,y0)(∇fi(y0))η(x0, y0) + ρ1i d1i (x0, y0) α1 i (x0, y0) < 0, (η1 = η). (5.9) Let us denote τ = p∑ i=1 u0i + ∑ j∈I v0j . Therefore from equations (5.8)-(5.9) and the convexity of C(x0,y0)(·), we get 0 > p∑ i=1 u0i τ C(x0,y0)(∇fi(y0))η(x0, y0) + ∑ j∈J v0j τ C(x0,y0)(∇gj(y0))η(x0, y0) + p∑ i=1 u0i τ ρ1i d1i (x0, y0) α1 i (x0, y0) + ∑ j∈J v0j τ ρ2j d2j (x0, y0) α2 j (x0, y0) ≧ C(x0,y0) 1 τ  p∑ i=1 u0i∇fi(y0) + ∑ j∈J v0j∇gj(y0)  η(x0, y0) + 1 τ  p∑ i=1 u0iρ 1 i d1i (x0, y0) α1 i (x0, y0) + ∑ j∈J v0jρ 2 j d2j (x0, y0) α2 j (x0, y0)  ≧ 0. Which gives a contradiction, i.e., our assumption is wrong. Therefore x0 = y0. 6 conclusion In this paper, we formulted generalized (C,α, η, ρ, d) invexities, which are the generalized version of (C,α, η, ρ, d) convexity. Using these definitions, we have stablished several optimality and duality results for multiobjective semiinfinite programming problems. The results of this paper are more general than the corresponding results present in the literature [19]. 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