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

Abstract

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.

1. Introduction

In this paper, we consider the following nondifferentiable nonconvex multiobjective fractional programming problem: (MFP)Minimize𝑓₁(𝑥)𝑔₁(𝑥),𝑓₂(𝑥)𝑔₂(𝑥)𝑓,…,^𝑚(𝑥)𝑔_𝑚(𝑥),subjectto𝑥∈𝑆=𝑥∈ℝ^𝑛∣ℎ_𝑗(𝑥)≤0,for1≤𝑗≤𝑝, (1) where (a1)𝑓_𝑖∶𝑋₀↦ℝ, 𝑔_𝑖∶𝑋₀↦ℝ, 𝑖=1,2,…,𝑚 and ℎ_𝑗∶𝑋₀↦ℝ, 𝑗=1,2,…,𝑝 are Lipschitz on 𝑋₀, and 𝑋₀ is an open subset of ℝ^𝑛; (a2)𝑓_𝑖(𝑥)≥0, 𝑔_𝑖(𝑥)>0, 𝑖=1,2,…,𝑚.Minimize means obtaining efficient solution in the following sense. A point 𝑥∈𝑆 is said to be an efficient solution for (MFP) if there is no 𝑥∈𝑆 such that 𝑓_𝑖(𝑥)𝑔_𝑖(𝑥)≤𝑓_𝑖(𝑥)𝑔_𝑖(𝑥),1≤𝑖≤𝑚, (2) with at least one strict inequality.

A point 𝑥∈𝑆 is said to be a  properly efficient solution for (MFP) which was introduced by Geoffrion [1] if and only if (a)𝑥 is an efficient solution for (MFP), (b) there exists a scalar 𝑀>0 such that for each 𝑖, we have 𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)−𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)≤𝑀, (3) for some 𝑗 such that 𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)<𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥), whenever 𝑥∈𝑆 and 𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)>𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥).

An efficient point for (MFP) that is not properly efficient is said to be improperly efficient. Thus, for 𝑥 to be improperly efficient for (MFP) means that to every scalar 𝑀>0, there is a point 𝑥∈𝑆 and an 𝑖 such that 𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)>𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥) and 𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)−𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)>𝑀, (4) for all 𝑗 such that 𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)<𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥).

Many papers have been devoted to the multiobjective fractional programming problem in recent decades; see for example [1–13].

In [8], Preda introduced generalized (ℱ,𝜌)-convexity, an extension of ℱ-convexity and generalized 𝜌-convexity defined by Vial [14, 15]. Bhatia and Jain [2] defined generalized (ℱ,𝜌)-convexity for nonsmooth functions, an extension of generalized (ℱ,𝜌)-convexity defined by Preda [8], and they derived some duality theorems for nonsmooth multiobjective programs. In [5, 6], Liu also established the Kuhn-Tucker type necessary and sufficient optimality conditions for multiobjective fractional programming involving nonsmooth pseudoinvex functions in [5] or (ℱ,𝜌)-convex functions in [6] and considered the parameter dual problem in the framework of generalized convex functions.

Recently, Zalmai [13, 16, 17] introduced generalized (ℱ,𝑏,𝜙,𝜌,𝜃)-univex 𝑛-set functions and he also established sufficient efficiency conditions in multiobjective fractional subset programming [13] and sufficient optimality conditions in minimax fractional subset programming [16, 17] under various generalized (ℱ,𝑏,𝜙,𝜌,𝜃)-univexity assumptions. In [10], Preda et al. obtained duality results for a dual model of Zalmai [13] replacing the assumptions of sublinearity or convexity by that of quasiconvexity in the third argument.

In this paper, we are inspired to consider the optimality and duality of properly efficient for (MFP) containing nonsmooth generalized (ℱ,𝑏,𝜙,𝜌,𝜃)-univex functions. We organize this paper as follows. Some definitions and notations are given in Section 2. In Section 3, we establish some properly efficient sufficient optimality conditions for (MFP) involving nonsmooth generalized (ℱ,𝑏,𝜙,𝜌,𝜃)-univex functions. We also derive some duality results in Section 4.

2. Notations and Preliminary

Throughout the paper, let ℝ^𝑛 be the 𝑛-dimensional Euclidean space and let ℝ^𝑛₊ be its nonnegative orthant.

Definition 1. The function 𝑓∶𝑋₀↦ℝ is said to be Lipschitz on 𝑋₀ if there exists 𝑐>0 such that for all 𝑦,𝑥∈𝑋₀, 𝑓𝑦−𝑓(𝑥)≤𝑐𝑦−𝑥, (5) where ‖⋅‖ denotes any norm in ℝ^𝑛.

For each 𝑑 in ℝ^𝑛, 𝑓^∘(𝑥;𝑑) is the generalized directional derivative of Clarke [18] defined by 𝑓^∘(𝑥;𝑑)=limsup_{𝑦→𝑥𝑡↓0}𝑓𝑦+𝑡𝑑−𝑓𝑦𝑡. (6) It then follows that 𝑓^∘(𝑥;𝑑)=max𝜉^⊤𝑑∣𝜉∈𝜕𝑓(𝑥)forany𝑥and𝑑. (7) where 𝜕𝑓(⋅) denotes Clarke’s generalized gradient [18].

Definition 2. A functional ℱ∶ℝ^𝑛↦ℝ is said to be sublinear (superlinear) if ℱ(𝑎₁+𝑎₂)≤(≥)ℱ(𝑎₁)+ℱ(𝑎₂) for all 𝑎₁,𝑎₂∈ℝ^𝑛, and ℱ(𝛼𝑎)=𝛼ℱ(𝑎) for all 𝛼∈ℝ₊ and 𝑎∈ℝ^𝑛.

Let 𝑋 be an open subset of ℝ^𝑛. We suppose that the function 𝑓∶𝑋↦ℝ is Lipschitz on 𝑋 and 𝑑(⋅,⋅)∶𝑋×𝑋↦ℝ such that 𝑑(𝑥,𝑥)=0 for any 𝑥.

Definition 3 (cf. [13, 16, 17, 19, 20]). (1) Let ℱ(𝑥,𝑥^∗;⋅)∶ℝ^𝑛↦ℝ be a sublinear function, 𝑏∶𝑋×𝑋↦ℝ a function with positive values, 𝜙∶ℝ↦ℝ a function, 𝜌 a real number, and 𝜃∶𝑋×𝑋↦𝑋×𝑋 a function such that 𝑥≠𝑥^∗⇒𝜃(𝑥,𝑥^∗)≠(0,0). The function 𝑓 is said to be (ℱ,𝑏,𝜙,𝜌,𝜃)-univex at 𝑥^∗ if 𝜙𝑓(𝑥)−𝑓𝑥^∗≥ℱ𝑥,𝑥^∗;𝑏𝑥,𝑥^∗𝜉+𝜌𝑑²𝜃𝑥,𝑥^∗ (8) for all 𝑥∈𝑋 and 𝜉∈𝜕𝑓(𝑥^∗).
(2) The function 𝑓 is said to be (ℱ,𝑏,𝜙,𝜌,𝜃)-pseudounivex at 𝑥^∗ if ℱ𝑥,𝑥^∗;𝑏𝑥,𝑥^∗𝜉≥−𝜌𝑑²𝜃𝑥,𝑥^∗⟹𝜙𝑓(𝑥)−𝑓𝑥^∗≥0 (9)  for all 𝑥∈𝑋 and 𝜉∈𝜕𝑓(𝑥^∗).
(3) The function 𝑓 is said to be (ℱ,𝑏,𝜙,𝜌,𝜃)-strictly-pseudounivex at 𝑥^∗ if ℱ𝑥,𝑥^∗;𝑏𝑥,𝑥^∗𝜉≥−𝜌𝑑²𝜃𝑥,𝑥^∗⟹𝜙𝑓(𝑥)−𝑓𝑥^∗>0 (10)  for all 𝑥∈𝑋 and 𝜉∈𝜕𝑓(𝑥^∗).
(4) The function 𝑓 is said to be (ℱ,𝑏,𝜙,𝜌,𝜃)-prestrictly-pseudounivex at 𝑥^∗ if ℱ𝑥,𝑥^∗;𝑏𝑥,𝑥^∗𝜉>−𝜌𝑑²𝜃𝑥,𝑥^∗⟹𝜙𝑓(𝑥)−𝑓𝑥^∗≥0 (11)  for all 𝑥∈𝑋 and 𝜉∈𝜕𝑓(𝑥^∗).
(5) The function 𝑓 is said to be (ℱ,𝑏,𝜙,𝜌,𝜃)-quasiunivex at 𝑥^∗ if 𝜙𝑓(𝑥)−𝑓𝑥^∗≤0⟹ℱ𝑥,𝑥^∗;𝑏𝑥,𝑥^∗𝜉≤−𝜌𝑑²𝜃𝑥,𝑥^∗ (12)  for all 𝑥∈𝑋 and 𝜉∈𝜕𝑓(𝑥^∗).
(6) The function 𝑓 is said to be (ℱ,𝑏,𝜙,𝜌,𝜃)-strictly-quasiunivex at 𝑥^∗ if 𝜙𝑓(𝑥)−𝑓𝑥^∗≤0⟹ℱ𝑥,𝑥^∗;𝑏𝑥,𝑥^∗𝜉<−𝜌𝑑²𝜃𝑥,𝑥^∗ (13)  for all 𝑥∈𝑋 and 𝜉∈𝜕𝑓(𝑥^∗).
(7) The function 𝑓 is said to be (ℱ,𝑏,𝜙,𝜌,𝜃)-prestrictly-quasiunivex at 𝑥^∗ if 𝜙𝑓(𝑥)−𝑓𝑥^∗<0⟹ℱ𝑥,𝑥^∗;𝑏𝑥,𝑥^∗𝜉≤−𝜌𝑑²𝜃𝑥,𝑥^∗ (14)  for all 𝑥∈𝑋 and 𝜉∈𝜕𝑓(𝑥^∗).

In order to establish some Kuhn-Tucker conditions for efficient optimality of (MFP), we give the scalar minimization problem as follows: (SFP)_𝑖Minimize𝑓_𝑖(𝑥)𝑔_𝑖(𝑥),subjectto𝑓_𝑘(𝑥)𝑔_𝑘(𝑥)≤𝑓_𝑘(𝑥)𝑔_𝑘(𝑥)ℎ,𝑘≠𝑖,_𝑗(𝑥)≤0,𝑗=1,2,…,𝑝. (15)

In [6], we established the following necessary efficient optimality conditions for (MFP).

Lemma 4. Let 𝑥∈𝑆. If   𝑥 is an efficient solution for (MFP) and satisfies an appropriate constraint qualification [21] for (𝑆𝐹𝑃)_𝑖, 𝑖=1,2,…,𝑚. Then there exist 𝑣_𝑖=𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)≥0, 𝑦_𝑖>0, 1≤𝑖≤𝑚, and 𝛽∈ℝ^𝑝₊ such that 0∈^𝑚_𝑖=1𝑦_𝑖𝑇_𝑖(𝑥)+^𝑝_𝑗=1𝛽_𝑗𝜕ℎ_𝑗(𝑥),𝛽_𝑗ℎ_𝑗(𝑥)=0,1≤𝑗≤𝑝, (16) where 𝑇_𝑖(𝑥)=𝜕𝑓_𝑖(𝑥)+𝑣_𝑖𝜕−𝑔_𝑖(𝑥)𝑔_𝑖(𝑥),1≤𝑖≤𝑚. (17)

If we choose 𝜆_𝑖=𝑦_𝑖/𝑔_𝑖(𝑥)>0, the result of Lemma 4 can be restated as follows.

Theorem 5 (necessary optimality conditions). Let 𝑥∈𝑆. If 𝑥 is an (properly) efficient solution for (MFP) and satisfies an appropriate constraint qualification [21] for (𝑆𝐹𝑃)_𝑖, 𝑖=1,2,…,𝑚. Then there exist 𝜆_𝑖>0, 1≤𝑖≤𝑚, 𝑣∈ℝ^𝑚₊, and 𝛽∈ℝ^𝑝₊ such that 0∈^𝑚_𝑖=1𝜆_𝑖𝜕𝑓_𝑖(𝑥)+𝑣_𝑖𝜕−𝑔_𝑖(𝑥)+^𝑝_𝑗=1𝛽_𝑗𝜕ℎ_𝑗(𝑥),𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)=0,1≤𝑖≤𝑚,𝛽_𝑗ℎ_𝑗(𝑥)=0,1≤𝑗≤𝑝. (18)(19)(20)

In order to simplify the complication of (MFP), we consider the following nonfractional multiobjective programming problem with a parameter 𝑣∈ℝ^𝑚₊: (MP)_𝑣minimize𝑓₁(𝑥)−𝑣₁𝑔₁(𝑥),𝑓₂(𝑥)−𝑣₂𝑔₂(𝑥)𝑓,…,_𝑚(𝑥)−𝑣_𝑚𝑔_𝑚(𝑥)subjectto𝑥∈𝑆. (21) We obtained the equivalence between (MFP) and (MP)_𝑣 as follows.

Lemma 6. Let 𝑥∈𝑆 and 𝑣_𝑖=𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥),1≤𝑖≤𝑚. Then 𝑥 is a properly efficient solution for (MFP) if and only of 𝑥 is a properly efficient solution for (𝑀𝑃)_𝑣.

Proof. (a) Let 𝑥 be an efficient solution for (MP)_𝑣. Assume that 𝑥 is not an efficient solution for (MFP). Then there exists 𝑥∈𝑆 such that 𝑓_𝑖(𝑥)𝑔_𝑖(𝑥)≤𝑓_𝑖(𝑥)𝑔_𝑖(𝑥),1≤𝑖≤𝑚, (22) with at least one strict inequality. Thus, we have 𝑓_𝑖(𝑥)−𝑓_𝑖(𝑥)𝑔_𝑖(𝑥)𝑔_𝑖(𝑥)≤0=𝑓_𝑖(𝑥)−𝑓_𝑖(𝑥)𝑔_𝑖(𝑥)𝑔_𝑖(𝑥),1≤𝑖≤𝑚, (23) or equivalently, 𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)≤𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥),1≤𝑖≤𝑚, (24) with at least one strict inequality which contradicts the efficient solution of 𝑥 for (MP)_𝑣. Therefore, we have 𝑥 as an efficient solution for (MFP). Conversely, if 𝑥 is an efficient solution for   (MFP), along with the same lines of above, we can obtain that 𝑥 is an efficient solution for   (MP)_𝑣.
(b) Let 𝑥 be a properly efficient solution for   (MFP). Assume that 𝑥 is not a properly efficient solution for   (MP)_𝑣. Then every scalar 𝑀>0, there is a point 𝑥∈𝑆 and an 𝑖 such that 𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)>𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥) and 𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥)−𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥𝑔)>𝑀^𝑖(𝑥)𝑔_𝑗(𝑥), (25) for all 𝑗 such that 𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥)<𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥).
It follows that for some 𝑖 such that 0>𝑓_𝑖(𝑥)−𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)𝑔_𝑖(𝑥) and 𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)𝑓_𝑗(𝑥)−𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)𝑔_𝑗(𝑥)𝑔>𝑀^𝑖(𝑥)𝑔_𝑗(𝑥), (26) for all 𝑗 such that 0<𝑓_𝑗(𝑥)−𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)𝑔_𝑗(𝑥), or equivalently, for some 𝑖 such that 𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)>𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥) and 𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥)𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)−𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)>𝑀, (27) for all 𝑗 such that 𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥)<𝑓_𝑗(𝑥)/𝑔_𝑗(𝑥), which contradicts the properly efficient solution of 𝑥 for (MFP). Therefore, we have 𝑥 is a properly efficient solution for   (MP)_𝑣. Conversely, if 𝑥 is a properly efficient solution for   (MP)_𝑣, along with the same lines of above, we can obtain that 𝑥 is a properly efficient solution for (MFP). Thus, the proof is complete.

Lemma 7. Let 𝑥∈𝑆 and 𝑣_𝑖=𝑓_𝑖(𝑥)/𝑔_𝑖(𝑥), 1≤𝑖≤𝑚. If there exists 𝜆∈ℝ^𝑚₊⧵{0} such that ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)≥0,𝑓𝑜𝑟𝑎𝑛𝑦𝑥∈𝑆, (28) then 𝑥 is a properly efficient solution to (MFP).

Proof. Assume that 𝑥 is not an efficient for (MFP). Then there exists a point 𝑥∈𝑆 such that 𝑓_𝑖(𝑥)𝑔_𝑖(𝑥)≤𝑓_𝑖(𝑥)𝑔_𝑖(𝑥),1≤𝑖≤𝑚, (29) with at least one strict inequality. Thus, we have 𝑓_𝑖(𝑥)−𝑓_𝑖(𝑥)𝑔_𝑖(𝑥)𝑔_𝑖(𝑥)≤0,1≤𝑖≤𝑚, (30) or equivalently, 𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)≤0,1≤𝑖≤𝑚, (31) with at least one strict inequality.
Thus, for any 𝜆∈ℝ^𝑚₊⧵{0}, we have ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)<0, (32) which contradicts the inequality (28).
It remains to show that 𝑥 is properly efficient. Assume that it is an improperly efficient point for the problem (MFP). Then, by Lemma 6, 𝑥 is an improperly efficient solution for  (MP)_𝑣. Then every scalar 𝑀>0, there is a point 𝑥∈𝑆 and an 𝑖 such that 𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)>𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥) and 𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥)−𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥)>𝑀, (33) for all 𝑗 such that 𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥)<𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥).
For any 𝜆∈ℝ^𝑚₊⧵{0}, we choose 𝑀=(𝑚−1)max_𝑖,𝑗{𝜆_𝑗/𝜆_𝑖}>0, (we may assume that 𝑚≥2) such that 𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥>()𝜆)𝑚−1^𝑗𝜆_𝑖𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥)−𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥), (34) for all 𝑗≠𝑖. These imply that 𝜆_𝑖𝑚−1𝑣_𝑖𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)>𝜆_𝑗𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥),∀𝑗≠𝑖. (35) Summing over 𝑗≠𝑖 yield 𝜆_𝑖𝑣_𝑖𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)>_𝑗≠𝑖𝜆_𝑗𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥), (36) or ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)<0, (37) which contradicts the inequality (28). Thus, the proof is complete.

3. Sufficient Optimality Conditions

Applying Lemmas 6 and 7, we can establish the following sufficient optimality conditions for properly efficient solution of (MFP).

Let 𝑃=1,2,…,𝑝,𝑃(𝑥)=𝑗∈𝑃∣ℎ_𝑗(𝑥)=0. (38)

Theorem 8 (sufficient optimality conditions). Let 𝑥∈𝑆, and let there exist 𝜆_𝑖>0, 1≤𝑖≤𝑚, 𝑣∈ℝ^𝑚₊, and 𝛽∈ℝ^𝑝₊ such that (18), (19), and (20). Let 𝐴(𝑥)=^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖(𝑥)+^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥),𝐵(𝑥)=^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖𝑥,𝐶(𝑥)=()_𝑝𝑗=1𝛽_𝑗ℎ_𝑗(𝑥). (39) If any one of the following conditions holds: (a)(i)𝑓_𝑖 is (ℱ,𝑏,𝜙₁,𝜌_1𝑖,𝜃)-univex at 𝑥, −𝑔_𝑖 is (ℱ,𝑏,𝜙₁,𝜌_2𝑖,𝜃)-univex at 𝑥, 1≤𝑖≤𝑚,𝜙₁ is superlinear, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii) for each 𝑗∈𝑃(𝑥),   ℎ_𝑗 is (ℱ,𝑏,𝜙_2𝑗,𝜌_3𝑗,𝜃)-quasiunivex at 𝑥,   𝜙_2𝑗 is increasing, and 𝜙_2𝑗(0)=0; (iii)∑^𝑚_𝑖=1𝜆_𝑖𝜌_1𝑖+∑^𝑚_𝑖=1𝜆_𝑖𝑣_𝑖𝜌_2𝑖+∑_{𝑗∈𝑃(𝑥)}𝛽_𝑗𝜌_3𝑗≥0; (b)(i)𝑓_𝑖 is (ℱ,𝑏,𝜙₁,𝜌_1𝑖,𝜃)-univex at 𝑥, −𝑔_𝑖 is (ℱ,𝑏,𝜙₁,𝜌_2𝑖,𝜃)-univex at 𝑥, 1≤𝑖≤𝑚, 𝜙₁ is superlinear, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii)𝐶 is (ℱ,𝑏,𝜙₂,𝜌₃,𝜃)-quasiunivex at 𝑥,    𝜙₂ is increasing, and 𝜙₂(0)=0; (iii)∑^𝑚_𝑖=1𝜆_𝑖𝜌_1𝑖+∑^𝑚_𝑖=1𝜆_𝑖𝑣_𝑖𝜌_2𝑖+𝜌₃≥0; (c)(i)𝐴 is (ℱ,𝑏,𝜙,0,𝜃)-pseudounivex at 𝑥 and 𝜙(𝑎)≥0⇒𝑎≥0; (d)(i)𝐵 is (ℱ,𝑏,𝜙₁,𝜌₁,𝜃)-pseudounivex at 𝑥, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii)𝐶 is (ℱ,𝑏,𝜙₂,𝜌₂,𝜃)-quasiunivex at 𝑥, 𝜙₂ is increasing, and 𝜙₂(0)=0; (iii)𝜌₁+𝜌₂≥0; (e)(i)𝐵 is (ℱ,𝑏,𝜙₁,𝜌₁,𝜃)-prestrictly-quasiunivex at 𝑥, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii)𝐶 is (ℱ,𝑏,𝜙₂,𝜌₂,𝜃)-quasiunivex at 𝑥, 𝜙₂ is increasing, and 𝜙₂(0)=0; (iii)𝜌₁+𝜌₂>0; (f)(i)𝐵 is (ℱ,𝑏,𝜙₁,𝜌₁,𝜃)-prestrictly-quasiunivex at 𝑥, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii)𝐶 is (ℱ,𝑏,𝜙₂,𝜌₂,𝜃)-strictly-pseudounivex at 𝑥, 𝜙₂ is increasing, and 𝜙₂(0)=0; (iii)𝜌₁+𝜌₂≥0; (g)(i)𝐵 is (ℱ,𝑏,𝜙₁,𝜌₁,𝜃)-pseudounivex at 𝑥, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii) for each 𝑗∈𝑃(𝑥), ℎ_𝑗 is (ℱ,𝑏,𝜙_2𝑗,𝜌_2𝑗,𝜃)-quasiunivex at 𝑥, 𝜙_2𝑗 is increasing, and 𝜙_2𝑗(0)=0; (iii)𝜌₁+∑_{𝑗∈𝑃(𝑥)}𝛽_𝑗𝜌_2𝑗≥0.

Then 𝑥 is a properly efficient solution for (MFP).

Proof. Let 𝑥 be any feasible solution of (MFP). By (18), there exist 𝜉_𝑖∈𝜕𝑓_𝑖(𝑥), 𝜍_𝑖∈𝜕(−𝑔_𝑖)(𝑥), for 1≤𝑖≤𝑚, and 𝜁_𝑗∈𝜕ℎ_𝑗(𝑥), for 1≤𝑗≤𝑝 such that ^𝑚_𝑖=1𝜆_𝑖𝜉_𝑖+𝑣_𝑖𝜍_𝑖+^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗=0. (40) From here it results ℱ𝑥,𝑥;^𝑚_𝑖=1𝜆_𝑖𝜉_𝑖+𝑣_𝑖𝜍_𝑖+^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≥0. (41) If hypothesis (a) holds, 𝜙₁𝑓_𝑖(𝑥)−𝑓_𝑖(𝑥)≥ℱ𝑥,𝑥;𝑏(𝑥,𝑥)𝜉_𝑖+𝜌_1𝑖𝑑²(𝜃(𝑥,𝑥)),1≤𝑖≤𝑚,𝜙₁−𝑔_𝑖(𝑥)+𝑔_𝑖(𝑥)≥ℱ𝑥,𝑥;𝑏(𝑥,𝑥)𝜍_𝑖+𝜌_2𝑖𝑑²(𝜃(𝑥,𝑥)),1≤𝑖≤𝑚. (42)(43) Now, multiplying (42) by 𝜆_𝑖, (43) by 𝜆_𝑖𝑣_𝑖, and adding the resulting inequalities, and then using the superlinearity of 𝜙₁ and sublinearity of ℱ(𝑥,𝑥;⋅), we obtain 𝜙₁^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)≥ℱ𝑥,𝑥;𝑏(𝑥,𝑥)^𝑚_𝑖=1𝜆_𝑖𝜉_𝑖+𝑣_𝑖𝜍_𝑖+^𝑚_𝑖=1𝜆_𝑖𝜌_1𝑖+^𝑚_𝑖=1𝜆_𝑖𝑣_𝑖𝜌_2𝑖𝑑²(𝜃(𝑥,𝑥)). (44) Since 𝑥∈𝑆, it is clear that ℎ_𝑗(𝑥)−ℎ_𝑗(𝑥)≤0,∀𝑗∈𝑃(𝑥). (45) Using the increasing property of 𝜙_2𝑗 and 𝜙_2𝑗(0)=0, we have 𝜙_2𝑗ℎ_𝑗(𝑥)−ℎ_𝑗(𝑥)≤0,∀𝑗∈𝑃(𝑥). (46) Using the (ℱ,𝑏,𝜙_2𝑗,𝜌_3𝑗,𝜃)-quasiunivexity of ℎ_𝑗 at 𝑥, we get from (46) ℱ𝑥,𝑥;𝑏(𝑥,𝑥)𝜁_𝑗≤−𝜌_3𝑗𝑑²(𝜃(𝑥,𝑥)),∀𝑗∈𝑃(𝑥). (47) Because 𝛽∈ℝ^𝑝₊ and (20) holds, 𝛽_𝑗=0 for each 𝑗∈𝑃⧵𝑃(𝑥), and ℱ(𝑥,𝑥;⋅) is sublinear, the above inequality yields ℱ𝑥,𝑥;𝑏(𝑥,𝑥)^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≤−_{(𝑗∈𝑃𝑥)}𝛽_𝑗𝜌_3𝑗𝑑²(𝜃(𝑥,𝑥)). (48) From the nonnegativity of 𝑏(𝑥,𝑥), sublinearity of ℱ(𝑥,𝑥;⋅), and (41), it follows that ℱ𝑥,𝑥;𝑏(𝑥,𝑥)^𝑚_𝑖=1𝜆_𝑖𝜉_𝑖+𝑣_𝑖𝜍_𝑖+ℱ𝑥,𝑥;𝑏(𝑥,𝑥)^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≥0. (49) Now adding (44) and (48), and then using (49) and ∑^𝑚_𝑖=1𝜆_𝑖𝜌_1𝑖+∑^𝑚_𝑖=1𝜆_𝑖𝑣_𝑖𝜌_2𝑖+∑_{𝑗∈𝑃(𝑥)}𝛽_𝑗𝜌_3𝑗≥0, we obtain 𝜙₁^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)≥0. (50) But 𝜙₁(𝑎)≥0⇒𝑎≥0, and so (50) yields ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)≥0. (51) Thus, we have ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖(𝑥)≥^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖(𝑥)=0. (52) Therefore, by Lemma 7, 𝑥 is a properly efficient solution for (MFP).
If hypothesis (b) holds, from (20), 𝑥∈𝑆, and 𝛽∈ℝ^𝑝₊, we have ^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)≤0=^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥). (53) and so using the properties of 𝜙₂, we obtain 𝜙₂^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)−ℎ_𝑗(𝑥)≤0. (54) Using the (ℱ,𝑏,𝜙₂,𝜌₃,𝜃)-quasiunivexity of 𝐶 at 𝑥, we get from (54) ℱ𝑥,𝑥;𝑏(𝑥,𝑥)^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≤−𝜌₃𝑑²(𝜃(𝑥,𝑥)). (55) Now combining (44), (49), and (55), and using (iii), we obtain (52). Therefore, by Lemma 7, 𝑥 is a properly efficient solution for (MFP).
If hypothesis (c) holds, using the (ℱ,𝑏,𝜙,0,𝜃)-pseudounivexity of 𝐴, we get from (41) 𝜙^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)+^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)+^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)≥0. (56) In view of the properties of 𝜙, we deduce from this inequality that ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)+^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)+^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)≥0. (57) Equation (53) along with (57) yields ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖(𝑥)≥^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖(𝑥)=0. (58) Therefore, by Lemma 7, 𝑥 is a properly efficient solution for (MFP).
If hypothesis (d) holds, using the properties of 𝜙₂ and (53), we obtain 𝜙₂^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)−ℎ_𝑗(𝑥)≤0, (59) which in view of (ii) implies that ℱ𝑥,𝑥;𝑏(𝑥,𝑥)^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≤−𝜌₂𝑑²(𝜃(𝑥,𝑥)). (60) Now combining (49), (60), and 𝜌₁+𝜌₂≥0, we obtain ℱ𝑥,𝑥;𝑏(𝑥,𝑥)^𝑚_𝑖=1𝜆_𝑖𝜉_𝑖+𝑣_𝑖𝜍_𝑖≥−ℱ𝑥,𝑥;𝑏(𝑥,𝑥)^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≥𝜌₂𝑑²(𝜃(𝑥,𝑥))≥−𝜌₁𝑑²(𝜃(𝑥,𝑥)). (61) Along with the fact that the 𝐵 is (ℱ,𝑏,𝜙₁,𝜌₁,𝜃)-pseudounivex at 𝑥, we get from (61) that 𝜙₁^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)≥0. (62) In view of the properties of 𝜙₁, we deduce from this inequality that ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖(𝑥)≥^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖(𝑥)=0. (63) Therefore, by Lemma 7, 𝑥 is a properly efficient solution for (MFP). (e), (f), and (g) follows along with the same lines of (d) and (a). Thus, the proof is complete.

4. Duality Theorem

From the optimality of properly efficient for the problem (MFP), we can formulate the following parametric dual problem: (𝐷)maximize𝑣=𝑣₁,𝑣₂,…,𝑣_𝑚subjectto0∈^𝑚_𝑖=1𝜆_𝑖𝜕𝑓_𝑖(𝑢)+𝑣_𝑖𝜕−𝑔_𝑖(𝑢+_𝑝𝑗=1)𝛽_𝑗𝜕ℎ_𝑗(𝑢),𝑓_𝑖(𝑢)−𝑣_𝑖𝑔_𝑖(𝑢)≥0,1≤𝑖≤𝑚,𝛽_𝑗ℎ_𝑗(𝑢)≥0,1≤𝑗≤𝑝,𝑢∈ℝ^𝑛,𝑣∈ℝ^𝑚₊,𝛽∈ℝ^𝑝₊,𝜆_𝑖>0,1≤𝑖≤𝑚. (64)(65)(66)(67) We denote by 𝐹_𝐷 the set of all feasible solutions (𝑢,𝑣,𝜆,𝛽) of problem (𝐷). The point (𝑢,𝑣,𝜆,𝛽)∈𝐹_𝐷 is called an efficient for (𝐷) if there is no (𝑢,𝑣,𝜆,𝛽)∈𝐹_𝐷 such that 𝑣_𝑖≥𝑣_𝑖,1≤𝑖≤𝑚, (68) with at least one strict inequality.

The point (𝑢,𝑣,𝜆,𝛽)∈𝐹_𝐷 is said to be a properly efficient for (𝐷) if and only if (a)(𝑢,𝑣,𝜆,𝛽) is an efficient solution for (𝐷), (b) there exists a scalar 𝑀>0 such that for each 𝑖, we have 𝑣_𝑖−𝑣_𝑖𝑣_𝑗−𝑣_𝑗≤𝑀, (69) for some 𝑗 such that 𝑣_𝑗<𝑣_𝑗, whenever (𝑢,𝑣,𝜆,𝛽)∈𝐹_𝐷 and 𝑣_𝑖>𝑣_𝑖.

If (𝑢,𝑣,𝜆,𝛽) was not properly efficient solution for (𝐷), for each fixed real number 𝑀>0, there would exist a feasible solution (𝑢,𝑣,𝜆,𝛽)∈𝐹_𝐷 and an index 𝑖 such that 𝑣_𝑖>𝑣_𝑖 and 𝑣_𝑖−𝑣_𝑖𝑣_𝑗−𝑣_𝑗>𝑀, (70) for all 𝑗 such that 𝑣_𝑗<𝑣_𝑗.

Theorem 9 (duality theorem). Let 𝑥∈𝑆, (𝑢,𝑣,𝜆,𝛽)∈𝐹_𝐷, and let 𝐴(⋅)=^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(⋅)+𝑣_𝑖−𝑔_𝑖(⋅)+^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(⋅),𝐵(⋅)=^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(⋅)+𝑣_𝑖−𝑔_𝑖⋅,𝐶(⋅)=()_𝑝𝑗=1𝛽_𝑗ℎ_𝑗(⋅). (71)

If any one of the following conditions holds: (a)(i)𝑓_𝑖 is (ℱ,𝑏,𝜙₁,𝜌_1𝑖,𝜃)-univex at 𝑢, −𝑔_𝑖 is (ℱ,𝑏,𝜙₁,𝜌_2𝑖,𝜃)-univex at 𝑢, 1≤𝑖≤𝑚, 𝜙₁ is superlinear, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii)𝛽_𝑗ℎ_𝑗 is (ℱ,𝑏,𝜙_2𝑗,𝜌_3𝑗,𝜃)-quasiunivex at 𝑢,1≤𝑗≤𝑝, 𝜙_2𝑗 is increasing, and 𝜙_2𝑗(0)=0; (iii)∑^𝑚_𝑖=1𝜆_𝑖𝜌_1𝑖+∑^𝑚_𝑖=1𝜆_𝑖𝑣_𝑖𝜌_2𝑖+∑^𝑝_𝑗=1𝜌_3𝑗≥0; (b)(i)𝑓_𝑖 is (ℱ,𝑏,𝜙₁,𝜌_1𝑖,𝜃)-univex at 𝑢, −𝑔_𝑖 is (ℱ,𝑏,𝜙₁,𝜌_2𝑖,𝜃)-univex at 𝑢, 1≤𝑖≤𝑚, 𝜙₁ is superlinear, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii)𝐶 is (ℱ,𝑏,𝜙₂,𝜌₃,𝜃)-quasiunivex at 𝑢,𝜙₂ is increasing, and 𝜙₂(0)=0; (iii)∑^𝑚_𝑖=1𝜆_𝑖𝜌_1𝑖+∑^𝑚_𝑖=1𝜆_𝑖𝑣_𝑖𝜌_2𝑖+𝜌₃≥0; (c)(i)𝐴 is (ℱ,𝑏,𝜙,0,𝜃)-pseudounivex at 𝑢 and 𝜙(𝑎)≥0⇒𝑎≥0. (d)(i)𝐵 is (ℱ,𝑏,𝜙₁,𝜌₁,𝜃)-pseudounivex at 𝑢, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii)𝐶 is (ℱ,𝑏,𝜙₂,𝜌₂,𝜃)-quasiunivex at 𝑢, 𝜙₂ is increasing, and 𝜙₂(0)=0; (iii)𝜌₁+𝜌₂≥0; (e)(i)𝐵 is (ℱ,𝑏,𝜙₁,𝜌₁,𝜃)-prestrictly-quasiunivex at 𝑢, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii)𝐶 is (ℱ,𝑏,𝜙₂,𝜌₂,𝜃)-quasiunivex at 𝑢, 𝜙₂ is increasing, and 𝜙₂(0)=0; (iii)𝜌₁+𝜌₂>0; (f)(i)𝐵 is (ℱ,𝑏,𝜙₁,𝜌₁,𝜃)-prestrictly-quasiunivex at 𝑢, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii)𝐶 is (ℱ,𝑏,𝜙₂,𝜌₂,𝜃)-strictly-pseudounivex at 𝑢, 𝜙₂ is increasing, and 𝜙₂(0)=0; (iii)𝜌₁+𝜌₂≥0; (g)(i)𝐵 is (ℱ,𝑏,𝜙₁,𝜌₁,𝜃)-pseudounivex at 𝑢, and 𝜙₁(𝑎)≥0⇒𝑎≥0; (ii)𝛽_𝑗ℎ_𝑗 is (ℱ,𝑏,𝜙_2𝑗,𝜌_2𝑗,𝜃)-quasiunivex at 𝑢,1≤𝑗≤𝑝, 𝜙_2𝑗 is increasing, and 𝜙_2𝑗(0)=0; (iii)𝜌₁+∑^𝑝_𝑗=1𝜌_2𝑗≥0.

Assume that 𝑥 is a properly efficient solution for (MFP) and satisfies an appropriate constraint qualification [21] for (SFP)_𝑖, 𝑖=1,2,…,𝑚. Then there exist 𝜆_𝑖>0, 1≤𝑖≤𝑚, 𝑣∈ℝ^𝑚₊, and 𝛽∈ℝ^𝑝₊ such that (𝑥,𝑣,𝜆,𝛽) is a properly efficient solution for (𝐷).

Proof. With Theorem 5, there exist 𝜆_𝑖>0, 1≤𝑖≤𝑚, 𝑣∈ℝ^𝑚₊, and 𝛽∈ℝ^𝑝₊ such that (𝑥,𝑣,𝜆,𝛽) is a feasible solution of (𝐷). By (64), there exist 𝜉_𝑖∈𝜕𝑓_𝑖(𝑢), 𝜍_𝑖∈𝜕(−𝑔_𝑖)(𝑢), for 1≤𝑖≤𝑚, and 𝜁_𝑗∈𝜕ℎ_𝑗(𝑢), for 1≤𝑗≤𝑝 such that ^𝑚_𝑖=1𝜆_𝑖𝜉_𝑖+𝑣_𝑖𝜍_𝑖+^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗=0. (72) From here it results ℱ𝑥,𝑢;^𝑚_𝑖=1𝜆_𝑖𝜉_𝑖+𝑣_𝑖𝜍_𝑖+^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≥0. (73) If hypothesis (a) holds, then 𝜙₁𝑓_𝑖(𝑥)−𝑓_𝑖(𝑢)≥ℱ𝑥,𝑢;𝑏(𝑥,𝑢)𝜉_𝑖+𝜌_1𝑖𝑑²(𝜃(𝑥,𝑢)),1≤𝑖≤𝑚,𝜙₁−𝑔_𝑖(𝑥)+𝑔_𝑖(𝑢)≥ℱ𝑥,𝑢;𝑏(𝑥,𝑢)𝜍_𝑖+𝜌_2𝑖𝑑²(𝜃(𝑥,𝑢)),1≤𝑖≤𝑚. (74)(75) Now, multiplying (74) by 𝜆_𝑖, (75) by 𝜆_𝑖𝑣_𝑖, and adding the resulting inequalities, and then using the superlinearity of 𝜙₁ and sublinearity of ℱ(𝑥,𝑢;⋅), we obtain 𝜙₁^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑢)−𝑣_𝑖𝑔_𝑖(𝑢)≥ℱ𝑥,𝑢;𝑏(𝑥,𝑢)^𝑚_𝑖=1𝜆_𝑖𝜉_𝑖+𝑣_𝑖𝜍_𝑖+^𝑚_𝑖=1𝜆_𝑖𝜌_1𝑖+^𝑚_𝑖=1𝜆_𝑖𝑣_𝑖𝜌_2𝑖𝑑²(𝜃(𝑥,𝑢)). (76) Since 𝑥∈𝑆, 𝛽∈ℝ^𝑝₊, and (66), it is clear that 𝛽_𝑗ℎ_𝑗(𝑥)−𝛽_𝑗ℎ_𝑗(𝑢)≤0,1≤𝑗≤𝑝. (77) Using the increasing property of 𝜙_2𝑗 and 𝜙_2𝑗(0)=0, we have 𝜙_2𝑗𝛽_𝑗ℎ_𝑗(𝑥)−𝛽_𝑗ℎ_𝑗(𝑢)≤0,1≤𝑗≤𝑝. (78) Using the (ℱ,𝑏,𝜙_2𝑗,𝜌_3𝑗,𝜃)-quasiunivexity of 𝛽_𝑗ℎ_𝑗 at 𝑢, we get from (78) ℱ𝑥,𝑢;𝑏(𝑥,𝑢)𝛽_𝑗𝜁_𝑗≤−𝜌_3𝑗𝑑²(𝜃(𝑥,𝑢)),1≤𝑗≤𝑝. (79) Because 𝛽∈ℝ^𝑝₊ the nonnegativity of 𝑏(𝑥,𝑢), and ℱ(𝑥,𝑢;⋅) is sublinear, the above inequality yields ℱ𝑥,𝑢;𝑏(𝑥,𝑢)^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≤−^𝑝_𝑗=1𝜌_3𝑗𝑑²(𝜃(𝑥,𝑢)). (80) From the nonnegativity of 𝑏(𝑥,𝑢), sublinearity of ℱ(𝑥,𝑢;⋅), and (73), it follows that ℱ𝑥,𝑢;𝑏(𝑥,𝑢)^𝑚_𝑖=1𝜆_𝑖𝜉_𝑖+𝑣_𝑖𝜍_𝑖+ℱ𝑥,𝑢;𝑏(𝑥,𝑢)^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≥0. (81) Now adding (76) and (80), and then using (81) and ∑^𝑚_𝑖=1𝜆_𝑖𝜌_1𝑖+∑^𝑚_𝑖=1𝜆_𝑖𝑣_𝑖𝜌_2𝑖+∑^𝑝_𝑗=1𝜌_3𝑗≥0, we obtain 𝜙₁^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑢)−𝑣_𝑖𝑔_𝑖(𝑢)≥0. (82) But 𝜙₁(𝑎)≥0⇒𝑎≥0, and so (82) yields ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑢)−𝑣_𝑖𝑔_𝑖(𝑢)≥0. (83) Equation (65) along with (83) yields ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖(𝑥)≥^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑢)+𝑣_𝑖−𝑔_𝑖(𝑢)≥0. (84) Assume that (𝑥,𝑣,𝜆,𝛽) is not an efficient for (𝐷). Then there exists a point (𝑢,𝑣,𝜆,𝛽)∈ℱ_𝐷 such that 𝑣_𝑖≥𝑣_𝑖=𝑓_𝑖(𝑥)𝑔_𝑖(𝑥),1≤𝑖≤𝑚, (85) with at least one strict inequality.
Thus, for any 𝜆∈ℝ^𝑚₊⧵{0}, we have ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)<0, (86) which contradicts inequality (84).
It remains to show that (𝑥,𝑣,𝜆,𝛽) is properly efficient. Assume that it is an improperly efficient point for the problem (𝐷) and choose 𝑀=(𝑚−1)(𝜆_𝑗/𝜆_𝑖)(𝑔_𝑗(𝑥)/𝑔_𝑖(𝑥))>0 (we may assume that 𝑚≥2), there would exist a feasible solution (𝑢,𝑣,𝜆,𝛽)∈𝐹_𝐷 and an index 𝑖 such that 𝑣_𝑖>𝑣_𝑖 and 𝑣_𝑖−𝑣_𝑖>𝑀𝑣_𝑗−𝑣_𝑗=()𝜆𝑚−1^𝑗𝜆_𝑖𝑔_𝑗(𝑥)𝑔_𝑖(𝑥)𝑣_𝑗−𝑣_𝑗, (87) for all 𝑗 such that 𝑣_𝑗<𝑣_𝑗.
These imply that 𝑔_𝑖(𝑥)𝑣_𝑖−𝑓_𝑖(𝑥)𝑔_𝑖(𝑥)>()𝜆𝑚−1^𝑗𝜆_𝑖𝑔_𝑗(𝑥𝑓)^𝑗(𝑥)𝑔_𝑗(𝑥)−𝑣_𝑗,∀𝑗≠𝑖. (88) Thus, we have 𝜆_𝑖𝑣𝑚−1^𝑖𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)>𝜆_𝑗𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥),∀𝑗≠𝑖. (89) Summing over 𝑗≠𝑖 yield 𝜆_𝑖𝑣_𝑖𝑔_𝑖(𝑥)−𝑓_𝑖(𝑥)>_𝑗≠𝑖𝜆_𝑗𝑓_𝑗(𝑥)−𝑣_𝑗𝑔_𝑗(𝑥), (90) or ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)<0, (91) which contradicts inequality (84). Thus, the proof of (a) is complete.
If hypothesis (b) holds, from (66), 𝑥∈𝑆, and 𝛽∈ℝ^𝑝₊, then we have ^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)≤0≤^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑢). (92) and so using the properties of 𝜙₂, we obtain 𝜙₂^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)−ℎ_𝑗(𝑢)≤0. (93) Using the (ℱ,𝑏,𝜙₂,𝜌₃,𝜃)-quasiunivexity of 𝐶 at 𝑢, then we get from (93) ℱ𝑥,𝑢;𝑏(𝑥,𝑢)^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≤−𝜌₃𝑑²(𝜃(𝑥,𝑢)). (94) Now combining (76), (81), and (94), and using (iii), we obtain (84). Therefore, along with the same lines of (a), we obtain (𝑥,𝑣,𝜆,𝛽) is a properly efficient solution for (𝐷).
If hypothesis (c) holds, using the (ℱ,𝑏,𝜙,0,𝜃)-pseudounivexity of 𝐴, we get from (73) 𝜙^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)+^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑢)−𝑣_𝑖𝑔_𝑖(𝑢)+^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑢)≥0. (95) In view of the properties of 𝜙, we deduce from this inequality that ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)+^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑢)−𝑣_𝑖𝑔_𝑖(𝑢)+^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑢)≥0. (96) Equation (92) and (65) along with (96) yields ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖(𝑥)≥^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑢)+𝑣_𝑖−𝑔_𝑖(𝑢)≥0. (97) Thus, we obtain (𝑥,𝑣,𝜆,𝛽) is a properly efficient solution for (𝐷).
If hypothesis (d) holds, using the properties of 𝜙₂ and (92), we obtain 𝜙₂^𝑝_𝑗=1𝛽_𝑗ℎ_𝑗(𝑥)−ℎ_𝑗(𝑢)≤0, (98) which in view of (ii) implies that ℱ𝑥,𝑢;𝑏(𝑥,𝑢)^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≤−𝜌₂𝑑²(𝜃(𝑥,𝑢)). (99) Now combining (81), (99), and 𝜌₁+𝜌₂≥0, we obtain ℱ𝑥,𝑢;𝑏(𝑥,𝑢)^𝑚_𝑖=1𝜆_𝑖𝜉_𝑖+𝑣_𝑖𝜍_𝑖≥−ℱ𝑥,𝑢;𝑏(𝑥,𝑢)^𝑝_𝑗=1𝛽_𝑗𝜁_𝑗≥𝜌₂𝑑²(𝜃(𝑥,𝑢))≥−𝜌₁𝑑²(𝜃(𝑥,𝑢)). (100) Along with the fact that the 𝐵 is (ℱ,𝑏,𝜙₁,𝜌₁,𝜃)-pseudounivex at 𝑢, we get from (100) 𝜙₁^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)−𝑣_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑢)−𝑣_𝑖𝑔_𝑖(𝑢)≥0. (101) In view of the properties of 𝜙₁, we deduce from this inequality that ^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑥)+𝑣_𝑖−𝑔_𝑖(𝑥)≥^𝑚_𝑖=1𝜆_𝑖𝑓_𝑖(𝑢)+𝑣_𝑖−𝑔_𝑖(𝑢)≥0. (102) Therefore, (𝑥,𝑣,𝜆,𝛽) is a properly efficient solution for (𝐷). (e), (f), and (g) follows along with the same lines of (d) and (a). Thus, the proof is complete.

Acknowledgments

The authors are thankful to the referees for their comments and suggestions in an earlier version of the paper. The research was partly supported by the National Science Council, Taiwan.