Duality of (ℎ,𝜑)-Multiobjective Programming Involving Generalized Invex Functions

Abstract

In the setting of Ben-Tal's generalized algebraic operations, this paper deals with Mond-Weir type dual theorems of multiobjective programming problems involving generalized invex functions. Two classes of functions, namely, (ℎ,𝜑)-pseudoinvex and (ℎ,𝜑)-quasi-invex, are defined for a vector function. By utilizing these two classes of functions, some dual theorems are established for conditionally proper efficient solution in (ℎ,𝜑)-multiobjective programming problems.

1. Introduction

The theory and applications of multiobjective programming problems have been closely tied with convex analysis. Optimality conditions and duality theorems were established for the class of problems involving the optimizations of convex objective functions over convex feasible regions. Such assumptions were very convenient because of the known separation theorems and the guarantee that necessary conditions for optimality were sufficient under convexity. However, not all practical problems, when formulated as multiobjective programs, fulfill the requirements of convexity. Fortunately, such problems were often found to have some characteristics in common with convex problems, and these properties could be exploited to establish theoretical results or develop algorithms. Many notions of generalized convexity having some useful properties shared with convexity have been defined by a sizeable number of researchers. A meaningful generalization of convex functions is the introduction of invex functions, which was given by Hanson [1], for the scalar case. Nowadays, with and without differentiability, the invex functions are extended to vector functions in finite dimensions or infinite dimensions abstract spaces, and sufficient optimality criteria and duality results are obtained for multiobjective programming or vector optimization, respectively, see [1–15].

In 1976, Ben-Tal [8] introduced certain generalized operations of addition and multiplication. This kind of generalized algebraic means has many applications in pure and applied mathematical fields, see [6, 7, 10–16]. The biggest advantage under Ben-Tal's generalized means is that the function has some transformable properties. As pointed out in literature [12] that a function is not convex or differentiable, however it may be transformed into convex function or differentiable function in the setting of Ben-Tal's generalized algebraic operations. In this way, Ben-Tal's generalized means provided a manner in extension of convexity. Recently, more and more interest has been paid on dealing with optimality and duality of multiobjective program problems involving generalized convexity under Ben-Tal's generalized means circumstances, for instance, see [10–16].

The properness of the efficient solution of the multiobjective programming problem is of importance. In 1991, Singh and Hanson [9] introduced conditionally properly efficiency for multiobjective programming problems. This kind of proper efficiency has specific significance in the optimal problem with multicriteria. In present paper, we first extend the notions of the conditionally proper efficiency for multiobjective programming problems, pseudoinvexity and quasi-invexity for vector functions in the setting of Ben-Tal's generalized means. Then, for a class of constraint multiobjective program problem, we will establish several duality results by using the new defined proper efficient solutions and generalized invex functions. This paper is organized as follows. In Section 2, we present some preliminaries and related results which will be used in the rest of the paper. In Section 3, some duality theorems are derived.

2. Preliminaries

Let ℝ^𝑛 be the 𝑛-dimensional Euclidean space and ℝ₊₊ be the set of all positive real numbers. Throughout this paper, the following convention for vector in ℝ^𝑛 will be used: 𝑥>𝑦iff𝑥_𝑖>𝑦_𝑖,𝑖=1,2,…,𝑛,𝑥≧𝑦iff𝑥_𝑖≧𝑦_𝑖,𝑖=1,2,…,𝑛,𝑥⩾𝑦iff𝑥_𝑖≧𝑦_𝑖,𝑖=1,2,…,𝑛,but𝑥≠𝑦. (2.1)

We first present the generalized algebraic operations given by Ben-Tal [8].

Definition 2.1 (see [6, 8]). Let ℎ∶ℝ^𝑛→ℝ^𝑛 be a continuous vector function. Suppose that the inverse function ℎ⁻¹ of ℎ exists. Then the ℎ-vector addition of 𝑥,𝑦∈ℝ^𝑛 defined by 𝑥⊕𝑦=ℎ⁻¹ℎ(𝑥)+ℎ𝑦, (2.2) and the ℎ-scalar multiplication of 𝑥∈ℝ^𝑛 and 𝛼∈ℝ is defined by 𝛼⊗𝑥=ℎ⁻¹(𝛼ℎ(𝑥)). (2.3)
Similarly, generalized algebraic operations for scalar-valued functions can be defined as follows.

Definition 2.2 (see [6, 8]). Let 𝜑∶ℝ→ℝ be a continuous and scalar function. Suppose that the inverse function 𝜑⁻¹ of 𝜑 exists. Then the 𝜑-addition of 𝛼∈ℝ and 𝛽∈ℝ, is given by 𝛼[+]𝛽=𝜑⁻¹𝜑(𝛼)+𝜑𝛽, (2.4) and the 𝜑-scalar multiplication of 𝛼∈ℝ and 𝛽∈ℝ as 𝛽[⋅]𝛼=𝜑⁻¹𝛽𝜑(𝛼). (2.5)

Definition 2.3 (see [6, 8]). The (ℎ,𝜑)-inner product of vector 𝑥,𝑦∈𝑅^𝑛 is defined as 𝑥^𝑇𝑦_ℎ,𝜑=𝜑⁻¹ℎ(𝑥)^𝑇ℎ𝑦. (2.6)
In this paper, we denote ^𝑚_𝑖=1𝑥_𝑖=𝑥₁⊕𝑥₂⊕⋯⊕𝑥_𝑚,𝑥_𝑖∈𝑅^𝑛,𝑖=1,2,…,𝑚,_𝑚𝑖=1𝛼_𝑖=𝛼₁[+]𝛼₂[+]⋯[+]𝛼_𝑚,𝛼_𝑖𝛼[−][+]⋅]𝛽∈ℝ,𝑖=1,2,…,𝑚,𝛽=𝛼(−1)[,𝛼,𝛽∈ℝ. (2.7)
For the differentiability of a real-valued function in the setting of generalized algebraic means, Avriel [6] introduced the following important concept.

Definition 2.4 (see [6]). Let 𝑓 be a real-valued function defined on ℝ^𝑛, denote 𝑓(𝑡)=𝜑(𝑓(ℎ⁻¹(𝑡))),𝑡∈𝑅^𝑛. For simplicity, write 𝑓(𝑡)=𝜑𝑓ℎ⁻¹(𝑡). The function 𝑓 is said to be (ℎ,𝜑)-differentiable at 𝑥∈𝑅^𝑛, if 𝑓(𝑡) is differentiable at 𝑡=ℎ(𝑥), and denoted by ∇^∗𝑓(𝑥)=ℎ⁻¹(∇𝑓(𝑡)∣_{𝑡=ℎ(𝑥)}). In addition, It is said that 𝑓 is (ℎ,𝜑)-differentiable on 𝑋⊂ℝ^𝑛 if it is (ℎ,𝜑)-differentiable at each 𝑥∈𝑋. A vector-valued function is called (ℎ,𝜑)-differentiable on 𝑋⊂ℝ^𝑛 if each of its components is (ℎ,𝜑)-differentiable at each 𝑥∈𝑋.
We collect some basic properties concerning Ben-Tal's generalized means from the literatures [12, 14], which will be used in the squeal.

Lemma 2.5 (see [12, 14]). Suppose that 𝑓,𝑓_𝑖 are real-valued functions defined on 𝑅^𝑛, for 𝑖=1,2,…,𝑚, and (ℎ,𝜑)-differentiable at 𝑥∈𝑅^𝑛. Then, the following statements hold:(1)∇^∗(𝜆[⋅]𝑓(𝑥))=𝜆⊗∇^∗𝑓(𝑥), for 𝜆∈ℝ, (2)((⨂^𝑚_𝑖=1𝜆_𝑖⊗∇^∗𝑓(𝑥))^𝑇𝑦)_ℎ,𝜑=[∑^𝑚_𝑖=1](∇^∗(𝜆_𝑖[⋅]𝑓(𝑥))^𝑇𝑦)_ℎ,𝜑, for 𝑦∈𝑅^𝑛, 𝜆_𝑖∈ℝ.

Lemma 2.6 (see [12, 14]). Let 𝑖=1,2,…,𝑚. The following statements hold:(1)𝜆[⋅](𝜇[⋅]𝛼)=𝜇[⋅](𝜆[⋅]𝛼)=𝜆𝜇[⋅]𝛼,for 𝜆,𝜇,𝛼∈ℝ;(2)𝜆[⋅](𝛼[−]𝛽)=𝜆[⋅]𝛼[−]𝜆[⋅]𝛽,for 𝜆,𝛼,𝛽∈ℝ;(3)[∑^𝑚_𝑖=1](𝛼_𝑖[−]𝛽_𝑖)=[∑^𝑚_𝑖=1]𝛼_𝑖[−][∑^𝑚_𝑖=1]𝛽_𝑖, for 𝛼_𝑖,𝛽_𝑖∈ℝ.

Lemma 2.7 (see [12, 14]). Suppose that function 𝜑, which appears in Ben-Tal generalized algebraic operations, is strictly monotone with 𝜑(0)=0. Then, the following statements hold:(1)let 𝜆≧0, 𝛼,𝛽∈ℝ, and 𝛼≦𝛽. Then 𝜆[⋅]𝛼≦𝜆[⋅]𝛽;(2)let 𝜆>0, 𝛼,𝛽∈ℝ, and 𝛼<𝛽. Then 𝜆[⋅]𝛼<𝜆[⋅]𝛽;(3)let 𝜆<0, 𝛼,𝛽∈ℝ, and 𝛼≦𝛽. Then 𝜆[⋅]𝛼≧𝜆[⋅]𝛽;(4)let 𝛼_𝑖,𝛽_𝑖∈ℝ, 𝑖=1,2,…,𝑚. If 𝛼_𝑖≦𝛽_𝑖 for any 𝑖∈𝑀, then ^𝑚_𝑖=1𝛼_𝑖≦^𝑚_𝑖=1𝛽_𝑖. (2.8)If 𝛼_𝑖≦𝛽_𝑖 for any 𝑖=1,2,…,𝑚, and there exists at least an index 𝑘 such that 𝑥_𝑘<𝑦_𝑘, then ^𝑚_𝑖=1𝛼_𝑖<^𝑚_𝑖=1𝛽_𝑖. (2.9)

Lemma 2.8 (see [12, 14]). Suppose that 𝜑 is a continuous one-to-one strictly monotone and onto function with 𝜑(0)=0. Let 𝛼,𝛽∈ℝ. Then,(1)𝛼<𝛽 if and only if 𝛼[−]𝛽<0,(2)𝛼[+]𝛽=0 if and only if 𝛼=(−1)[⋅]𝛽.

Throughout the rest of this paper, one further assumes that ℎ is a continuous one-to-one and onto function with ℎ(0)=0. Similarly, suppose that 𝜑 is a continuous one-to-one strictly monotone and onto function with 𝜑(0)=0. Under the above assumptions, it is clear that 0[⋅]𝛼=𝛼[⋅]0=0.

Let 𝑋 be a nonempty subset of ℝ^𝑛 and the functions 𝑓=(𝑓₁,…,𝑓_𝑝)^𝑇∶𝑋→ℝ^𝑝 and 𝑔=(𝑔₁,…,𝑔_𝑚)^𝑇∶𝑋→ℝ^𝑚 are (ℎ,𝜑)-differentiable on the set 𝑋 with respect to the same (ℎ,𝜑). Consider the following (ℎ,𝜑)-multiobjective programming problem: min𝑓(𝑥)=𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑝(𝑥)^𝑇,𝑥∈𝑋⊂ℝ^𝑛s.t.𝑔(𝑥)≦0. (MOP)_ℎ,𝜑

Definition 2.9. A point 𝑥 is said to be an efficient solution for (MOP)_ℎ,𝜑 if 𝑥∈𝑋 and 𝑓(𝑥)≰𝑓(𝑥) for all 𝑥∈𝑋.
Singh and Hanson [9] introduced the concept of conditionally properly efficient for multiobjective optimization. Now, we extend this notion under Ben-Tal's generalized algebraic operations as follows.

Definition 2.10. The point 𝑥 is said to be (ℎ,𝜑)-conditionally proper efficient solution for (MOP)_ℎ,𝜑 if 𝑥 is an efficient solution and there exists a positive function 𝑀(𝑥) such that, for 𝑖, one has 𝑓_𝑖(𝑥)[−]𝑓_𝑖(𝑥)𝑓_𝑗(𝑥)[−]𝑓_𝑗(𝑥)≦𝑀(𝑥), (2.10) for some 𝑗 such that 𝑓_𝑗(𝑥)>𝑓_𝑗(𝑥), whenever 𝑥∈𝑋 and 𝑓_𝑖(𝑥)<𝑓_𝑖(𝑥). (2.11)

Example 2.11. Consider the following multiobjective problem: min𝑓(𝑥)=𝑓₁(𝑥),𝑓₂(𝑥)^𝑇=𝑥₁𝑥₂,𝑥₂𝑥₁^𝑇s.t.𝑔(𝑥)=𝑔₁(𝑥),𝑔₂(𝑥)^𝑇=𝑥₁,𝑥₂^𝑇𝑥≧1𝑥=₁,𝑥₂^𝑇∈ℝ². (MOP)^′_ℎ,𝜑 Taking ℎ(𝑥)=𝑥, 𝜑(𝑡)=𝑡³, it can be shown that every point of the feasible region is efficient. Let 𝑥^∗=(𝑎,𝑏)^𝑇 be an efficient solution. Choosing 𝑀(𝑥)≧(𝑏𝑥₂/𝑎𝑥₁), where 𝑥=(𝑥₁,𝑥₂)^𝑇. For 𝑖=1, we get 𝑓₂𝑥^∗[−]𝑓₂(𝑥)𝑓₁(𝑥)[−]𝑓₁(𝑥^∗)=³(𝑏/𝑎)³−𝑥₂/𝑥₁³₃𝑥₁/𝑥₂³−(𝑎/𝑏)³=𝑏𝑥₂𝑎𝑥₁≦𝑀(𝑥), (2.12) for 𝑗=2 such that 𝑓₂(𝑥)=𝑥₂/𝑥₁>𝑏/𝑎=𝑓₂(𝑥^∗) whenever 𝑥=(𝑥₁,𝑥₂)^𝑇 is feasible and 𝑓₁(𝑥)=𝑥₁𝑥₂<𝑎𝑏=𝑓₁𝑥^∗. (2.13) Thus, 𝑥^∗ is (ℎ,𝜑)-conditionally proper efficient solution.
Xu and Liu [10] introduced (ℎ,𝜑)-Kuhn-Tucker constraint qualification and used it to establish Kuhn-Tucker necessary condition for (ℎ,𝜑)-multiobjective programming problems, for more details concerning (ℎ,𝜑)-Kuhn-Tucker constraint qualification, please see [10]. We now state this result as the following (Lemma 2.12).

Lemma 2.12 (Kuhn-Tucker-type necessary condition). Let 𝑓_𝑖 for 𝑖=1,2,…,𝑝, 𝑔_𝑗 for 𝑗=1,2,…,𝑚 be (ℎ,𝜑)-differentiable on ℝ^𝑛, 𝑥 be an efficient solution of (MOP)_ℎ,𝜑 and the (ℎ,𝜑)-Kuhn-Tucker constraint qualification be satisfied at 𝑥. Then there exist 𝜏=(𝜏₁,𝜏₂,…,𝜏_𝑝)^𝑇>0 and 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑚)^𝑇≧0 such that ^𝑚_𝑖=1𝜏_𝑖⊗∇^∗𝑓_𝑖(𝑥)⊕^𝑚_𝑗=1𝜆_𝑗⊗∇^∗𝑔_𝑗(𝑥)=0,𝜆_𝑗[⋅]𝑔_𝑗(𝑥)=0,𝑗=1,2,…,𝑚. (2.14)

Jeyakumar and Mond [2] introduced the notion of 𝑉-invexity for a vector function 𝑓=(𝑓₁,𝑓₂,…,𝑓_𝑝) and discussed its applications to a class of constrained multi-objective optimization problems. One now gives the definitions of generalized 𝑉-invexity for a vector function in the setting of Ben-Tal's generalized algebraic operations as follows.

Definition 2.13. A vector function 𝑓∶𝑋⊂ℝ^𝑛→ℝ^𝑝 is said to be (ℎ,𝜑)-𝑉-invex at 𝑥∈𝑋 if there exist functions 𝜂∶𝑋×𝑋→ℝ^𝑛 and 𝛼_𝑖∶𝑋×𝑋→ℝ₊₊ such that for each 𝑥∈𝑋 and for 𝑖=1,2,…,𝑝, 𝑓_𝑖(𝑥)[−]𝑓_𝑖(𝑥)≧𝛼_𝑖(𝑥,𝑥)[⋅]∇^∗𝑓_𝑖(𝑥)^𝑇𝜂(𝑥,𝑥)_ℎ,𝜑. (2.15)
If we take ℎ and 𝜑 as the identity functions, the above definitions reduce to the 𝑉-invex function given by Jeyakumar and Mond [2].

Example 2.14. The functions 𝑓∶ℝ→ℝ², 𝑓(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥))^𝑇=(|𝑥|,|𝑥|)^𝑇. Let ℎ(𝑥)=𝑥 and 𝜑(𝑡)=𝑡³. Then, 𝑓 is (ℎ,𝜑)-𝑉-invex function at 𝑥=0 with respect to any 𝜂(𝑥,𝑥) and 𝛼_𝑖(𝑥,𝑥), 𝑖=1,2.

Definition 2.15. A vector function 𝑓∶𝑋⊂ℝ^𝑛→𝑅^𝑝 is said to be (ℎ,𝜑)-𝑉-pseudoinvex at 𝑥∈𝑋 if there exist functions 𝜂∶𝑋×𝑋→𝑅^𝑛 and 𝛽_𝑖∶𝑋×𝑋→ℝ₊₊ such that for each 𝑥∈𝑋 and for 𝑖=1,2,…,𝑝, ^𝑝_𝑖=1∇^∗𝑓_𝑖(𝑥)^𝑇𝜂(𝑥,𝑥)_ℎ,𝜑≧0⇒^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑥)[⋅]𝑓_𝑖(𝑥)≧^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑥)[⋅]𝑓_𝑖(𝑥). (2.16) If in the above definition 𝑥≠𝑥 and (2.16) is satisfied as ^𝑝_𝑖=1∇^∗𝑓_𝑖(𝑥)^𝑇𝜂(𝑥,𝑥)_ℎ,𝜑≧0⇒^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑥)[⋅]𝑓_𝑖(𝑥)>^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑥)[⋅]𝑓_𝑖(𝑥), (2.17) then we say that 𝑓 is strictly (ℎ,𝜑)-𝑉-pseudoinvex at 𝑥∈𝑋.

Example 2.16. The functions 𝑓∶(0,1]→𝑅², 𝑓(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥))=(cos²(𝑥),−sin²(𝑥)). Let ℎ(𝑡)=𝑡 and 𝜑(𝛼)=arctan(𝛼). Then, 𝑓 is (ℎ,𝜑)-𝑉-quasi-invex function at 𝑥=1 with respect to 𝜂(𝑥,𝑥)=0 and any 𝛽_𝑖(𝑥,𝑥)>0 (𝑖=1,2). In fact, observing that 𝜑⁻¹(𝛼)=tan(𝛼) and ℎ(0)=0, 𝜑(0)=𝜑⁻¹(0)=0. In this case, we have 0=²_𝑖=1∇^∗𝑓_𝑖(1)^𝑇0_ℎ,𝜑≧0, (2.18) and for 𝑥∈(0,1], it follows that 𝑓₁(𝑥)[−]𝑓₁(1)=tanarctancos²(𝑥)−arctancos²(1)=cos²(𝑥)−cos²(1)1+cos²(𝑥)cos²(1)𝑓≧0,₂(𝑥)[−]𝑓₂(1)=tanarctan−sin²(𝑥)−arctan−sin²(1)=−sin²(𝑥)−−sin²(1)1+sin²(𝑥)sin²(1)≧0. (2.19) Thus, we get from Lemmas 2.6 and 2.7 that ²_𝑖=1𝛽_𝑖(𝑥,𝑥)[⋅]𝑓_𝑖(𝑥)≧²_𝑖=1𝛽_𝑖(𝑥,𝑥)[⋅]𝑓_𝑖(𝑥). (2.20) By Definition 2.15, we have shown that 𝑓 is (ℎ,𝜑)-𝑉-pseudoinvex at 𝑥=1.

Definition 2.17. A vector function 𝑓∶𝑋⊂ℝ^𝑛→𝑅^𝑝 is said to be (ℎ,𝜑)-𝑉-quasi-invex at 𝑥∈𝑋 if there exist functions 𝜂∶𝑋×𝑋→𝑅^𝑛 and 𝛿_𝑖∶𝑋×𝑋→𝑅⁺⁺ such that for each 𝑥∈𝑋 and for 𝑖=1,2,…,𝑝, ^𝑝_𝑖=1𝛿_𝑖(𝑥,𝑥)[⋅]𝑓_𝑖(𝑥)≦^𝑝_𝑖=1𝛿_𝑖(𝑥,𝑥)[⋅]𝑓_𝑖(𝑥)⇒^𝑝_𝑖=1∇^∗𝑓_𝑖(𝑥)^𝑇𝜂(𝑥,𝑥)_ℎ,𝜑≦0. (2.21)

Example 2.18. The function 𝑓∶ℝ→ℝ is defined as 𝑓(𝑥)=𝑥³. Taking ℎ(𝑥)=𝑥³ and 𝜑(𝑡)=𝑡, then, 𝑓 is (ℎ,𝜑)-𝑉-quasi-invex at 𝑥=0 with respect to 𝜂(𝑥,𝑥)=𝑥⊖𝑥 and any 𝛿(𝑥,𝑥)>0.

3. Duality

In this section, we will establish the weak and strong duality theorems under the generalized (ℎ,𝜑)-𝑉-invexity assumptions for Mond and Weir type dual model in relation to (MOP)_ℎ,𝜑 Considering the following dual problem: max𝑓(𝑢)=𝑓₁(𝑢),𝑓₂(𝑢),…,𝑓_𝑚(𝑢)^𝑇s.t.^𝑝_𝑖=1𝜏_𝑖⊗∇^∗𝑓_𝑖(𝑢)⊕^𝑝_𝑗=1𝜆_𝑗⊗∇^∗𝑔_𝑗(𝑢)=0,𝜆_𝑗[⋅]𝑔_𝑗(𝑢)≧0,𝜏>0,𝜏=𝜏₁,𝜏₂,…,𝜏_𝑝^𝑇,𝜆≧0,𝜆=𝜆₁,𝜆₂,…,𝜆_𝑚^𝑇𝑢∈𝑋⊂ℝ^𝑛. (DMOP)_ℎ,𝜑(3.1)(3.2)(3.3)(3.4)(3.5)

Theorem 3.1 (weak duality). Let 𝑥 and (𝑢,𝜏,𝜆) be any feasible solutions for (MOP)_ℎ,𝜑 and (DMOP)_ℎ,𝜑, respectively. Let either (a) or (b) below hold:(a)(𝜏₁[⋅]𝑓₁,𝜏₂[⋅]𝑓₂,…,𝜏_𝑝[⋅]𝑓_𝑝)^𝑇 is (ℎ,𝜑)-𝑉-pseudoinvex and (𝜆₁[⋅]𝑔₁,𝜆₂[⋅]𝑔₂,…,𝜆_𝑚[⋅]𝑔_𝑚)^𝑇 is (ℎ,𝜑)-𝑉-quasi-invex at 𝑢 with respect to same 𝜂; (b)(𝜏₁[⋅]𝑓₁,𝜏₂[⋅]𝑓₂,…,𝜏_𝑝[⋅]𝑓_𝑝)^𝑇 is (ℎ,𝜑)-𝑉-quasi-invex and (𝜆₁[⋅]𝑔₁,𝜆₂[⋅]𝑔₂,…,𝜆_𝑚[⋅]𝑔_𝑚)^𝑇 is strictly (ℎ,𝜑)-𝑉-pseudoinvex at 𝑢 with respect to same 𝜂. Then 𝑓(𝑥)𝑓(𝑢). (3.6)

Proof. Since (𝑢,𝜏,𝜆) is a feasible solution for (DMOP)_ℎ,𝜑, by Lemma 2.5 and (3.1), for all 𝑥^′∈ℝ^𝑛 we obtain that ^𝑝_𝑖=1∇^∗𝜏_𝑖[⋅]𝑓_𝑖(𝑢)^𝑇𝜂𝑥^′,𝑢_ℎ,𝜑[+]^𝑚_𝑗=1∇^∗𝜆_𝑗[⋅]𝑔_𝑗(𝑢)^𝑇𝜂𝑥^′,𝑢_ℎ,𝜑=0. (3.7)(a) Let 𝑥 be feasible for (MOP)_ℎ,𝜑 and 𝑓(𝑥)⩽𝑓(𝑢). Since 𝜏>0 and 𝛽_𝑖(𝑥,𝑢)>0,  for all  𝑖=1,…,𝑝, it follows from Lemmas 2.6 and 2.7 that ^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑥)<^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑢), (3.8) and (ℎ,𝜑)-𝑉-pseudoinvexity at 𝑢 of (𝜏₁[⋅]𝑓₁,…,𝜏_𝑝[⋅]𝑓_𝑝)^𝑇 implies ^𝑝_𝑖=1∇^∗𝜏_𝑖[⋅]𝑓_𝑖(𝑢)^𝑇𝜂(𝑥,𝑢)_ℎ,𝜑<0. (3.9) Observing that 𝑥 and (𝑢,𝜏,𝜆) are feasible of (MOP)_ℎ,𝜑 and (DMOP)_ℎ,𝜑, respectively, we get from Lemma 2.7 that 𝜆_𝑗[⋅]𝑔_𝑗(𝑢)≧0≧𝜆_𝑗[⋅]𝑔_𝑗(𝑥),∀𝑗=1,2,…𝑚. (3.10) Again, since 𝛿_𝑗(𝑥,𝑢)>0, for all 𝑗=1,2,…,𝑚, it follows from Lemma 2.7 that ^𝑚_𝑗=1𝛿_𝑗(𝑥,𝑢)[⋅]𝜆_𝑗[⋅]𝑔_𝑗(𝑥)≦^𝑚_𝑗=1𝛿_𝑗(𝑥,𝑢)[⋅]𝜆_𝑗[⋅]𝑔_𝑗(𝑢). (3.11) Now, (ℎ,𝜑)-𝑉-quasi-invexity at 𝑢 of (𝜆₁[⋅]𝑔₁,…,𝜆_𝑚[⋅]𝑔_𝑚)^𝑇 implies that ^𝑚_𝑗=1∇^∗𝜆_𝑗[⋅]𝑔_𝑗(𝑢)^𝑇𝜂(𝑥,𝑢)_ℎ,𝜑≦0. (3.12) Together with (3.9) and (3.12), it yields from Lemma 2.7 that ^𝑝_𝑖=1∇^∗𝜏_𝑖[⋅]𝑓_𝑖(𝑢)^𝑇𝜂(𝑥,𝑢)_ℎ,𝜑[+]^𝑚_𝑗=1∇^∗𝜆_𝑗[⋅]𝑔_𝑗(𝑢)^𝑇𝜂(𝑥,𝑢)_ℎ,𝜑<0, (3.13) which contradicts to (3.7)(b) Let 𝑥 be feasible for (MOP)_ℎ,𝜑 and (𝑢,𝜏,𝜆) feasible for (DMOP)_ℎ,𝜑. Suppose that 𝑓(𝑥)⩽𝑓(𝑢). Since 𝛿_𝑖(𝑥,𝑢)>0, for all  𝑖=1,…,𝑝, and 𝜏>0, we get from Lemmas 2.6 and 2.7 that ^𝑝_𝑖=1𝛿_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑥)<^𝑝_𝑖=1𝛿_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑢). (3.14) The (ℎ,𝜑)-𝑉-quasi-invexity at 𝑢 of (𝜏₁[⋅]𝑓₁,…,𝜏_𝑝[⋅]𝑓_𝑝)^𝑇 implies that ^𝑝_𝑖=1∇^∗𝜏_𝑖[⋅]𝑓_𝑖(𝑢)^𝑇𝜂(𝑥,𝑢)_ℎ,𝜑≦0. (3.15) By (3.7), we get from Lemmas 2.7 and 2.8 that ^𝑚_𝑗=1∇^∗𝜆_𝑗[⋅]𝑔_𝑗(𝑢)^𝑇𝜂(𝑥,𝑢)_ℎ,𝜑≧0. (3.16) and since (𝜆₁[⋅]𝑔₁,…,𝜆_𝑚[⋅]𝑔_𝑚)^𝑇 is strictly (ℎ,𝜑)-𝑉-pseudoinvex, we have ^𝑚_𝑗=1𝛽(𝑥,𝑢)_𝑗[⋅]𝜆_𝑗[⋅]𝑔_𝑗(𝑥)>^𝑚_𝑗=1𝛽(𝑥,𝑢)_𝑗[⋅]𝜆_𝑗[⋅]𝑔_𝑗(𝑢). (3.17) According to Lemma 2.7, this is a contradiction, since 𝜆_𝑗[⋅]𝑔_𝑗(𝑥)≦0, 𝜆_𝑗[⋅]𝑔_𝑗(𝑢)≧0 and 𝛽(𝑥,𝑢)_𝑗>0, for all 𝑗=1,2,…,𝑚.

Theorem 3.2. If 𝑥 is feasible for (MOP)_ℎ,𝜑 and (𝑢,𝜏,𝜆) feasible for (DMOP)_ℎ,𝜑 such that 𝑓(𝑥)=𝑓(𝑢). Let neither (a′) or (b′) bellow hold:(a′)(𝜏₁[⋅]𝑓₁,𝜏₂[⋅]𝑓₂,…,𝜏_𝑝[⋅]𝑓_𝑝)^𝑇 is (ℎ,𝜑)-𝑉-pseudoinvex and (𝜆₁[⋅]𝑔₁,𝜆₂[⋅]𝑔₂,…,𝜆_𝑚[⋅]𝑔_𝑚)^𝑇 is (ℎ,𝜑)-𝑉-quasi-invex at 𝑢 with respect to same 𝜂; (b′)(𝑢₁[⋅]𝑓₁,𝑢₂[⋅]𝑓₂,…,𝑢_𝑝[⋅]𝑓_𝑝)^𝑇 is (ℎ,𝜑)-𝑉-quasi-invex and (𝜆₁[⋅]𝑔₁,𝜆₂[⋅]𝑔₂,…,𝜆_𝑚[⋅]𝑔_𝑚)^𝑇 is strictly (ℎ,𝜑)-𝑉-pseudoinvex at 𝑢 with respect to same 𝜂.

Then 𝑥 is (ℎ,𝜑)-conditionally properly efficient for (MOP)_ℎ,𝜑 and (𝑢,𝜏,𝜆) is (ℎ,𝜑)-conditionally properly efficient solution for (DMOP)_ℎ,𝜑.

Proof. Suppose 𝑥 is not an efficient solution for (MOP)_ℎ,𝜑, then there exists 𝑥 feasible for (MOP)_ℎ,𝜑 such that 𝑓(𝑥)⩽𝑓(𝑥). (3.18) Using the assumption 𝑓(𝑥)=𝑓(𝑢), a contradiction to Theorem 3.1 is obtained. Hence, 𝑥 is an efficient solution for (MOP)_ℎ,𝜑. Similarly it can be ensured that (𝑢,𝜏,𝜆)) is an efficient solution for (DMOP)_ℎ,𝜑.

Now suppose that 𝑥 is not (ℎ,𝜑)-conditionally properly efficient solution for (MOP)_ℎ,𝜑. Therefore, for every positive function 𝑀(𝑥)>0, there exists 𝑥∈𝑋 feasible for (MOP)_ℎ,𝜑 and an index 𝑖 such that 𝑓_𝑖(𝑥)[−]𝑓_𝑖(𝑥)>𝑀(𝑥)[⋅]𝑓_𝑗(𝑥)[−]𝑓_𝑗(𝑥), (3.19) for all 𝑗 satisfying 𝑓_𝑗(𝑥)>𝑓_𝑗(𝑥), whenever 𝑓_𝑖(𝑥)<𝑓_𝑖(𝑥). This shows that 𝑓_𝑖(𝑥)[−]𝑓_𝑖(𝑥) can be made arbitrarily large and hence for 𝜏>0 and 𝛽_𝑖(𝑥,𝑢)>0, for all 𝑖=1,2,…,𝑝, the inequality ^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑥)[−]𝑓_𝑖(𝑥)>0. (3.20) is obtained. Consequently, we ge from Lemmas 2.7 and 2.8 that ^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑥)>^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑥). (3.21) Now from feasibility conditions, we have 𝜆_𝑗[⋅]𝑔_𝑗(𝑥)≦𝜆_𝑗[⋅]𝑔_𝑗(𝑢),∀𝑗=1,…,𝑚. (3.22) Since 𝛿_𝑗(𝑥,𝑢)>0, for all 𝑗=1,…,𝑚, ^𝑚_𝑗=1𝛿_𝑗(𝑥,𝑢)[⋅]𝜆_𝑗[⋅]𝑔_𝑗(𝑥)≦^𝑚_𝑗=1𝛿_𝑗(𝑥,𝑢)[⋅]𝜆_𝑗[⋅]𝑔_𝑗(𝑢). (3.23) Suppose that the hypothesis (a′) holds at 𝑢, we can get from (ℎ,𝜑)-𝑉-quasi-invexity at 𝑢 of (𝜆₁[⋅]𝑔₁,𝜆₂[⋅]𝑔₂,…,𝜆[⋅]𝑔_𝑚)^𝑇 that ^𝑚_𝑗=1∇^∗𝜆_𝑗[⋅]𝑔_𝑗(𝑢)^𝑇𝜂(𝑥,𝑢)_ℎ,𝜑≦0. (3.24) Therefore, from (3.1), we get from Lemmas 2.5, 2.7, and 2.8 that ^𝑝_𝑖=1∇^∗𝜏_𝑖[⋅]𝑓_𝑖(𝑢)^𝑇𝜂(𝑥,𝑢)_ℎ,𝜑≧0. (3.25) Since (𝜏₁[⋅]𝑓₁,𝜏₂[⋅]𝑓₂,…,𝜏_𝑝[⋅]𝑓_𝑝)^𝑇 is (ℎ,𝜑)-𝑉-pseudoinvex and (𝜆₁[⋅]𝑔₁,𝜆₂[⋅]𝑔₂,…,𝜆_𝑚[⋅]𝑔_𝑚)^𝑇 is (ℎ,𝜑)-𝑉-quasi-invex at 𝑢, we have ^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑥)≧^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑢). (3.26) On using the assumption 𝑓(𝑥)=𝑓(𝑢) in the above equation, we get ^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑥)≧^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑥), (3.27) which is a contradiction to (3.21). Hence 𝑥 is a (ℎ,𝜑)-conditionally properly efficient solution for (MOP)_ℎ,𝜑.

We now suppose that (𝑢,𝜏,𝜆) is not (ℎ,𝜑)-conditionally properly efficient solution for (DMOP)_ℎ,𝜑. Therefore, for every positive function 𝑀(𝑥)>0, there exists a feasible (𝑢,𝜏,𝜆) feasible for (DMOP)_ℎ,𝜑 and an index 𝑖 such that 𝑓_𝑖(𝑢)[−]𝑓_𝑖(𝑢)>𝑀(𝑥)[⋅]𝑓_𝑗(𝑢)[−]𝑓_𝑗(𝑢), (3.28) for all 𝑗 satisfying 𝑓_𝑗(𝑢)[−]𝑓_𝑗(𝑢)>0 whenever 𝑓_𝑖(𝑢)[−]𝑓_𝑖(𝑢)<0. This means 𝑓_𝑖(𝑢)[−]𝑓_𝑖(𝑢) can be made arbitrarily large and hence for 𝜏>0 and 𝛽_𝑖(𝑥,𝑢)>0, for all  𝑖=1,2,…,𝑝, the inequality ^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑢)>^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑢) (3.29) is obtained. Since 𝑥 and (𝑢,𝜏,𝜆) are feasible for (MOP)_ℎ,𝜑 and (DMOP)_ℎ,𝜑, respectively, it follows that as in first part: ^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑢)≦^𝑝_𝑖=1𝛽_𝑖(𝑥,𝑢)[⋅]𝜏_𝑖[⋅]𝑓_𝑖(𝑢), (3.30) which contradicts (3.29). Hence, (𝑢,𝜏,𝜆) is (ℎ,𝜑)-conditionally properly efficient solution for (DMOP)_ℎ,𝜑.

Assuming that the hypothesis (b^′) holds, we can finish the proof with the similar argument.

Theorem 3.3 (strong duality). Let 𝑥 be an efficient solution for (MOP)_ℎ,𝜑. If the (ℎ,𝜑)-Kuhn-Tucker constraint qualification is satisfied, then there are 𝜏>0,𝜆≧0 such that (𝑥,𝜏,𝜆) is feasible for (DMOP)_ℎ,𝜑 and the objective values of (MOP)_ℎ,𝜑 and (DMOP)_ℎ,𝜑 are equal at 𝑥. Furthermore, if the hypothesis (a^′) or (b^′) of Theorem 3.2 hold at 𝑥, then (𝑥,𝜏,𝜆) is (ℎ,𝜑)-conditionally properly efficient for the problem (DMOP)_ℎ,𝜑.

Proof. Since 𝑥 is an efficient solution for (MOP)_ℎ,𝜑 at which the (ℎ,𝜑)-Kuhn-Tucker-type necessary conditions are satisfied, it follows from Lemma 2.12 that there exist 𝜏>0,𝜆≧0 such that (𝑥,𝜏,𝜆) is feasible for (DMOP)_ℎ,𝜑. Evidently, the objective values of (MOP)_ℎ,𝜑 and (DMOP)_ℎ,𝜑 are equal at 𝑥, since the objective functions for both problems are the same. The (ℎ,𝜑)-conditionally proper efficiency of (𝑥,𝜏,𝜆) for the problem (DMOP)_ℎ,𝜑 yields from Theorem 3.2.

Acknowledgment

This research is supported by Zizhu Science Foundation of Beifang University of Nationalities (no. 2011ZQY024); Natural Science Foundation for the Youth (no. 10901004); Natural Science Foundation of Ningxia (no. NZ12207); Ministry of Education Science and technology key projects (no. 212204).