JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 540746 10.1155/2012/540746 540746 Research Article Duality of (h,φ)-Multiobjective Programming Involving Generalized Invex Functions Yu GuoLin Shahzad Naseer Research Institute of Information and System Computation Science Beifang University of Nationalities Yinchuan 750021 China 2012 28 11 2012 2012 01 03 2012 02 10 2012 2012 Copyright © 2012 GuoLin Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , 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 .

In 1976, Ben-Tal  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, 1016]. The biggest advantage under Ben-Tal's generalized means is that the function has some transformable properties. As pointed out in literature  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 .

The properness of the efficient solution of the multiobjective programming problem is of importance. In 1991, Singh and Hanson  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 n be the n-dimensional Euclidean space and ++ be the set of all positive real numbers. Throughout this paper, the following convention for vector in n will be used: (2.1)x>yiffxi>yi,i=1,2,,n,xyiffxiyi,i=1,2,,n,xyiffxiyi,i=1,2,,n,but  xy.

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

Definition 2.1 (see [<xref ref-type="bibr" rid="B1">6</xref>, <xref ref-type="bibr" rid="B3">8</xref>]).

Let h:nn be a continuous vector function. Suppose that the inverse function h-1 of h exists. Then the h-vector addition of x,yn defined by (2.2)xy=h-1(h(x)+h(y)), and the h-scalar multiplication of xn and α is defined by (2.3)αx=h-1(αh(x)).

Similarly, generalized algebraic operations for scalar-valued functions can be defined as follows.

Definition 2.2 (see [<xref ref-type="bibr" rid="B1">6</xref>, <xref ref-type="bibr" rid="B3">8</xref>]).

Let φ: be a continuous and scalar function. Suppose that the inverse function φ-1 of φ exists. Then the φ-addition of α and β, is given by (2.4)α[+]β=φ-1(φ(α)+φ(β)), and the φ-scalar multiplication of α and β as (2.5)β[·]α=φ-1(βφ(α)).

Definition 2.3 (see [<xref ref-type="bibr" rid="B1">6</xref>, <xref ref-type="bibr" rid="B3">8</xref>]).

The (h,φ)-inner product of vector x,yRn is defined as (2.6)(xTy)h,φ=φ-1(h(x)Th(y)).

In this paper, we denote (2.7)i=1mxi=x1x2xm,      xiRn,i=1,2,,m,[i=1m]αi=α1[+]α2[+][+]αm,      αi,i=1,2,,m,α[-]β=α[+]((-1)[·]β),      α,β.

For the differentiability of a real-valued function in the setting of generalized algebraic means, Avriel  introduced the following important concept.

Definition 2.4 (see [<xref ref-type="bibr" rid="B1">6</xref>]).

Let f be a real-valued function defined on n, denote f^(t)=φ(f(h-1(t))),tRn. For simplicity, write f^(t)=φfh-1(t). The function f is said to be (h,φ)-differentiable at xRn, 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 Xn if it is (h,φ)-differentiable at each xX. A vector-valued function is called (h,φ)-differentiable on Xn if each of its components is (h,φ)-differentiable at each xX.

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 [<xref ref-type="bibr" rid="B12">12</xref>, <xref ref-type="bibr" rid="B14">14</xref>]).

Suppose that f,fi are real-valued functions defined on Rn, for i=1,2,,m, and (h,φ)-differentiable at x-Rn. Then, the following statements hold:

*(λ[·]f(x-))=λ*f(x-), for λ,

((i=1mλi*f(x-))Ty)h,φ=[i=1m](*(λi[·]f(x-))Ty)h,φ, for yRn, λi.

Lemma 2.6 (see [<xref ref-type="bibr" rid="B12">12</xref>, <xref ref-type="bibr" rid="B14">14</xref>]).

Let i=1,2,,m. The following statements hold:

λ[·](μ[·]α)=μ[·](λ[·]α)=λμ[·]α,for λ,μ,α;

λ[·](α[-]β)=λ[·]α[-]λ[·]β,for λ,α,β;

[i=1m](αi[-]βi)=[i=1m]αi[-][i=1m]βi, for αi,βi.

Lemma 2.7 (see [<xref ref-type="bibr" rid="B12">12</xref>, <xref ref-type="bibr" rid="B14">14</xref>]).

Suppose that function φ, which appears in Ben-Tal generalized algebraic operations, is strictly monotone with φ(0)=0. Then, the following statements hold:

let λ0, α,β, and αβ. Then λ[·]αλ[·]β;

let λ>0, α,β, and α<β. Then λ[·]α<λ[·]β;

let λ<0, α,β, and αβ. Then λ[·]αλ[·]β;

let αi,βi, i=1,2,,m. If αiβi for any iM, then (2.8)[i=1m]αi[i=1m]βi.

If αiβi for any i=1,2,,m, and there exists at least an index k such that xk<yk, then (2.9)[i=1m]αi<[i=1m]βi.

Lemma 2.8 (see [<xref ref-type="bibr" rid="B12">12</xref>, <xref ref-type="bibr" rid="B14">14</xref>]).

Suppose that φ is a continuous one-to-one strictly monotone and onto function with φ(0)=0. Let α,β. Then,

α<β if and only if α[-]β<0,

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 n and the functions f=(f1,,fp)T:Xp and g=(g1,,gm)T:Xm are (h,φ)-differentiable on the set X with respect to the same (h,φ). Consider the following (h,φ)-multiobjective programming problem: (MOP)h,φminf(x)=(f1(x),f2(x),,fp(x))T,xXns.t.  g(x)0.

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 xX.

Singh and Hanson  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 (2.10)fi(x-)[-]fi(x)fj(x)[-]fj(x-)M(x), for some j such that fj(x)>fj(x-), whenever xX and (2.11)fi(x)<fi(x-).

Example 2.11.

Consider the following multiobjective problem: (MOP)h,φminf(x)=(f1(x),f2(x))T=(x1x2,x2x1)Ts.t.  g(x)=(g1(x),g2(x))T=(x1,x2)T1x=(x1,x2)T2. 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 (2.12)f2(x*)[-]f2(x)f1(x)[-]f1(x*)=(b/a)3-(x2/x1)33(x1/x2)3-(a/b)33=bx2ax1M(x), for j=2 such that f2(x)=x2/x1>b/a=f2(x*) whenever x=(x1,x2)T is feasible and (2.13)f1(x)=x1x2<ab=f1(x*). Thus, x* is (h,φ)-conditionally proper efficient solution.

Xu and Liu  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 . We now state this result as the following (Lemma 2.12).

Lemma 2.12 (Kuhn-Tucker-type necessary condition).

Let fi for i=1,2,,p, gj for j=1,2,,m be (h,φ)-differentiable on 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-=(τ-1,τ-2,,τ-p)T>0 and λ-=(λ-1,λ-2,,λ-m)T0 such that (2.14)(i=1mτ-i*fi(x-))(j=1mλ-j*gj(x-))=0,λ-j[·]gj(x-)=0,j=1,2,,m.

Jeyakumar and Mond  introduced the notion of V-invexity for a vector function f=(f1,f2,,fp) 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:Xnp is said to be (h,φ)-V-invex at x-X if there exist functions η:X×Xn and αi:X×X++ such that for each xX and for i=1,2,,p, (2.15)fi(x)[-]fi(x-)αi(x,x-)[·](*fi(x-)Tη(x,x-))h,φ.

If we take h and φ as the identity functions, the above definitions reduce to the V-invex function given by Jeyakumar and Mond .

Example 2.14.

The functions f:2, 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 αi(x,x-), i=1,2.

Definition 2.15.

A vector function f:XnRp is said to be (h,φ)-V-pseudoinvex at x-X if there exist functions η:X×XRn and βi:X×X++ such that for each xX and for i=1,2,,p, (2.16)[i=1p](*fi(x-)Tη(x,x-))h,φ0[i=1p]βi(x,x-)[·]fi(x)[i=1p]βi(x,x-)[·]fi(x-). If in the above definition xx- and (2.16) is satisfied as (2.17)[i=1p](*fi(x-)Tη(x,x-))h,φ0[i=1p]βi(x,x-)[·]fi(x)>[i=1p]βi(x,x-)[·]fi(x-), 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 βi(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 (2.18)0=[i=12](*fi(1)T0)h,φ0, and for x(0,1], it follows that (2.19)f1(x)[-]f1(1)=tan(arctan(cos2(x))-arctan(cos2(1)))=cos2(x)-cos2(1)1+cos2(x)cos2(1)0,f2(x)[-]f2(1)=tan(arctan(-sin2(x))-arctan(-sin2(1)))=-sin2(x)-(-sin2(1))1+sin2(x)sin2(1)0. Thus, we get from Lemmas 2.6 and 2.7 that (2.20)[i=12]βi(x,x-)[·]fi(x)[i=12]βi(x,x-)[·]fi(x-). By Definition 2.15, we have shown that f is (h,φ)-V-pseudoinvex at x-=1.

Definition 2.17.

A vector function f:XnRp is said to be (h,φ)-V-quasi-invex at x-X if there exist functions η:X×XRn and δi:X×XR++ such that for each xX and for i=1,2,,p, (2.21)[i=1p]δi(x,x-)[·]fi(x)[i=1p]δi(x,x-)[·]fi(x-)[i=1p](*fi(x-)Tη(x,x-))h,φ0.

Example 2.18.

The function f: 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-)=xx- 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: (DMOP)h,φmaxf(u)=(f1(u),f2(u),,fm(u))T(3.1)s.t. (i=1pτi*fi(u))(j=1pλj*gj(u))=0,(3.2)λj[·]gj(u)0,(3.3)τ>0,τ=(τ1,τ2,,τp)T,(3.4)λ0,λ=(λ1,λ2,,λm)T(3.5)uXn.

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:

(τ1[·]f1,τ2[·]f2,,τp[·]fp)T is (h,φ)-V-pseudoinvex and (λ1[·]g1,λ2[·]g2,,λm[·]gm)T is (h,φ)-V-quasi-invex at u with respect to same η;

(τ1[·]f1,τ2[·]f2,,τp[·]fp)T is (h,φ)-V-quasi-invex and (λ1[·]g1,λ2[·]g2,,λm[·]gm)T is strictly (h,φ)-V-pseudoinvex at u with respect to same η. Then (3.6)f(x)f(u).

Proof.

Since (u,τ,λ) is a feasible solution for (DMOP)h,φ, by Lemma 2.5 and (3.1), for all x'n we obtain that (3.7)[i=1p](*(τi[·]fi(u))Tη(x,u))h,φ[+][j=1m](*(λj[·]gj(u))Tη(x,u))h,φ=0.

Let x be feasible for (MOP)h,φ and f(x)f(u). Since τ>0 and βi(x,u)>0,  for all  i=1,,p, it follows from Lemmas 2.6 and 2.7 that (3.8)[i=1p]βi(x,u)[·]τi[·]fi(x)<[i=1p]βi(x,u)[·]τi[·]fi(u), and (h,φ)-V-pseudoinvexity at u of (τ1[·]f1,,τp[·]fp)T implies (3.9)[i=1p](*(τi[·]fi(u))Tη(x,u))h,φ<0. 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 (3.10)λj[·]gj(u)0λj[·]gj(x),j=1,2,m. Again, since δj(x,u)>0, for all j=1,2,,m, it follows from Lemma 2.7 that (3.11)[j=1m]δj(x,u)[·]λj[·]gj(x)[j=1m]δj(x,u)[·]λj[·]gj(u). Now, (h,φ)-V-quasi-invexity at u of (λ1[·]g1,,λm[·]gm)T implies that (3.12)[j=1m](*(λj[·]gj(u))Tη(x,u))h,φ0. Together with (3.9) and (3.12), it yields from Lemma 2.7 that (3.13)[i=1p](*(τi[·]fi(u))Tη(x,u))h,φ[+][j=1m](*(λj[·]gj(u))Tη(x,u))h,φ<0, which contradicts to (3.7)

Let x be feasible for (MOP)h,φ and (u,τ,λ) feasible for (DMOP)h,φ. Suppose that f(x)f(u). Since δi(x,u)>0, for all  i=1,,p, and τ>0, we get from Lemmas 2.6 and 2.7 that (3.14)[i=1p]δi(x,u)[·]τi[·]fi(x)<[i=1p]δi(x,u)[·]τi[·]fi(u). The (h,φ)-V-quasi-invexity at u of (τ1[·]f1,,τp[·]fp)T implies that (3.15)[i=1p](*(τi[·]fi(u))Tη(x,u))h,φ0. By (3.7), we get from Lemmas 2.7 and 2.8 that (3.16)[j=1m](*(λj[·]gj(u))Tη(x,u))h,φ0. and since (λ1[·]g1,,λm[·]gm)T is strictly (h,φ)-V-pseudoinvex, we have (3.17)[j=1m]β(x,u)j[·]λj[·]gj(x)>[j=1m]β(x,u)j[·]λj[·]gj(u). According to Lemma 2.7, this is a contradiction, since λj[·]gj(x)0, λj[·]gj(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:

(τ-1[·]f1,τ-2[·]f2,,τ-p[·]fp)T is (h,φ)-V-pseudoinvex and (λ-1[·]g1,λ-2[·]g2,,λ-m[·]gm)T is (h,φ)-V-quasi-invex at u- with respect to same η;

(u-1[·]f1,u-2[·]f2,,u-p[·]fp)T is (h,φ)-V-quasi-invex and (λ-1[·]g1,λ-2[·]g2,,λ-m[·]gm)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 (3.18)f(x)f(x-). 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 (3.19)fi(x-)[-]fi(x^)>M(x)[·](fj(x^)[-]fj(x-)), for all j satisfying fj(x^)>fj(x-), whenever fi(x^)<fi(x-). This shows that fi(x-)[-]fi(x^) can be made arbitrarily large and hence for τ->0 and βi(x^,u-)>0, for all i=1,2,,p, the inequality (3.20)[i=1p]βi(x^,u-)[·]τ-i[·](fi(x-)[-]fi(x^))>0. is obtained. Consequently, we ge from Lemmas 2.7 and 2.8 that (3.21)[i=1p]βi(x^,u-)[·]τ-i[·]fi(x-)>[i=1p]βi(x^,u-)[·]τ-i[·]fi(x^). Now from feasibility conditions, we have (3.22)λ-j[·]gj(x^)λ-j[·]gj(u-),j=1,,m. Since δj(x^,u-)>0, for all j=1,,m, (3.23)[j=1m]δj(x^,u-)[·]λ-j[·]gj(x^)[j=1m]δj(x^,u-)[·]λ-j[·]gj(u-). Suppose that the hypothesis (a') holds at u-, we can get from (h,φ)-V-quasi-invexity at u- of (λ-1[·]g1,λ-2[·]g2,,λ-[·]gm)T that (3.24)[j=1m](*(λ-j[·]gj(u-))Tη(x^,u-))h,φ0. Therefore, from (3.1), we get from Lemmas 2.5, 2.7, and 2.8 that (3.25)[i=1p](*(τ-i[·]fi(u-))Tη(x^,u-))h,φ0. Since (τ-1[·]f1,τ-2[·]f2,,τ-p[·]fp)T is (h,φ)-V-pseudoinvex and (λ-1[·]g1,λ-2[·]g2,,λ-m[·]gm)T is (h,φ)-V-quasi-invex at u-, we have (3.26)[i=1p]βi(x^,u-)[·]τ-i[·]fi(x^)[i=1p]βi(x^,u-)[·]τ-i[·]fi(u-). On using the assumption f(x-)=f(u-) in the above equation, we get (3.27)[i=1p]βi(x^,u-)[·]τ-i[·]fi(x^)[i=1p]βi(x^,u-)[·]τ-i[·]fi(x-), 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 (3.28)fi(u^)[-]fi(u-)>M(x)[·](fj(u-)[-]fj(u^)), for all j satisfying fj(u^)[-]fj(u-)>0 whenever fi(u^)[-]fi(u-)<0. This means fi(u^)[-]fi(u-) can be made arbitrarily large and hence for τ->0 and βi(x^,u-)>0, for all  i=1,2,,p, the inequality (3.29)[i=1p]βi(x^,u-)[·]τ-i[·]fi(u^)>[i=1p]βi(x^,u-)[·]τ-i[·]fi(u-) 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: (3.30)[i=1p]βi(x^,u-)[·]τ-i[·]fi(u^)[i=1p]βi(x^,u-)[·]τ-i[·]fi(u-), 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).

Hanson M. A. On sufficiency of the Kuhn-Tucker conditions Journal of Mathematical Analysis and Applications 1981 80 2 545 550 10.1016/0022-247X(81)90123-2 614849 ZBL0463.90080 Jeyakumar V. Mond B. On generalised convex mathematical programming Australian Mathematical Society B 1992 34 1 43 53 10.1017/S0334270000007372 1168574 ZBL0773.90061 Mishra S. K. On multiple-objective optimization with generalized univexity Journal of Mathematical Analysis and Applications 1998 224 1 131 148 10.1006/jmaa.1998.5992 1632974 ZBL0911.90292 Mishra S. K. Giorgi G. Wang S. Y. Duality in vector optimization in Banach spaces with generalized convexity Journal of Global Optimization 2004 29 4 415 424 10.1023/B:JOGO.0000047911.03061.88 2104407 ZBL1119.90049 Mishra S. K. Noor M. A. Some nondifferentiable multiobjective programming problems Journal of Mathematical Analysis and Applications 2006 316 2 472 482 10.1016/j.jmaa.2005.04.067 2206684 ZBL1148.90012 Avriel M. Nonlinear Programming 1976 Englewood Cliffs, NJ, USA Prentice-Hall xv+512 0489892 ZBL0361.90035 Dos Santos L. B. Osuna-Gómez R. Rojas-Medar M. A. Rufián-Lizana A. Preinvex functions and weak efficient solutions for some vectorial optimization problem in Banach spaces Computers & Mathematics with Applications 2004 48 5-6 885 895 10.1016/j.camwa.2003.05.013 2105260 ZBL1095.90105 Ben-Tal A. On generalized means and generalized convex functions Journal of Optimization Theory and Applications 1977 21 1 1 13 0442166 10.1007/BF00932539 ZBL0325.26007 Singh C. Hanson M. A. Generalized proper efficiency in multiobjective programming Journal of Information & Optimization Sciences 1991 12 1 139 144 1108295 ZBL0738.90065 Xu Y. Liu S. Kuhn-Tucker necessary conditions for (h,φ)-multiobjective optimization problems Journal of Systems Science and Complexity 2004 17 4 472 484 2106778 Xu Y. H. Liu S. Y. Some properties for (h,φ)-generalized invex functions and optimality and duality of (h,φ)-generalized invex multiobjective programming Acta Mathematicae Applicatae Sinica 2003 26 4 726 736 2047135 Yu G. Some (h,φ)-differentiable multiobjective programming problems International Journal of Optimization: Theory, Methods and Applications 2009 136 157 Yu G. Liu S. Some vector optimization problems in Banach spaces with generalized convexity Computers & Mathematics with Applications 2007 54 11-12 1403 1410 10.1016/j.camwa.2007.05.002 2368222 ZBL1137.90663 Yu G. Liu S. Optimality for (h,φ)-multiobjective programming involving generalized type-I functions Journal of Global Optimization 2008 41 1 147 161 10.1007/s10898-007-9196-3 2386601 Zhang Q. X. On sufficiency and duality of solutions of nonsmooth (h,φ)-semi-infinite programming problems Acta Mathematicae Applicatae Sinica 2001 24 1 129 138 1831967 Yuan D. Chinchuluun A. Liu X. Pardalos P. M. Generalized convexities and generalized gradients based on algebraic operations Journal of Mathematical Analysis and Applications 2006 321 2 675 690 10.1016/j.jmaa.2005.08.093 2241147 ZBL1093.49014