# Optimality Conditions and Duality of Three Kinds of Nonlinear Fractional Programming Problems

Authors: Xiaomin Zhang; Zezhong Wu
Journal: Advances in Operations Research
 (2013)

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## Abstract

Some assumptions for the objective functions and constraint functions are given under the conditions of convex and generalized convex, which are based on theF-convex, ρ-convex, and (F,ρ)-convex. The sufficiency of Kuhn-Tucker optimality conditions and appropriate duality results are proved involving (F,ρ)-convex, (F,α,ρ,d)-convex, and generalized (F,α,ρ,d)-convex functions.


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## Body

## 1. Introduction

Multiobjective optimization theory is a development of numerical optimization and related to many subjects, such as nonsmooth analysis, convex analysis, nonlinear analysis, and the theory of set value. It has a wide range of applications in the fields of industrial design, economics, engineering, military, management sciences, financial investment, transport, and so forth, and now it is an interdisciplinary science branch between applied mathematics and decision sciences. Convexity plays an important role in optimization theory, and it becomes an important theoretical basis and useful tool for mathematical programming and optimization theory.Convex function theory can be traced back to the works of Holder, Jensen, and Minkowski in the beginning of this century, but the real work that caught the attention of people is the research on game theory and mathematical programming by von Neumann and Morgenstern [1], Dantzing, and Kuhn and Tucker in the forties to fifties, and people have done a lot of intensive research about convex functions from the fifties to sixties. In the middle of the sixties convex analysis was produced, and the concept of convex function is promoted in a variety of ways, and the notion of generalized convex is given.Fractional programming has an important significance in the optimization problems; for instance, in order to measure the production or the efficiency of a system, we should minimize a ratio of functions between a given period of time and a utilized resource in engineering and economics.Preda [2] has established the concept of (F,ρ)-convex based on F-convex [3] and ρ-convex [4] and obtained some results, which are the expansion of F-convex and ρ-convex. Motivated by various concepts of convexity, Liang et al. [5] have put forward a generalized convexity, which was called (F,α,ρ,d)-convex, which extended (F,ρ)-convex, and Liang et al. [6], Weir and Mond [7], Weir [8], Jeyakumar and Mond [9], Egudo [10], Preda [2], and Gulati and Islam [3] obtained some corresponding optimality conditions and applied these optimality conditions to define dual problems and derived duality theorems for single objective fractional problems and multiobjective problems. Then the definition of generalized (F,α,ρ,d)-convex is given under the condition of (F,α,ρ,d)-convex. However, in general, fractional programming problems are nonconvex and the Kuhn-Tucker optimality conditions are only necessary. Under what conditions are the Kuhn-Tucker conditions sufficient for the optimality of problems? This question appeals to the interests of many researchers, and those are what we should probe. Based on the former conclusions, by adding conditions to objective functions and constraint functions and by changing K-T conditions [11], the optimality conditions and dual are given involving weaker convexity conditions. The main results in this paper are based on convex and generalized convex functions and the properties of sublinear functions.In this paper, we will discuss sufficient optimality conditions and dual problems for three kinds of nonlinear fractional programming problems, and the paper is organized as follows.In Sections3.1 and 3.2, we present the Kuhn-Tucker sufficient optimality conditions and dual for nonlinear fractional programming problem and multiobjective fractional programming problem based on generalized (F,α,ρ,d)-convex. Section 3.3 contains optimality conditions and dual for multiobjective fractional programming problem under (F,ρ)-convex. In these sections, I present some assumptions for the objective functions and constraint functions such that the Kuhn-Tucker optimality conditions are sufficient and obtain the corresponding duality theorem.


## 2. Preliminaries

LetEn be the n-dimensional real vector space, that is, n-dimensional Euclidean space, where y=(y1,y2,…,yn)T, z=(z1,z2,…,zn)T∈En, and provides as follows, (see [12]),
(1)y=z⟺yi=zi,i=1,2,…,n;y>z⟺yi>zi,i=1,2,…,n;y≧z⟺yi≧zi,i=1,2,…,n.Definition 1 (see [12]).
Suppose thatx0∈X0; that is, if x∉X0, such that f(x)≦f(x0), x0 is an efficient solution of multiobjective programming problem.Definition 2 (see [12]).
Suppose thatx0∈X0; that is, if x∉X0, such that f(x)<f(x0), x0 is a weakly efficient solution of multiobjective programming problem.Definition 3 (see [5]).
Given an open setX0⊂Rn, a functional F:X0×X0×Rn→R is called sublinear if, for any x, x0∈X0,
(2)F(x,x0;a1+a2)≦F(x,x0;a1)+F(x,x0;a2),∀a1,a2∈Rn,F(x,x0;αa)=αF(x,x0;a),∀α∈R,α≧0,∀a∈Rn.
It follows from the second equality that
(3)F(x,x0;0)=F(x,x0;0×a)=0×F(x,x0;a)=0,foranya∈Rn.
LetF:X0×X0×Rn→R be a sublinear function, and let α:X0×X0→R+∖{0}, ρ=(ρ1,ρ2,…,ρm)T, ρi∈R, d:X0×X0→R, and the function f=(f1,f2,…,fm):X0→Rm is differentiable at x0∈X0.Definition 4 (see [3]).
Letϕ(x) be a differentiable function defined on X0⊂En. The function ϕ(x) is said to be F-convex on X0 with respect to F, if ϕ(x)-ϕ(y)≧Fx,y[∇ϕ(y)].Definition 5 (see [4, 12]).
Letf(x) be a real-valued function defined on the convex set X0⊂En, if there exists a real number ρ∈R, such that
(4)f(λx1+(1-λ)x2)≦λf(x1)+(1-λ)f(x2)-ρλ(1-λ)∥x1-x2∥2
for any x1,x2∈X0 and any λ∈[0,1], then the function f(x) is said to be ρ-convex on X0.
Especially, ifρ=0, then we obtain the definition of convex.
Ifρ>0(orρ<0) in the above definition, then we have strong convex (or weak convex).Definition 6 (see [2]).
The functionfi:X0→R is said to be (F,ρ)-convex at x0∈X0, if for any x0∈X0, fi(x) satisfies the following condition:
(5)fi(x)-fi(x0)≧F(x,x0;∇fi(x0))+ρid2(x,x0).Definition 7 (see [5]).
The functionfi is said to be (F,α,ρi,d)-convex at x0∈X0, if
(6)fi(x)-fi(x0)≧F(x,x0;α(x,x0)∇fi(x0))+ρid2(x,x0),∀x∈X0.
The functionf is said to be (F,α,ρ,d)-convex at x0, if each component fi of f is (F,α,ρi,d)-convex at x0.
The functionf is said to be (F,α,ρ,d)-convex on X0, if it is (F,α,ρ,d)-convex at every point in X0.Definition 8.
The functionfi is said to be (F,α,ρi,d)-quasiconvex at x0, if fi(x)≦fi(x0)⇒F(x,x0;α(x,x0)∇fi(x0))≦-ρid2(x,x0).
The functionf is said to be (F,α,ρ,d)-quasiconvex at x0, if each component fi of f is (F,α,ρi,d)-quasiconvex at x0.Definition 9.
The functionfi is said to be (F,α,ρi,d)-pseudoconvex at x0, if for all x∈X0, fi(x)<fi(x0)⇒F(x,x0;α(x,x0)∇fi(x0))<-ρid2(x,x0).
The functionf is said to be (F,α,ρ,d)-pseudoconvex at x0, if each component fi of f is (F,α,ρi,d)-pseudoconvex at x0.Definition 10.
The functionf is said to be strictly (F,α,ρi,d)-pseudoconvex at x0∈X0, if f(x)≦f(x0)⇒F(x,x0;α(x,x0)∇f(x0))<-ρd2(x,x0), where F(x,x0;α(x,x0)∇f(x0))=(F(x,x0;α(x,x0)∇f1(x0)),…,F(x,x0;α(x,x0)∇fm(x0))).
Further,f is said to be weakly strictly (F,α,ρ,d)-pseudoconvex at x0∈X0, if f(x)≦f(x0)⇒F(x,x0;α(x,x0)∇f(x0))<-ρd2(x,x0).In order to prove our main result, we need a lemma which we present in this section.Lemma 11 (see [13]).
Suppose that differentiable real-valued functionshj(x)(j=1,2,…,m) are (F,α,ρj,d)-quasiconvex at x-∈S; then VTh(x) is (F,α,∑j=1mvjρj,d)-quasiconvex at x-∈S, where ρj∈R, V≧0 and VT denote the transpose of the m-dimensional column vector V; that is, VT=(v1,v2,…,vm).


## 3. Optimality Conditions and Duality

### 3.1. Nonlinear Fractional Programming Problem Involved Inequality and Equality Constraints Based on Generalized(F,α,ρ,d)-Convex

Consider the nonlinear fractional programming problem(FP)(7)minf(x)g(x)s.t.h(x)≦0,l(x)=0,x∈X0,
where X0 is an open set of Rn, f(x) and g(x) are real-valued functions defined on X0, h(x) is an m-dimensional vector-valued functions defined also on X0, and l(x) a q-dimensional vector-valued function.Let(8)S={x∈X0∣h(x)≦0,l(x)=0}
denotes the set of all feasible solutions for (FP) and assume that f(x), g(x), hj(x)(j=1,2,…,m), and li(x)(i=1,2,…,q) are continuously differentiable over X0 and that f(x)≧0, g(x)>0, for all x∈X0.Ifx-∈X0 is a solution for problem (FP) and if a constraint qualification [14] holds, then the Kuhn-Tucker necessary conditions are given below: there exists V0∈Rm and W0∈Rq such that
(9)∇(f(x-)g(x-))+∇h(x-)V0+∇l(x-)W0=0,V0Th(x-)=0,V0≧0,h(x-)≦0,l(x-)=0.Theorem 12.
Suppose thatx- is a feasible solution of (FP), that the Kuhn-Tucker conditions hold at x-, that f(x)/g(x) in problem (FP) is (F,α,ρ,d)-pseudoconvex on S, hj(x)(j=1,2,…,m) are (F,α,ρj,d)-quasiconvex on S, and that li(x)(i=1,2,…,q) are (F,α,ρi,d)-quasiconvex over S, ρ, ρi, ρj∈R, ρ+V0Tρ′+W0Tρ′′≧0, where V0Tρ′ is the inner product about V0 and ρ′ and W0Tρ′′ the inner product about W0 and ρ′′. Then, x- is an optimality solution for problem (FP).Proof.
Suppose thatx- is not an optimality solution of (FP). Then, there exists a feasible solution x∈S such that f(x)/g(x)<f(x-)/g(x-).
By the(F,α,ρ,d)-pseudoconvexity assumption of f(x)/g(x), we have
(10)F(x,x-;α(x,x-)∇(f(x-)g(x-)))<-ρd2(x,x-).
For each j(j=1,2,…,m), by the (F,α,ρj,d)-quasiconvexity assumption of hj(x) and Lemma 11, we have that V0Th(x) is (F,α,∑j=1mvjρj,d)-quasiconvex on S. Therefore,
(11)F(x,x-;α(x,x-)∇h(x-)V0)≦-∑j=1m‍ρjvjd2(x,x-),
that is, F(x,x-;α(x,x-)∇h(x-)V0)≦-V0Tρ′d2(x,x-).
By the(F,α,ρi,d)-quasiconvexity of li(x)(i=1,2,…,q) over S, we have that W0Tl(x) is (F,α,∑i=1qwiρi,d)-quasiconvexity on S.
Then we obtainF(x,x-;α(x,x-)∇l(x-)W0)≦-∑i=1qρiwid2(x,x-), that is,
(12)F(x,x-;α(x,x-)∇l(x-)W0)≦-W0Tρ′′d2(x,x-).
By (10), (11), and (12), and based on the sublinearity of F, we have
(13)F(x,x-;α(x,x-)(∇(f(x-)g(x-))+∇h(x-)V0+∇l(x-)W0))<-(ρ+V0Tρ′+W0Tρ′′)d2(x,x-).
Considering that ρ+V0Tρ′+W0Tρ′′≧0, we get
(14)F(x,x-;α(x,x-)(∇(f(x-)g(x-))+∇h(x-)V0+∇l(x-)W0))<0.
By the K-T conditions, we have ∇(f(x-)/g(x-))+∇h(x-)V0+∇l(x-)W0=0.
Hence, based on the sublinearity ofF, we obtain
(15)F(x,x-;α(x,x-)(∇(f(x-)g(x-))+∇h(x-)V0+∇l(x-)W0))=0,
which contradicts (14). The proof is complete.Consider the dual problem of(FP):
(FD)maxf(y)g(y)+uTh(y)+vTl(y)s.t.λT∇(f(y)g(y))+uT∇h(y)+vT∇l(y)=0,uTh(y)≧0,vTl(y)=0,u≧0,v≧0.Theorem 13.
Suppose thatλT(f(y)/g(y)) is (F,α,ρ1,d)-pseudoconvex at y, uTh(y)+vTl(y) is (F,α,ρ2,d)-quasiconvex at y in problem (FP) and (FD), and that ρ1+ρ2≧0; then λT(f(x)/g(x))≧λT(f(y)/g(y)), for any feasible solution x of (FP) and (y,λ,u,v) of (FD).Proof.
Assume that the conclusion is not true; that is,λT(f(x)/g(x))<λT(f(y)/g(y)).
By the(F,α,ρ1,d)-pseudoconvex of λT(f(x)/g(x)) at y, we get
(16)F(x,y;α(x,y)λT∇(f(y)g(y)))<-ρ1d2(x,y).
Using uTh(x)≦0, vTl(x)=0, uTh(y)≧0, vTl(y)=0, we have
(17)uTh(x)+vTl(x)≦uTh(y)+vTl(y).
By the (F,α,ρ2,d)-quasiconvex of uTh(y)+vTl(y), we get
(18)F(x,y;α(x,y)(uT∇h(y)+vT∇l(y)))≦-ρ2d2(x,y).
By (16) and (18) and based on the sublinearity of F, we have
(19)F(x,y;α(x,y)(λT∇(f(y)g(y))+(f(y)g(y))uT∇h(y)+vT∇l(y)))<-(ρ1+ρ2)d2(x,y).
Since (y,λ,u,v) is the feasible solution of (FD), so
(20)λ-T∇(f(y)g(y))+uT∇h(y)+vT∇l(y)=0.
Hence,
(21)0=F(x,y;α(x,y)(λT∇(f(y)g(y))+(f(y)g(y))uT∇h(y)+vT∇l(y)))<-(ρ1+ρ2)d2(x,y),
which contradicts the known condition ρ1+ρ2≧0. The proof is complete.


### 3.2. Nonlinear Multiobjective Fractional Programming Problem Involved Inequality and Equality Constraints Based on(F,α,ρ,d)-Convex and Generalized (F,α,ρ,d)-Convex

Consider the nonlinear multiobjective fractional programming problem(VFP)(22)minM(x)=f(x)g(x)=[f1(x)g(x),f2(x)g(x),…,fp(x)g(x)]s.t.h(x)≦0,x∈X0l(x)=0,x∈X0,
where X0 is an open set of Rn, fi(x)(i=1,2,…,p), g(x), hj(x):X0→R(j=1,2,…,m) are real-valued functions defined on X0, lk(x):X0→R(k=1,2,…,q) are real-valued functions defined also on X0, f(x)=(f1(x),f2(x),…,fp(x))T, h(x)=(h1(x),h2(x),…,hm(x))T, and l(x)=(l1(x),l2(x),…,lq(x))T.Let(23)S={x∣x∈X0,h(x)≦0,l(x)=0}
denotes the set of all feasible solutions of (VFP) and assume that fi(x)(i=1,2,…,p), g(x), hj(x)(j=1,2,…,m), and lk(x)(k=1,2,…,q) are continuously differentiable over X0 and that g(x)>0, for all x∈X0.Theorem 14.
Suppose thatf(x)/g(x) is weakly strictly (F,α,ρ1,d)-pseudoconvex at x-∈X0, hj(x)(j=1,2,…,m) are (F,α,ρ2,d)-quasiconvex with respect to x-∈X0, lk(x)(k=1,2,…,q) are (F,α,ρ3,d)-convex with respect to x-∈X0, and that there exists λ-∈∧++ (or ∧+), u-∈R+m, v-∈R+q satisfying
(24)∑i=1p‍λ-i∇(f(x-)g(x-))+∑j=1m‍u-j∇hj(x-)+∑k=1q‍v-k∇lk(x-)=0,u-jhj(x-)=0,hj(x)≦0,j=1,2,…,m,lk(x)=0,k=1,2,…,q
and ρ=∑i=1pλ-iρ1+∑j=1mu-jρ2+∑k=1qv-kρ3≧0. Then, x- is an efficient solution of (VFP), where
(25)∧+={λ-=(λ-1,λ-2,…,λ-p)T∣λ-i≧0,i=1,2,…,p·∑i=1p‍λ-i=1},∧++={λ-=(λ-1,λ-2,…,λ-p)T∣λ-i>0,i=1,2,…,p·∑i=1p‍λ-i=1}.Proof.
Suppose thatx- is not an efficient solution of (VFP); then there exists a feasible solution x∈X0 such that M(x)≦M(x-), that is, f(x)/g(x)≦f(x-)/g(x-).
By the weakly strict(F,α,ρ1,d)-pseudoconvexity of f(x)/g(x) at x-∈X0, we get
(26)F(x,x-;α(x,x-)∇(f(x-)g(x-)))<-ρ1d2(x,x-).
Using λ-∈∧++ (or ∧+), then we have
(27)∑i=1p‍λ-iF(x,x-;α(x,x-)∇(f(x-)g(x-)))<-∑i=1p‍λ-iρ1d2(x,x-).
Based on the sublinearity of F, we obtain
(28)F(x,x-;α(x,x-)∑i=1p‍λ-i∇(f(x-)g(x-)))<-∑i=1p‍λ-iρ1d2(x,x-).
Since u-Th(x-)=0, u-≧0 and hj(x)≦0, we have u-Th(x)-u-Th(x-)≦0; that is, u-Th(x)≦u-Th(x-).
Sincehj(x)(j=1,2,…,m) are (F,α,ρ2,d)-quasiconvex at x-∈X0, by Lemma 11, we have u-Th is (F,α,ρ2∑j=1mu-j,d)-quasiconvex at x-∈X0.
Hence, we obtain the following inequality:(29)F(x,x-;α(x,x-)∑j=1m‍u-j∇hj(x-))≦-∑j=1m‍u-jρ2d2(x,x-).
By the (F,α,ρ3,d)-convexity of lk(x) at x-∈X0, we have
(30)lk(x)-lk(x-)≧F(x,x-;α(x,x-)∇lk(x-))+ρ3d2(x,x-).
Since v-≧0, we have
(31)0=v-l(x)-v-l(x-)≧∑k=1q‍v-kF(x,x-;α(x,x-)∇lk(x-))+∑k=1q‍v-kρ3d2(x,x-).
By the sublinearity of F, we obtain
(32)F(x,x-;α(x,x-)∑k=1q‍v-k∇lk(x-))≦-∑k=1q‍v-kρ3d2(x,x-).
By the known conditions, we have
(33)F(x,x-;α(x,x-)(∑i=1p‍λ-i∇(f(x-)g(x-))+∑j=1m‍u-j∇hj(x-)+∑k=1q‍v-k∇lk(x-)))=0.
By (28), (29), and (32) and by the sublinearity of F, we obtain
(34)F(x,x-;α(x,x-)(∑i=1p‍λ-i∇(f(x-)g(x-))+∑j=1m‍u-j∇hj(x-)+∑k=1q‍v-k∇lk(x-)))<-(∑i=1p‍λ-iρ1+∑j=1m‍u-jρ2+∑k=1q‍v-kρ3)d2(x,x-)=-ρd2(x,x-)≦0
which contradicts the fact of (33). Therefore, x- is an efficient solution of (VFP). The proof is complete.Consider the dual problem of(VFP)(35)max(f1(y)g(y)+uTh(y)+vTl(y),…,fp(y)g(y)+uTh(y)+vTl(y))s.t.∑i=1p‍λi∇(fi(y)g(y))+∑j=1m‍uj∇hj(y)+∑k=1q‍vk∇lk(y)=0λ∈∧++,u∈R+m,v∈R+q.Theorem 15.
(
f
i
/
g
)
(
i
=
1,2
,
…
,
p
) is (F,α,ρ1,d)-convex at y, hj(j=1,2,…,m) is (F,α,ρ2,d)-convex at y, lk(k=1,2,…,q) is (F,α,ρ3,d)-convex at y, and ∑i=1pλiρ1+∑j=1mujρ2+∑k=1qvkρ3≧0, then
(36)∑i=1p‍λi(fi(x)g(x))≧∑i=1p‍λi(fi(y)g(y))+uTh(y)+vTl(y).Proof.
By the(F,α,ρ1,d)-convex of fi/g at y, we get
(37)fi(x)g(x)-fi(y)g(y)≧F(x,y;α(x,y)∇(fi(y)g(y)))+ρ1d2(x,y),i=1,2,…,p.
By the (F,α,ρ2,d)-convex of hj at y, we get
(38)hj(x)-hj(y)≧F(x,y;α(x,y)∇hj(y))+ρ2d2(x,y),j=1,2,…,m.
By the (F,α,ρ3,d)-convex of lk at y, we get
(39)lk(x)-lk(y)≧F(x,y;α(x,y)∇lk(y))+ρ3d2(x,y),k=1,2,…,q.
Since λ∈∧++, u∈R+m, v∈R+q, and by the previous three inequalities, we have that
(40)(∑i=1p‍λi(fi(x)g(x))+uTh(x)+vTl(x))-(∑i=1p‍λi(fi(y)g(y))+uTh(y)+vTl(y))≧∑i=1p‍λiF(x,y;α(x,y)∇(fi(y)g(y)))+∑j=1m‍ujF(x,y;α(x,y)∇hj(y))+∑k=1q‍vkF(x,y;α(x,y)∇lk(y))+(∑i=1p‍λiρ1+∑j=1m‍ujρ2+∑k=1q‍vkρ3)d2(x,y).
By the sublinearity of F, we obtain
(41)(∑i=1p‍λi(fi(x)g(x))+uTh(x)+vTl(x))-(∑i=1p‍λi(fi(y)g(y))+uTh(y)+vTl(y))≧F(x,y;α(x,y)(∑i=1p‍λi∇(fi(y)g(y))+∑j=1m‍uj∇hj(y)+∑k=1q‍vk∇lk(y)(fi(y)g(y))))+(∑i=1p‍λiρ1+∑j=1m‍ujρ2+∑k=1q‍vkρ3)d2(x,y).
By the feasibility of (y,λ,u,v), we have
(42)∑i=1p‍λi∇(fi(y)g(y))+∑j=1m‍uj∇hj(y)+∑k=1q‍vk∇lk(y)=0.
Since ∑i=1pλiρ1+∑j=1mujρ2+∑k=1qvkρ3≧0 and by (41), we get
(43)(∑i=1p‍λi(fi(x)g(x))+uTh(x)+vTl(x))-(∑i=1p‍λi(fi(y)g(y))+uTh(y)+vTl(y))≧0.
Since uTh(x)≦0, vTl(x)=0, we obtain
(44)∑i=1p‍λi(fi(x)g(x))≧∑i=1p‍λi(fi(y)g(y))+uTh(y)+vTl(y).
The proof is complete.


### 3.3. Nonlinear Multiobjective Fractional Programming Problem Involved Inequality and Equality Constraints under(F,ρ)-Convex

Consider the multiobjective fractional programming problem(MFP)(45)min(f1(x)g1(x),f2(x)g2(x),…,fp(x)gp(x))s.t.hj(x)≦0,j=1,2,…,m,x∈X0,lk(x)=0,k=1,2,…,q,x∈X0,
where X0 is an open set of Rn, fi(x)(i=1,2,…,p):X0→R, fi(x)≧0, gi(x)(i=1,2,…,p):X0→R, gi(x)>0, and hj(x)(j=1,2,…,m):X0→R, lk(x)(k=1,2,…,q):X0→R are continuously differentiable over X0.Denote byG the set of all feasible solutions for (MFP); that is,
(46)G={x∈X0∣hj(x)≦0,j=1,2,…,m;lk(x)=0,k=1,2,…,q}
and let φi(x)=fi(x)/gi(x), φ(x)=(φ1(x),φ2(x),…,φp(x)).Theorem 16.
Assume that there exists(x-,α-,λ-,v-) and α-=(α-1,α-2,…,α-p)∈R+p, λ-=(λ-1,λ-2,…,λ-m), v-=(v-1,v-2,…,v-q) such that(i)
∑i=1pα-iTi(x-)+∑j=1mλ-j∇hj(x-)+∑k=1qv-k∇lk(x-)=0,λ-jhj(x-)=0, j=1,2,…,m,hj(x-)≦0, j=1,2,…,m, lk(x-)=0, k=1,2,…,q,whereTi(x-)=(1/gi(x-))[∇fi(x-)-φi(x-)∇gi(x-)];(ii)
fi and -gi(i=1,2,…,p) are (F,ρ)-convex at x-, and ρ>0;(iii)
hj are (F,ρ)-convex at x- for all j, j=1,2,…,m, and ρ>0;(iv)
lk are (F,ρ)-convex at x- for all k, k=1,2,…,q, and ρ>0.Then x- is a Pareto optimality solution of (MFP).Proof.
Suppose thatx- is not a Pareto optimality solution of (MFP); then there exists a feasible solution x∈G such that fi(x)/gi(x)≦fi(x-)/gi(x-), i=1,2,…,p, that is, fi(x)-(fi(x-)gi(x))/gi(x-)≦0, that is, fi(x)-(fi(x-)/gi(x-))gi(x)≦fi(x-)-(fi(x-)/gi(x-))gi(x-); it follows that
(47)fi(x)-φi(x-)gi(x)≦fi(x-)-φi(x-)gi(x-).
By the (F,ρ)-convexity of fi and -gi, i=1,2,…,p, we have
(48)fi(x)-fi(x-)≧F(x,x-;∇fi(x-))+ρd2(x,x-),∀x∈X0,-gi(x)+gi(x-)≧F(x,x-;-∇gi(x-))+ρd2(x,x-),∀x∈X0.
Using the conditions fi(x-)≧0, gi(x-)>0, we see that φi(x-)=fi(x-)/gi(x-)≧0.
By the properties of the sublinear functionalF, we obtain
(49)-φi(x-)gi(x)+φi(x-)gi(x-)≧φi(x-)F(x,x-;-∇gi(x-))+φi(x-)ρd2(x,x-)=F(x,x-;-φi(x-)∇gi(x-))+φi(x-)ρd2(x,x-).
By (48) and (49) and based on the sublinearity of F, we have
(50)fi(x)-φi(x-)gi(x)-[fi(x-)-φi(x-)gi(x-)]≧F(x,x-;∇fi(x-)-φi(x-)∇gi(x-))+[1+φi(x-)]ρd2(x,x-).
By (47), we have
(51)F(x,x-;∇fi(x-)-φi(x-)∇gi(x-))+[1+φi(x-)]ρd2(x,x-)≦0.
If we sum up after multiplying by α-i(1/gi(x-))≧0(i=1,2,…,p) in the above inequality and by using the sublinearity of F, we have
(52)F(x,x-;∑i=1p‍α-i(1gi(x-))[∇fi(x-)-φi(x-)∇gi(x-)])+∑i=1p‍α-i(1gi(x-))[1+φi(x-)]ρd2(x,x-)≦0.
Since Ti(x-)=(1/gi(x-))[∇fi(x-)-φi(x-)∇gi(x-)], we get
(53)F(x,x-;∑i=1p‍α-iTi(x-))+∑i=1p‍α-i(1gi(x-))[1+φi(x-)]ρd2(x,x-)≦0.
On the other hand, for j=1,2,…,m, by the (F,ρ)-convexity of hj at x-, we have
(54)hj(x)-hj(x-)≧F(x,x-;∇hj(x-))+ρd2(x,x-),∀x∈X0.
On multiplying the inequality (54) by λ-j≥0 and using the sublinearity of F, we have
(55)λ-jhj(x)-λ-jhj(x-)≧F(x,x-;λ-j∇hj(x-))+λ-jρd2(x,x-),
which together with λ-jhj(x)≦0 and λ-jhj(x-)=0 yields
(56)F(x,x-;λ-j∇hj(x-))+λ-jρd2(x,x-)≦0.
By accumulating the inequality (56) with j, we have
(57)∑j=1m‍F(x,x-;λ-j∇hj(x-))+∑j=1m‍λ-jρd2(x,x-)≦0,
that is,
(58)F(x,x-;∑j=1m‍λ-j∇hj(x-))+∑j=1m‍λ-jρd2(x,x-)≦0.
For k=1,2,…,q, by the (F,ρ)-convexity of lk at x-, we have that
(59)lk(x)-lk(x-)≧F(x,x-;∇lk(x-))+ρd2(x,x-),∀x∈X0.
On multiplying the inequality (59) with v-k≧0, we get
(60)v-klk(x)-v-klk(x-)≧F(x,x-;v-k∇lk(x-))+v-kρd2(x,x-)
which together with v-klk(x)=0 and v-klk(x-)=0 yields
(61)F(x,x-;v-k∇lk(x-))+v-kρd2(x,x-)≦0.
By accumulating the inequality (61) with k, we have
(62)∑k=1q‍F(x,x-;v-k∇lk(x-))+∑k=1q‍v-kρd2(x,x-)≦0.
The inequality (62) along with the sublinearity of F implies
(63)F(x,x-;∑k=1q‍v-k∇lk(x-))+∑k=1q‍v-kρd2(x,x-)≦0.
The sublinearity of F, (53), (58), and (63) yields
(64)F(x,x-;∑i=1p‍α-iTi(x-)+∑j=1m‍λ-j∇hj(x-)+∑k=1q‍v-k∇lk(x-))+∑i=1p‍α-i(1gi(x-))[1+φi(x-)]ρd2(x,x-)+∑j=1m‍λ-jρd2(x,x-)+∑k=1q‍v-kρd2(x,x-)≦0.
According to the assumption and the sublinearity of F, we obtain
(65)F(x,x-;∑i=1p‍α-iTi(x-)+∑j=1m‍λ-j∇hj(x-)+∑k=1q‍v-k∇lk(x-))+∑i=1p‍α-i(1gi(x-))[1+φi(x-)]ρd2(x,x-)+∑j=1m‍λ-jρd2(x,x-)+∑k=1q‍v-kρd2(x,x-)>0,
which contradicts (64) obviously.
Therefore,x- is a Pareto optimality solution of (MFP).
The proof is complete.Consider the dual problem of(MFP)(66)max(f1(y)g1(y)+uTh(y)+vTl(y),…,fp(y)gp(y)+uTh(y)+vTl(y))s.t.∑i=1p‍λi∇(fi(y)gi(y))+∑j=1m‍uj∇hj(y)+∑k=1q‍vk∇lk(y)=0,uTh(y)≧0,vTl(y)=0,u≧0,v≧0.Theorem 17.
f
i
(
y
)
/
g
i
(
y
)  (i=1,2,…,p) is (F,ρ)-convex at y, hj(y)  (j=1,2,…,m) is (F,ρ)-convex at y, lk(y)  (k=1,2,…,q) is (F,ρ)-convex at y, and λ∈∧++, u∈R+m, v∈R+q, ∑i=1pλiρi+∑j=1mujρj+∑k=1qvkρk≧0, then λT(fi(x)/gi(x))≧λT(fi(y)/gi(y)).Proof.
By the(F,ρ)-convexity of fi(y)/gi(y), hj(y), and lk(y), the sublinearity of F, and since λ∈∧++, u∈R+m, v∈R+q, we have that
(67)∑i=1p‍λi(fi(x)gi(x))-∑i=1p‍λi(fi(y)gi(y))≧F(x,y;∑i=1p‍λi∇(fi(y)gi(y)))+∑i=1p‍λiρid2(x,y),∑j=1m‍ujhj(x)-∑j=1m‍ujhj(y)≧F(x,y;∑j=1m‍uj∇hj(y))+∑j=1m‍ujρjd2(x,y),∑k=1q‍vklk(x)-∑k=1q‍vklk(y)≧F(x,y;∑k=1q‍vk∇lk(y))+∑k=1q‍vkρkd2(x,y).
By (67) and based on the sublinearity of F, we have
(68)∑i=1p‍λi(fi(x)gi(x))-∑i=1p‍λi(fi(y)gi(y))+∑j=1m‍ujhj(x)-∑j=1m‍ujhj(y)+∑k=1q‍vklk(x)-∑k=1q‍vklk(y)≧F(x,y;(∑i=1p‍λi∇(fi(y)gi(y))+∑j=1m‍uj∇hj(y)+∑k=1q‍vk∇lk(y)))+(∑i=1p‍λiρi+∑j=1m‍ujρj+∑k=1q‍vkρk)d2(x,y).
Since ∑i=1pλi∇(fi(y)/gi(y))+∑j=1muj∇hj(y)+∑k=1qvk∇lk(y)=0, we have
(69)∑i=1p‍λi(fi(x)gi(x))-∑i=1p‍λi(fi(y)gi(y))≧∑j=1m‍ujhj(y)-∑j=1m‍ujhj(x)+∑k=1q‍vklk(y)-∑k=1q‍vklk(x)+(∑i=1p‍λiρi+∑j=1m‍ujρj+∑k=1q‍vkρk)d2(x,y).
Since hj(x)≦0, (j=1,2,…,m), lk(x)=0, (k=1,2,…,q), uTh(y)≧0, vTl(y)=0, ∑i=1pλiρi+∑j=1mujρj+∑k=1qvkρk≧0, we have ∑i=1pλi(fi(x)/gi(x))≧∑i=1pλi(fi(y)/gi(y)). That is, λT(fi(x)/gi(x))≧λT(fi(y)/gi(y)).


## 3.1. Nonlinear Fractional Programming Problem Involved Inequality and Equality Constraints Based on Generalized(F,α,ρ,d)-Convex

Consider the nonlinear fractional programming problem(FP)(7)minf(x)g(x)s.t.h(x)≦0,l(x)=0,x∈X0,
where X0 is an open set of Rn, f(x) and g(x) are real-valued functions defined on X0, h(x) is an m-dimensional vector-valued functions defined also on X0, and l(x) a q-dimensional vector-valued function.Let(8)S={x∈X0∣h(x)≦0,l(x)=0}
denotes the set of all feasible solutions for (FP) and assume that f(x), g(x), hj(x)(j=1,2,…,m), and li(x)(i=1,2,…,q) are continuously differentiable over X0 and that f(x)≧0, g(x)>0, for all x∈X0.Ifx-∈X0 is a solution for problem (FP) and if a constraint qualification [14] holds, then the Kuhn-Tucker necessary conditions are given below: there exists V0∈Rm and W0∈Rq such that
(9)∇(f(x-)g(x-))+∇h(x-)V0+∇l(x-)W0=0,V0Th(x-)=0,V0≧0,h(x-)≦0,l(x-)=0.Theorem 12.
Suppose thatx- is a feasible solution of (FP), that the Kuhn-Tucker conditions hold at x-, that f(x)/g(x) in problem (FP) is (F,α,ρ,d)-pseudoconvex on S, hj(x)(j=1,2,…,m) are (F,α,ρj,d)-quasiconvex on S, and that li(x)(i=1,2,…,q) are (F,α,ρi,d)-quasiconvex over S, ρ, ρi, ρj∈R, ρ+V0Tρ′+W0Tρ′′≧0, where V0Tρ′ is the inner product about V0 and ρ′ and W0Tρ′′ the inner product about W0 and ρ′′. Then, x- is an optimality solution for problem (FP).Proof.
Suppose thatx- is not an optimality solution of (FP). Then, there exists a feasible solution x∈S such that f(x)/g(x)<f(x-)/g(x-).
By the(F,α,ρ,d)-pseudoconvexity assumption of f(x)/g(x), we have
(10)F(x,x-;α(x,x-)∇(f(x-)g(x-)))<-ρd2(x,x-).
For each j(j=1,2,…,m), by the (F,α,ρj,d)-quasiconvexity assumption of hj(x) and Lemma 11, we have that V0Th(x) is (F,α,∑j=1mvjρj,d)-quasiconvex on S. Therefore,
(11)F(x,x-;α(x,x-)∇h(x-)V0)≦-∑j=1m‍ρjvjd2(x,x-),
that is, F(x,x-;α(x,x-)∇h(x-)V0)≦-V0Tρ′d2(x,x-).
By the(F,α,ρi,d)-quasiconvexity of li(x)(i=1,2,…,q) over S, we have that W0Tl(x) is (F,α,∑i=1qwiρi,d)-quasiconvexity on S.
Then we obtainF(x,x-;α(x,x-)∇l(x-)W0)≦-∑i=1qρiwid2(x,x-), that is,
(12)F(x,x-;α(x,x-)∇l(x-)W0)≦-W0Tρ′′d2(x,x-).
By (10), (11), and (12), and based on the sublinearity of F, we have
(13)F(x,x-;α(x,x-)(∇(f(x-)g(x-))+∇h(x-)V0+∇l(x-)W0))<-(ρ+V0Tρ′+W0Tρ′′)d2(x,x-).
Considering that ρ+V0Tρ′+W0Tρ′′≧0, we get
(14)F(x,x-;α(x,x-)(∇(f(x-)g(x-))+∇h(x-)V0+∇l(x-)W0))<0.
By the K-T conditions, we have ∇(f(x-)/g(x-))+∇h(x-)V0+∇l(x-)W0=0.
Hence, based on the sublinearity ofF, we obtain
(15)F(x,x-;α(x,x-)(∇(f(x-)g(x-))+∇h(x-)V0+∇l(x-)W0))=0,
which contradicts (14). The proof is complete.Consider the dual problem of(FP):
(FD)maxf(y)g(y)+uTh(y)+vTl(y)s.t.λT∇(f(y)g(y))+uT∇h(y)+vT∇l(y)=0,uTh(y)≧0,vTl(y)=0,u≧0,v≧0.Theorem 13.
Suppose thatλT(f(y)/g(y)) is (F,α,ρ1,d)-pseudoconvex at y, uTh(y)+vTl(y) is (F,α,ρ2,d)-quasiconvex at y in problem (FP) and (FD), and that ρ1+ρ2≧0; then λT(f(x)/g(x))≧λT(f(y)/g(y)), for any feasible solution x of (FP) and (y,λ,u,v) of (FD).Proof.
Assume that the conclusion is not true; that is,λT(f(x)/g(x))<λT(f(y)/g(y)).
By the(F,α,ρ1,d)-pseudoconvex of λT(f(x)/g(x)) at y, we get
(16)F(x,y;α(x,y)λT∇(f(y)g(y)))<-ρ1d2(x,y).
Using uTh(x)≦0, vTl(x)=0, uTh(y)≧0, vTl(y)=0, we have
(17)uTh(x)+vTl(x)≦uTh(y)+vTl(y).
By the (F,α,ρ2,d)-quasiconvex of uTh(y)+vTl(y), we get
(18)F(x,y;α(x,y)(uT∇h(y)+vT∇l(y)))≦-ρ2d2(x,y).
By (16) and (18) and based on the sublinearity of F, we have
(19)F(x,y;α(x,y)(λT∇(f(y)g(y))+(f(y)g(y))uT∇h(y)+vT∇l(y)))<-(ρ1+ρ2)d2(x,y).
Since (y,λ,u,v) is the feasible solution of (FD), so
(20)λ-T∇(f(y)g(y))+uT∇h(y)+vT∇l(y)=0.
Hence,
(21)0=F(x,y;α(x,y)(λT∇(f(y)g(y))+(f(y)g(y))uT∇h(y)+vT∇l(y)))<-(ρ1+ρ2)d2(x,y),
which contradicts the known condition ρ1+ρ2≧0. The proof is complete.


## 3.2. Nonlinear Multiobjective Fractional Programming Problem Involved Inequality and Equality Constraints Based on(F,α,ρ,d)-Convex and Generalized (F,α,ρ,d)-Convex

Consider the nonlinear multiobjective fractional programming problem(VFP)(22)minM(x)=f(x)g(x)=[f1(x)g(x),f2(x)g(x),…,fp(x)g(x)]s.t.h(x)≦0,x∈X0l(x)=0,x∈X0,
where X0 is an open set of Rn, fi(x)(i=1,2,…,p), g(x), hj(x):X0→R(j=1,2,…,m) are real-valued functions defined on X0, lk(x):X0→R(k=1,2,…,q) are real-valued functions defined also on X0, f(x)=(f1(x),f2(x),…,fp(x))T, h(x)=(h1(x),h2(x),…,hm(x))T, and l(x)=(l1(x),l2(x),…,lq(x))T.Let(23)S={x∣x∈X0,h(x)≦0,l(x)=0}
denotes the set of all feasible solutions of (VFP) and assume that fi(x)(i=1,2,…,p), g(x), hj(x)(j=1,2,…,m), and lk(x)(k=1,2,…,q) are continuously differentiable over X0 and that g(x)>0, for all x∈X0.Theorem 14.
Suppose thatf(x)/g(x) is weakly strictly (F,α,ρ1,d)-pseudoconvex at x-∈X0, hj(x)(j=1,2,…,m) are (F,α,ρ2,d)-quasiconvex with respect to x-∈X0, lk(x)(k=1,2,…,q) are (F,α,ρ3,d)-convex with respect to x-∈X0, and that there exists λ-∈∧++ (or ∧+), u-∈R+m, v-∈R+q satisfying
(24)∑i=1p‍λ-i∇(f(x-)g(x-))+∑j=1m‍u-j∇hj(x-)+∑k=1q‍v-k∇lk(x-)=0,u-jhj(x-)=0,hj(x)≦0,j=1,2,…,m,lk(x)=0,k=1,2,…,q
and ρ=∑i=1pλ-iρ1+∑j=1mu-jρ2+∑k=1qv-kρ3≧0. Then, x- is an efficient solution of (VFP), where
(25)∧+={λ-=(λ-1,λ-2,…,λ-p)T∣λ-i≧0,i=1,2,…,p·∑i=1p‍λ-i=1},∧++={λ-=(λ-1,λ-2,…,λ-p)T∣λ-i>0,i=1,2,…,p·∑i=1p‍λ-i=1}.Proof.
Suppose thatx- is not an efficient solution of (VFP); then there exists a feasible solution x∈X0 such that M(x)≦M(x-), that is, f(x)/g(x)≦f(x-)/g(x-).
By the weakly strict(F,α,ρ1,d)-pseudoconvexity of f(x)/g(x) at x-∈X0, we get
(26)F(x,x-;α(x,x-)∇(f(x-)g(x-)))<-ρ1d2(x,x-).
Using λ-∈∧++ (or ∧+), then we have
(27)∑i=1p‍λ-iF(x,x-;α(x,x-)∇(f(x-)g(x-)))<-∑i=1p‍λ-iρ1d2(x,x-).
Based on the sublinearity of F, we obtain
(28)F(x,x-;α(x,x-)∑i=1p‍λ-i∇(f(x-)g(x-)))<-∑i=1p‍λ-iρ1d2(x,x-).
Since u-Th(x-)=0, u-≧0 and hj(x)≦0, we have u-Th(x)-u-Th(x-)≦0; that is, u-Th(x)≦u-Th(x-).
Sincehj(x)(j=1,2,…,m) are (F,α,ρ2,d)-quasiconvex at x-∈X0, by Lemma 11, we have u-Th is (F,α,ρ2∑j=1mu-j,d)-quasiconvex at x-∈X0.
Hence, we obtain the following inequality:(29)F(x,x-;α(x,x-)∑j=1m‍u-j∇hj(x-))≦-∑j=1m‍u-jρ2d2(x,x-).
By the (F,α,ρ3,d)-convexity of lk(x) at x-∈X0, we have
(30)lk(x)-lk(x-)≧F(x,x-;α(x,x-)∇lk(x-))+ρ3d2(x,x-).
Since v-≧0, we have
(31)0=v-l(x)-v-l(x-)≧∑k=1q‍v-kF(x,x-;α(x,x-)∇lk(x-))+∑k=1q‍v-kρ3d2(x,x-).
By the sublinearity of F, we obtain
(32)F(x,x-;α(x,x-)∑k=1q‍v-k∇lk(x-))≦-∑k=1q‍v-kρ3d2(x,x-).
By the known conditions, we have
(33)F(x,x-;α(x,x-)(∑i=1p‍λ-i∇(f(x-)g(x-))+∑j=1m‍u-j∇hj(x-)+∑k=1q‍v-k∇lk(x-)))=0.
By (28), (29), and (32) and by the sublinearity of F, we obtain
(34)F(x,x-;α(x,x-)(∑i=1p‍λ-i∇(f(x-)g(x-))+∑j=1m‍u-j∇hj(x-)+∑k=1q‍v-k∇lk(x-)))<-(∑i=1p‍λ-iρ1+∑j=1m‍u-jρ2+∑k=1q‍v-kρ3)d2(x,x-)=-ρd2(x,x-)≦0
which contradicts the fact of (33). Therefore, x- is an efficient solution of (VFP). The proof is complete.Consider the dual problem of(VFP)(35)max(f1(y)g(y)+uTh(y)+vTl(y),…,fp(y)g(y)+uTh(y)+vTl(y))s.t.∑i=1p‍λi∇(fi(y)g(y))+∑j=1m‍uj∇hj(y)+∑k=1q‍vk∇lk(y)=0λ∈∧++,u∈R+m,v∈R+q.Theorem 15.
(
f
i
/
g
)
(
i
=
1,2
,
…
,
p
) is (F,α,ρ1,d)-convex at y, hj(j=1,2,…,m) is (F,α,ρ2,d)-convex at y, lk(k=1,2,…,q) is (F,α,ρ3,d)-convex at y, and ∑i=1pλiρ1+∑j=1mujρ2+∑k=1qvkρ3≧0, then
(36)∑i=1p‍λi(fi(x)g(x))≧∑i=1p‍λi(fi(y)g(y))+uTh(y)+vTl(y).Proof.
By the(F,α,ρ1,d)-convex of fi/g at y, we get
(37)fi(x)g(x)-fi(y)g(y)≧F(x,y;α(x,y)∇(fi(y)g(y)))+ρ1d2(x,y),i=1,2,…,p.
By the (F,α,ρ2,d)-convex of hj at y, we get
(38)hj(x)-hj(y)≧F(x,y;α(x,y)∇hj(y))+ρ2d2(x,y),j=1,2,…,m.
By the (F,α,ρ3,d)-convex of lk at y, we get
(39)lk(x)-lk(y)≧F(x,y;α(x,y)∇lk(y))+ρ3d2(x,y),k=1,2,…,q.
Since λ∈∧++, u∈R+m, v∈R+q, and by the previous three inequalities, we have that
(40)(∑i=1p‍λi(fi(x)g(x))+uTh(x)+vTl(x))-(∑i=1p‍λi(fi(y)g(y))+uTh(y)+vTl(y))≧∑i=1p‍λiF(x,y;α(x,y)∇(fi(y)g(y)))+∑j=1m‍ujF(x,y;α(x,y)∇hj(y))+∑k=1q‍vkF(x,y;α(x,y)∇lk(y))+(∑i=1p‍λiρ1+∑j=1m‍ujρ2+∑k=1q‍vkρ3)d2(x,y).
By the sublinearity of F, we obtain
(41)(∑i=1p‍λi(fi(x)g(x))+uTh(x)+vTl(x))-(∑i=1p‍λi(fi(y)g(y))+uTh(y)+vTl(y))≧F(x,y;α(x,y)(∑i=1p‍λi∇(fi(y)g(y))+∑j=1m‍uj∇hj(y)+∑k=1q‍vk∇lk(y)(fi(y)g(y))))+(∑i=1p‍λiρ1+∑j=1m‍ujρ2+∑k=1q‍vkρ3)d2(x,y).
By the feasibility of (y,λ,u,v), we have
(42)∑i=1p‍λi∇(fi(y)g(y))+∑j=1m‍uj∇hj(y)+∑k=1q‍vk∇lk(y)=0.
Since ∑i=1pλiρ1+∑j=1mujρ2+∑k=1qvkρ3≧0 and by (41), we get
(43)(∑i=1p‍λi(fi(x)g(x))+uTh(x)+vTl(x))-(∑i=1p‍λi(fi(y)g(y))+uTh(y)+vTl(y))≧0.
Since uTh(x)≦0, vTl(x)=0, we obtain
(44)∑i=1p‍λi(fi(x)g(x))≧∑i=1p‍λi(fi(y)g(y))+uTh(y)+vTl(y).
The proof is complete.


## 3.3. Nonlinear Multiobjective Fractional Programming Problem Involved Inequality and Equality Constraints under(F,ρ)-Convex

Consider the multiobjective fractional programming problem(MFP)(45)min(f1(x)g1(x),f2(x)g2(x),…,fp(x)gp(x))s.t.hj(x)≦0,j=1,2,…,m,x∈X0,lk(x)=0,k=1,2,…,q,x∈X0,
where X0 is an open set of Rn, fi(x)(i=1,2,…,p):X0→R, fi(x)≧0, gi(x)(i=1,2,…,p):X0→R, gi(x)>0, and hj(x)(j=1,2,…,m):X0→R, lk(x)(k=1,2,…,q):X0→R are continuously differentiable over X0.Denote byG the set of all feasible solutions for (MFP); that is,
(46)G={x∈X0∣hj(x)≦0,j=1,2,…,m;lk(x)=0,k=1,2,…,q}
and let φi(x)=fi(x)/gi(x), φ(x)=(φ1(x),φ2(x),…,φp(x)).Theorem 16.
Assume that there exists(x-,α-,λ-,v-) and α-=(α-1,α-2,…,α-p)∈R+p, λ-=(λ-1,λ-2,…,λ-m), v-=(v-1,v-2,…,v-q) such that(i)
∑i=1pα-iTi(x-)+∑j=1mλ-j∇hj(x-)+∑k=1qv-k∇lk(x-)=0,λ-jhj(x-)=0, j=1,2,…,m,hj(x-)≦0, j=1,2,…,m, lk(x-)=0, k=1,2,…,q,whereTi(x-)=(1/gi(x-))[∇fi(x-)-φi(x-)∇gi(x-)];(ii)
fi and -gi(i=1,2,…,p) are (F,ρ)-convex at x-, and ρ>0;(iii)
hj are (F,ρ)-convex at x- for all j, j=1,2,…,m, and ρ>0;(iv)
lk are (F,ρ)-convex at x- for all k, k=1,2,…,q, and ρ>0.Then x- is a Pareto optimality solution of (MFP).Proof.
Suppose thatx- is not a Pareto optimality solution of (MFP); then there exists a feasible solution x∈G such that fi(x)/gi(x)≦fi(x-)/gi(x-), i=1,2,…,p, that is, fi(x)-(fi(x-)gi(x))/gi(x-)≦0, that is, fi(x)-(fi(x-)/gi(x-))gi(x)≦fi(x-)-(fi(x-)/gi(x-))gi(x-); it follows that
(47)fi(x)-φi(x-)gi(x)≦fi(x-)-φi(x-)gi(x-).
By the (F,ρ)-convexity of fi and -gi, i=1,2,…,p, we have
(48)fi(x)-fi(x-)≧F(x,x-;∇fi(x-))+ρd2(x,x-),∀x∈X0,-gi(x)+gi(x-)≧F(x,x-;-∇gi(x-))+ρd2(x,x-),∀x∈X0.
Using the conditions fi(x-)≧0, gi(x-)>0, we see that φi(x-)=fi(x-)/gi(x-)≧0.
By the properties of the sublinear functionalF, we obtain
(49)-φi(x-)gi(x)+φi(x-)gi(x-)≧φi(x-)F(x,x-;-∇gi(x-))+φi(x-)ρd2(x,x-)=F(x,x-;-φi(x-)∇gi(x-))+φi(x-)ρd2(x,x-).
By (48) and (49) and based on the sublinearity of F, we have
(50)fi(x)-φi(x-)gi(x)-[fi(x-)-φi(x-)gi(x-)]≧F(x,x-;∇fi(x-)-φi(x-)∇gi(x-))+[1+φi(x-)]ρd2(x,x-).
By (47), we have
(51)F(x,x-;∇fi(x-)-φi(x-)∇gi(x-))+[1+φi(x-)]ρd2(x,x-)≦0.
If we sum up after multiplying by α-i(1/gi(x-))≧0(i=1,2,…,p) in the above inequality and by using the sublinearity of F, we have
(52)F(x,x-;∑i=1p‍α-i(1gi(x-))[∇fi(x-)-φi(x-)∇gi(x-)])+∑i=1p‍α-i(1gi(x-))[1+φi(x-)]ρd2(x,x-)≦0.
Since Ti(x-)=(1/gi(x-))[∇fi(x-)-φi(x-)∇gi(x-)], we get
(53)F(x,x-;∑i=1p‍α-iTi(x-))+∑i=1p‍α-i(1gi(x-))[1+φi(x-)]ρd2(x,x-)≦0.
On the other hand, for j=1,2,…,m, by the (F,ρ)-convexity of hj at x-, we have
(54)hj(x)-hj(x-)≧F(x,x-;∇hj(x-))+ρd2(x,x-),∀x∈X0.
On multiplying the inequality (54) by λ-j≥0 and using the sublinearity of F, we have
(55)λ-jhj(x)-λ-jhj(x-)≧F(x,x-;λ-j∇hj(x-))+λ-jρd2(x,x-),
which together with λ-jhj(x)≦0 and λ-jhj(x-)=0 yields
(56)F(x,x-;λ-j∇hj(x-))+λ-jρd2(x,x-)≦0.
By accumulating the inequality (56) with j, we have
(57)∑j=1m‍F(x,x-;λ-j∇hj(x-))+∑j=1m‍λ-jρd2(x,x-)≦0,
that is,
(58)F(x,x-;∑j=1m‍λ-j∇hj(x-))+∑j=1m‍λ-jρd2(x,x-)≦0.
For k=1,2,…,q, by the (F,ρ)-convexity of lk at x-, we have that
(59)lk(x)-lk(x-)≧F(x,x-;∇lk(x-))+ρd2(x,x-),∀x∈X0.
On multiplying the inequality (59) with v-k≧0, we get
(60)v-klk(x)-v-klk(x-)≧F(x,x-;v-k∇lk(x-))+v-kρd2(x,x-)
which together with v-klk(x)=0 and v-klk(x-)=0 yields
(61)F(x,x-;v-k∇lk(x-))+v-kρd2(x,x-)≦0.
By accumulating the inequality (61) with k, we have
(62)∑k=1q‍F(x,x-;v-k∇lk(x-))+∑k=1q‍v-kρd2(x,x-)≦0.
The inequality (62) along with the sublinearity of F implies
(63)F(x,x-;∑k=1q‍v-k∇lk(x-))+∑k=1q‍v-kρd2(x,x-)≦0.
The sublinearity of F, (53), (58), and (63) yields
(64)F(x,x-;∑i=1p‍α-iTi(x-)+∑j=1m‍λ-j∇hj(x-)+∑k=1q‍v-k∇lk(x-))+∑i=1p‍α-i(1gi(x-))[1+φi(x-)]ρd2(x,x-)+∑j=1m‍λ-jρd2(x,x-)+∑k=1q‍v-kρd2(x,x-)≦0.
According to the assumption and the sublinearity of F, we obtain
(65)F(x,x-;∑i=1p‍α-iTi(x-)+∑j=1m‍λ-j∇hj(x-)+∑k=1q‍v-k∇lk(x-))+∑i=1p‍α-i(1gi(x-))[1+φi(x-)]ρd2(x,x-)+∑j=1m‍λ-jρd2(x,x-)+∑k=1q‍v-kρd2(x,x-)>0,
which contradicts (64) obviously.
Therefore,x- is a Pareto optimality solution of (MFP).
The proof is complete.Consider the dual problem of(MFP)(66)max(f1(y)g1(y)+uTh(y)+vTl(y),…,fp(y)gp(y)+uTh(y)+vTl(y))s.t.∑i=1p‍λi∇(fi(y)gi(y))+∑j=1m‍uj∇hj(y)+∑k=1q‍vk∇lk(y)=0,uTh(y)≧0,vTl(y)=0,u≧0,v≧0.Theorem 17.
f
i
(
y
)
/
g
i
(
y
)  (i=1,2,…,p) is (F,ρ)-convex at y, hj(y)  (j=1,2,…,m) is (F,ρ)-convex at y, lk(y)  (k=1,2,…,q) is (F,ρ)-convex at y, and λ∈∧++, u∈R+m, v∈R+q, ∑i=1pλiρi+∑j=1mujρj+∑k=1qvkρk≧0, then λT(fi(x)/gi(x))≧λT(fi(y)/gi(y)).Proof.
By the(F,ρ)-convexity of fi(y)/gi(y), hj(y), and lk(y), the sublinearity of F, and since λ∈∧++, u∈R+m, v∈R+q, we have that
(67)∑i=1p‍λi(fi(x)gi(x))-∑i=1p‍λi(fi(y)gi(y))≧F(x,y;∑i=1p‍λi∇(fi(y)gi(y)))+∑i=1p‍λiρid2(x,y),∑j=1m‍ujhj(x)-∑j=1m‍ujhj(y)≧F(x,y;∑j=1m‍uj∇hj(y))+∑j=1m‍ujρjd2(x,y),∑k=1q‍vklk(x)-∑k=1q‍vklk(y)≧F(x,y;∑k=1q‍vk∇lk(y))+∑k=1q‍vkρkd2(x,y).
By (67) and based on the sublinearity of F, we have
(68)∑i=1p‍λi(fi(x)gi(x))-∑i=1p‍λi(fi(y)gi(y))+∑j=1m‍ujhj(x)-∑j=1m‍ujhj(y)+∑k=1q‍vklk(x)-∑k=1q‍vklk(y)≧F(x,y;(∑i=1p‍λi∇(fi(y)gi(y))+∑j=1m‍uj∇hj(y)+∑k=1q‍vk∇lk(y)))+(∑i=1p‍λiρi+∑j=1m‍ujρj+∑k=1q‍vkρk)d2(x,y).
Since ∑i=1pλi∇(fi(y)/gi(y))+∑j=1muj∇hj(y)+∑k=1qvk∇lk(y)=0, we have
(69)∑i=1p‍λi(fi(x)gi(x))-∑i=1p‍λi(fi(y)gi(y))≧∑j=1m‍ujhj(y)-∑j=1m‍ujhj(x)+∑k=1q‍vklk(y)-∑k=1q‍vklk(x)+(∑i=1p‍λiρi+∑j=1m‍ujρj+∑k=1q‍vkρk)d2(x,y).
Since hj(x)≦0, (j=1,2,…,m), lk(x)=0, (k=1,2,…,q), uTh(y)≧0, vTl(y)=0, ∑i=1pλiρi+∑j=1mujρj+∑k=1qvkρk≧0, we have ∑i=1pλi(fi(x)/gi(x))≧∑i=1pλi(fi(y)/gi(y)). That is, λT(fi(x)/gi(x))≧λT(fi(y)/gi(y)).



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*Source: 708979-2013-11-27.xml*