On ϵ solutions for robust fractional optimization problems

R E S E A R C H

Open Access

On

-solutions for robust fractional

optimization problems

Jae Hyoung Lee and Gue Myung Lee

*

*_{Correspondence:}

[email protected] Department of Applied Mathematics, Pukyong National University, 45, Yongso-ro, Nam-Gu, Busan, 608-737, Korea

Abstract

We consider

-solutions (approximate solutions) for a fractional optimization problem

in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish optimality theorems and duality theorems for

-solutions for

the fractional optimization problem. Moreover, we give an example illustrating our duality theorems.

MSC: Primary 90C25; 90C32; secondary 90C46

Keywords: fractional programming under uncertainty; convex programming under uncertainty; strong duality; robust optimization

1 Introduction

A robust fractional optimization problem is to optimize an objective fractional function over the constrained set defined by functions with data uncertainty.

To get the -solution (approximate solution), many authors have established -opti-mality conditions and-duality theorems for several kinds of optimization problems [–]. Especially, Lee and Lee [] gave an-duality theorems for a convex semidefinite optimiza-tion problem with conic constraints. Also, they [] established optimality theorems and duality theorems for-solutions for convex optimization problems with uncertainty data. In [–], many authors have treated fractional programming problems in the absence of data uncertainty. Recently, many authors have studied robust optimization problems [, –]. Very recently, Jeyakumar and Li [] established duality theorems for a fractional programming problem in the face of data uncertainty via robust optimization.

The purpose of the paper is to extend the-optimality theorems and-duality theorems in [] to fractional optimization problems with uncertainty data.

Consider the following standard form of fractional programming problem with a geo-metric constraint set:

(FP) min f(x) g(x)

s.t. hi(x),i= , . . . ,m,

x∈C,

wheref,hi:Rn→R,i= , . . . ,m, are convex functions,Cis a closed convex cone ofRn, andg:Rn_{→Ris a concave function such that, for anyx∈C,f(x) andg(x) > .}

The fractional programming problem (FP) in the face of data uncertainty in the con-straints can be captured by the problem:

(UFP) min max

(u,v)∈U×V

f(x,u)

g(x,v)

s.t. hi(x,wi),i= , . . . ,m,

x∈C,

wheref :Rn_×Rp_→R,h

i:Rn×Rq→R,f(·,u) andhi(·,wi) are convex, andg:Rn×

Rp_{→R,g(·,v) is concave, andu∈R}p_,v∈Rp_{, andw}

i∈Rqare uncertain parameters which belong to the convex and compact uncertainty setsU ⊂Rp_,V⊂Rp_{, andW}

i⊂Rq,i= , . . . ,m, respectively.

We study-optimality theorems and-duality theorems for the uncertain fractional programming model problem (UFP) by examining its robust (worst-case) counterpart []:

(RFP) min max

(u,v)∈U×V

f(x,u)

g(x,v)

s.t. hi(x,wi),∀wi∈Wi,i= , . . . ,m,

x∈C.

Clearly,A:={x∈C|hi(x,wi),∀wi∈Wi,i= , . . . ,m}is a feasible set of (RFP). Let. Thenx¯is called an-solution of (RFP) if, for anyx∈A,

max

(u,v)∈U×V

f(x,u)

g(x,v) (u,vmax)∈U×V

f(x¯,u)

g(x¯,v)–.

Using the parametric approach, we transform the problem (RFP) into the robust non-fractional convex optimization problem (RNCP)rwith a parametricr∈R+:

(RNCP)r min max

u∈Uf(x,u) –rminv∈Vg(x,v)

s.t. hi(x,wi),∀wi∈Wi, i= , . . . ,m,

x∈C.

Let. Thenx¯is called an-solution of (RNCP)rif, for anyx∈A,

max

u∈Uf(x,u) –rminv∈Vg(x,v)maxu∈Uf(x¯,u) –rminv∈Vg(x¯,v) –.

In this paper, we consider -solutions for (RFP), and we establish optimality theo-rems and duality theotheo-rems for-solutions for the robust fractional optimization problem. Moreover, we give an example for our duality theorems.

2 Preliminaries

Let us first recall some notation and preliminary results which will be used throughout this paper.Rndenotes the Euclidean space with dimensionn. The nonnegative orthant ofRnis denoted byRn

wheneverμa+ ( –μ)a∈Afor allμ∈[, ],a,a∈A. A functionf :Rn→R∪ {+∞} is said to be convex if, for allμ∈[, ],

f( –μ)x+μy≤( –μ)f(x) +μf(y)

for allx,y∈Rn_{. The functionf is said to be concave whenever –f is convex. Letg:R}n_→

R∪ {+∞}be a convex function. The subdifferential ofgata∈domgis defined by

∂g(a) :=v∈Rn|g(x)g(a) +v,x–a∀x∈domg,

where·,·is the inner product onRn_{anddomg:={x∈R}n_{:g(x) < +∞}. Let. Then} the-subdifferential ofgata∈domgis defined by

∂g(a) :=

v∈Rn|g(x)g(a) +v,x–a–∀x∈domg.

The functionf is said to be proper iff(x) > –∞for allx∈Rn_{. We sayf is a lower} semi-continuous function iflim inf_y→xf(y)f(x) for allx∈Rn_{. As usual, for any proper convex} functiongonRn_{, its conjugate functiong}∗_:Rn_{→R∪ {+∞}is defined, for anyx}∗_∈Rn_, byg∗(x∗) =sup{x∗,x–g(x)|x∈Rn_{}. The epigraph of a functiong:R}n_{→R∪ {+∞},epig,} is defined byepig={(x,r)∈Rn_{×R|g(x)r}. We denote the convex hull of a subsetAof}

Rn_{bycoA, and denote the closure of the setAbyclA. LetCbe a closed convex set inR}n andx∈C. Then the normal coneNC(x) toCatxis defined by

NC(x) =

v∈Rn| v,y–x, for ally∈C,

and we let, then the-normal coneN

C(x) toCatxis defined by

N

C(x) =

v∈Rn| v,y–x, for ally∈C.

WhenCis a closed convex cone inRn_{, we denoteN}

C() byC∗and call it the negative dual cone ofC.

Proposition .[] Let f :Rn→Rbe a convex function and letδCbe the indicator

func-tion with respect to a closed convex subset C ofRn,that is,δC(x) = if x∈C,andδC(x) = +∞

if x∈/C.Let.Then

∂(f+δC)(x¯) =

, +=

∂f(x¯) +∂δC(x¯)

.

Proposition .[, ] If f :Rn→R∪ {+∞}is a proper lower semicontinuous convex function and if a∈domf :={x∈Rn_{|f(x) < +∞},then}

epif∗=



v,v,a+–f(a)|v∈∂f(a)

.

Proposition .[] Let f :Rn_{→Rbe a convex function and g:R}n_{→R∪ {+∞}be a}

proper lower semicontinuous convex function.Then

Moreover,if f,g:Rn_{→R∪ {+∞}are proper lower semicontinuous convex functions,and}

ifdomf ∩domg=∅,then

epi(f+g)∗=clepif∗+epig∗.

Proposition .[, ] Let hi:Rn→R∪ {+∞},i∈I(where I is an arbitrary index set),

be a proper lower semicontinuous convex function.Suppose that there exists x∈Rnsuch

thatsup_i∈Ihi(x) < +∞.Then

epi

sup

i∈I

hi

∗

=cl co

i∈I

epih∗_i

.

Proposition .[] Let hi:Rn→R∪{+∞},i= , . . . ,m,be proper lower semicontinuous

convex functions.Let.Ifm_i=ridomhi=∅,where ridomhiis the relative interior of

domhi,then for all x∈

n

i=domhi,

∂

_m

i=

h

(x) =

_m

i=

∂ihi(x)

i,i= , . . . ,m, m

i=

i=

.

Proposition .[] Let hi:Rn×Rq→R,i= , . . . ,m,be continuous functions such that,

for all wi∈Rq,hi(·,wi)is a convex function and let C be a closed convex cone ofRn.

Sup-pose that eachWi,i= , . . . ,m,is compact and convex,and there exists x∈C such that

hi(x,wi) < ,for all wi∈Wi,i= , . . . ,m.Then

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+

is closed.

Proposition .[] Let hi:Rn×Rq→R,i= , . . . ,m,be continuous functions and let C

be a closed convex cone ofRn_{.Suppose that eachW}

i⊆Rq,i= , . . . ,m,is convex,for all

wi∈Rq,hi(·,wi)is a convex function,and,for each x∈Rn,hi(x,·)is concave onWi.Then

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+

is convex.

Now we give the following relation between the-solutions of (RFP) and (RNCP)r¯.

Lemma . Letx¯∈A and let.Ifmax_(u,v)∈U×Vf(x¯,u)

g(x¯,v)–,then the following

state-ments are equivalent:

(i) x¯is an-solution of(RFP);

(ii) x¯is an¯-solution of(RNCP)r¯,where¯r=max(u,v)∈U×Vfg((x¯x¯,,uv))–and ¯

Proof (⇒) Letx¯∈Abe an-solution of (RFP). Then for anyx∈A,max(u,v)∈U×Vfg((xx,,uv))

max_(u,v)∈U×Vf(x¯,u)

g(¯x,v)–. Putr¯=max(u,v)∈U×V f(¯x,u)

g(x¯,v)–and¯=minv∈Vg(x¯,v). Then we have, for anyx∈A,maxu∈Uf(x,u) –minv∈Vrg¯ (x,v). Sincemaxu∈Uf(x¯,u) –¯rminv∈Vg(x¯,v) –

min_v∈Vg(x¯,v) = , for anyx∈A,

max

u∈Uf(x,u) –r¯minv∈Vg(x,v)maxu∈Uf(x¯,u) –¯rminv∈Vg(x¯,v) –minv∈Vg(x¯,v) =max

u∈Uf(x¯,u) –¯rminv∈Vg(x¯,v) –¯. Hencex¯is an¯-solution of (RNCP)¯r.

(⇐) Let x¯ ∈A be an ¯-solution of (RNCP)r¯. Then for any x∈A, maxu∈Uf(x,u) – ¯

rmin_v∈Vg(x,v)max_u∈Uf(x¯,u) –r¯min_v∈Vg(x¯,v) –¯. Sincemax_u∈Uf(x¯,u) –¯rmin_v∈Vg(x¯,

v) –minv∈Vg(x¯,v) = , for anyx∈A,maxu∈Uf(x,u) –¯rminv∈Vg(x,v). So, we have

max_(u,v)∈U×Vf(x,u)

g(x,v)¯r. Sincer¯=max(u,v)∈U×V f(x¯,u) g(x¯,v)–,

max

(u,v)∈U×V

f(x,u)

g(x,v) (u,vmax)∈U×V

f(x¯,u)

g(x¯,v)–.

Hencex¯is an-solution of (RFP).

3

-Optimality theorems

In this section, we establish-optimality theorems for-solutions for the robust fractional optimization problem.

Now we give the following lemma which is the robust version of Farkas lemma for non-fractional convex functions.

Lemma . Let f :Rn_×Rp_{→Rand h}

i:Rn×Rq→R,i= , . . . ,m,be functions such

that,for any u∈U,f(·,u)and,for each wi∈Wi, hi(·,wi)are convex functions,and,for

any x∈Rn_{,f(x,·)is concave function.Let g:R}n_×Rq_{→Rbe a function such that,for any}

v∈V,g(·,v)is a concave function,and,for all x∈R,g(x,·)is a convex function.LetU⊂Rp_, V⊂Rp_,andW

i⊂Rq,i= , . . . ,m be convex and compact sets.Let rand let C be a

closed convex cone ofRn_{.Assume that A:={x∈C|h}

i(x,wi),∀wi∈Wi,i= , . . . ,m} =∅.

Then the following statements are equivalent:

(i) {x∈C|hi(x,wi),∀wi∈Wi,i= , . . . ,m} ⊆ {x∈Rn|

maxu∈Uf(x,u) –rminv∈Vg(x,v)}; (ii) there existu¯∈U andv¯∈Vsuch that

x∈C|hi(x,wi),∀wi∈Wi,i= , . . . ,m

⊆x∈Rn|f(x,u¯) –rg(x,v¯);

(iii)

(, )∈ u∈U

epif(·,u)∗+ v∈V

epi–rg(·,v)∗

+cl co

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+

(iv)

(, )∈ epimax

u∈Uf(·,u)

∗

+epi–rmin

v∈Vg(·,v)

∗

+cl co

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+

.

Proof LetD:={x∈Rn_|h

i(x,wi),∀wi∈Wi,i= , . . . ,m}. ThenA=C∩D. We will prove thatepiδ_A∗=cl co(_w

i∈Wi,λiepi( m

i=λihi(·,wi))∗+C∗×R+). For anyx∈Rn,

δA(x) =δC(x) +δD(x) and δD(x) = sup wi∈Wi,λi

m

i=

λihi(x,wi).

Thus, by Propositions . and ., we have

epiδ_A∗=epi(δD+δC)∗=cl

epiδ_D∗+epiδ∗_C

=cl

epi

sup

wi∈Wi,λi

m

i=

λihi(·,wi)

∗

+epiδ_C∗

=cl

cl co

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+epiδ∗_C

=cl co

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+

.

[(i)⇔(iv)] Now we assume that the statement (iv) holds. Then, by Proposition ., the statement (iv) is equivalent to

(, )∈epi

max

u∈Uf(·,u)

∗

+epi

–rmin

v∈Vg(·,v)

∗

+epiδ∗_A

=epi

max

u∈Uf(·,u) –rminv∈Vg(·,v) +δA

∗

.

Equivalently, by definition of epigraph of (max_u∈Uf(·,u) –rmin_v∈Vg(·,v) +δA)∗,

max

u∈Uf(·,u) –rminv∈Vg(·,v) +δA

∗

().

From the definition of a conjugate function, for anyx∈Rn_,

max

u∈Uf(·,u) –rminv∈Vg(·,v) +δA

(x).

It is equivalent to the statement that, for anyx∈A,

max

[(ii)⇔(iii)] Now we assume that the statement (iii) holds. Then the statement (iii) is equivalent to

(, )∈ u∈U

epif(·,u)∗+ v∈V

epi–rg(·,v)∗+epiδ_A∗.

It means that there existu¯∈U andv¯∈Vsuch that

(, )∈epif(·,u¯) –rg(·,v¯) +δA

∗

.

It is equivalent to the statement that there existu¯∈U andv¯∈Vsuch that

f(·,u¯) –rg(·,v¯) +δA

∗

().

From the definition of a conjugate function, there existu¯∈Uandv¯∈Vsuch that, for any

x∈Rn_,

f(·,u¯) –rg(·,¯v) +δA

(x).

It means that there existu¯∈U andv¯∈Vsuch that, for anyx∈A,

f(x,u¯) –rg(x,¯v).

[(iii)⇔(iv)] To get the desired result, it suffices to show that

u∈U

epif(·,u)∗=epi

max

u∈Uf(·,u)

∗

, ()

v∈V

epi–rg(·,v)∗=epi

–rmin

v∈Vg(·,v)

∗

. ()

By Proposition ., epi(max_u∈Uf(·,u))∗ =cl co_u∈Uepi(f(·,u))∗. Let (z,α), (z,α) ∈

u∈Uepi(f(·,u))∗ and let μ ∈ [, ]. Then there exist u,u ∈ U such that (z,α) ∈

epi(f(·,u))∗and (z,α)∈epi(f(·,u))∗, that is, (f(·,u))∗(z)α and (f(·,u))∗(z)α. Using the definition of a conjugate function, we have, for allx∈Rn_,

z,x–f(x,u)α and z,x–f(x,u)α. ()

Since, for all x∈Rn_{, f(x,·) is concave, we havef(x,μu}

+ ( –μ)u)μf(x,u) + ( –

μ)f(x,u),i.e.,

–fx,μu+ ( –μ)u

–μf(x,u) – ( –μ)f(x,u). ()

So, from () and (), we have, for allx∈Rn_,

μz+ ( –μ)z,x

–fx,μu+ ( –μ)u

and so (f(·,μu+ ( –μ)u))∗(μz+ ( –μ)z)μα+ ( –μ)α. Hence, we have

μz+ ( –μ)z,μα+ ( –μ)α

∈epif·,μu+ ( –μ)u

∗

.

So,_u∈Uepi(f(·,u))∗is convex.

Now we show that_u∈Uepi(f(·,u))∗is closed. Let

(zn,αn)∈

u∈U

epif(·,u)∗

with (zn,αn)→(z∗,α∗) asn→ ∞. Then there existsun∈U such that (f(·,un))∗(zn)αn. SinceUis compact, we may assume thatun→u∗∈Uasn→ ∞. So, for eachx∈Rn,

zn,x–f(x,un)αn.

Since, for allx∈Rn_{,f(x,·) is concave,f(x,·) is continuous. Passing to the limit asn→ ∞,} we get, for eachx∈Rn_,z∗_,x–f(x,u∗_)α∗_{. Hence, we have}

z∗,α∗∈epif·,u∗∗.

So,_u∈Uepi(f(·,u))∗is closed. Thus, () holds.

Moreover, since, for all x∈Rn_{, g(x,·) is convex and r, for allx∈R}n_{, –rg(x,·) is} concave. So, similarly, we can prove that () holds.

Remark . Using convex-concave minimax theorem (Corollary .. in []), we can prove that the statement (i) in Lemma . is equivalent to the statement (ii) in Lemma ..

Remark . From proving in Lemma . that the statement (i) is equivalent to the

state-ment (iv), we see that we can prove the equivalent relation without the assumptions that, for allx∈Rn_{,f(x,·), andg(x,·) are concave and convex, respectively.}

From Lemmas . and ., we can get the following theorem.

Theorem . Let f :Rn×Rp→Rand hi:Rn×Rq→R,i= , . . . ,m,be functions such

that,for any u∈U,f(·,u),and,for each wi∈Wi,hi(·,wi)are convex functions,and,for

any x∈Rn, f(x,·)is concave function.Let g:Rn×Rp→Rbe a function such that,for any v∈V,g(·,v)is a concave function,and,for all x∈Rn_{,g(x,·)is a convex function.Let} U ⊂Rp_,V⊂Rp_,andW

i⊂Rq,i= , . . . ,m.Letx¯∈A and let¯r=max(u,v)∈U×Vfg((x¯x¯,,uv)) –.

Suppose that_w

i∈Wi,λiepi( m

i=λihi(·,wi))∗+C∗×R+is closed and convex.Then the

following statements are equivalent: (i) x¯is an-solution of(RFP);

(ii) There existu¯∈U,v¯∈V,w¯i∈Wi,andλ¯i,i= , . . . ,msuch that,for anyx∈C,

f(x,u¯) –¯rg(x,v¯) + m

i= ¯

Proof (⇒) Let x¯ be an-solution of (RFP). Then, by Lemma ., equivalently,x¯ is an ¯

-solution of (RNCP)¯r, where ¯r=max(u,v)∈U×Vfg((¯xx¯,,uv)) – and ¯ =minv∈Vg(x¯,v), that is, for any x∈A, maxu∈Uf(x,u) –¯rminv∈Vg(x,v)maxu∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) –

minv∈Vg(x¯,v). Sincemaxu∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) –minv∈Vg(x¯,v) = , we haveA⊆ {x∈C|maxu∈Uf(x,u) –r¯minv∈Vg(x,v)}. Then, by Lemma ., we have

(, )∈ u∈U

epif(·,u)∗+ v∈V

epi–¯rg(·,v)∗

+cl co

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+

.

Moreover, by assumption,

(, )∈ u∈U

epif(·,u)∗+ v∈V

epi–rg¯ (·,v)∗+ wi∈Wi,λi

epi _m

i=

λihi(·,wi)

∗

+C∗×R+.

So, there existu¯∈U,v¯∈V,w¯i∈Wi, andλ¯i,i= , . . . ,msuch that

(, )∈epif(·,u¯)∗+epi–rg¯ (·,¯v)∗+epi _m

i= ¯

λihi(·,w¯i)

∗

+C∗×R+.

Then there exists∈Rn_,η,t∈Rn_,μ,z

i∈Rn,ρi,i= , . . . ,m,c∗∈C∗, and

γ ∈R+such that

(, ) =s,f(·,u¯)∗(s) +η+t,–¯rg(·,v¯)∗(t) +μ

+ m

i=

zi,

_¯ λihi(·,w¯i)

∗

(zi) +ρi

+c∗,γ.

So,  =s+t+m_i=zi+c∗and  = (f(·,u¯))∗(s) +η+ (–¯rg(·,v¯))∗(t) +μ+

m

i=((λ¯ihi(·,w¯i))∗(zi) +

ρi) +γ. Hence, for anyx∈Rn,

–

_m

i=

zi,x

–c∗,x–f(x,u¯) ––rg¯ (x,v¯)

=s,x+t,x–f(x,u¯) ––¯rg(x,v¯)

f(·,u¯)∗(s) +–rg¯ (·,v)∗(t)

= –η–μ– m

i=

_¯ λihi(·,w¯i)

∗

(zi) +ρi

–γ. ()

Sinceη,μ,ρi,i= , . . . ,m, andc∗∈C∗, from (), for anyx∈C,



_m

i=

zi,x

+c∗,x+f(x,u¯) +–rg¯ (x,v¯)–η–μ

– m

i=

_¯

λihi(·,w¯)

∗

(zi) – m

i= ¯

_m

i=

zi,x

+f(x,u¯) –rg¯ (x,v¯) – m

i=

_¯ λihi(·,w¯i)

∗

(zi)

f(x,u¯) –rg¯ (x,v¯) + m

i=

_¯

λihi(x,w¯i)

.

(⇐) Suppose that there existu¯ ∈U,v¯∈V,w¯i∈Wi, andλ¯i,i= , . . . ,m, such that, for anyx∈C,

f(x,u¯) –¯rg(x,v¯) + m

i= ¯

λihi(x,w¯i). ()

Since¯r=max_(u,v)∈U×Vf(x¯,u)

g(x¯,v)–, we havemaxu∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) –minv∈Vg(x¯,v) = . So, from (), we have, for anyx∈A,

max

u∈Uf(x,u) –r¯minv∈Vg(x,v)maxu∈Uf(x,u) –r¯minv∈Vg(x,v) + m

i= ¯

λihi(x,w¯i)

f(x,u¯) –rg¯ (x,¯v) + m

i= ¯

λihi(x,w¯i)



=max

u∈Uf(x¯,u) –¯rminv∈Vg(x¯,v) –minv∈Vg(x¯,v).

Hence, for anyx∈A,maxu∈Uf(x,u) –¯rminv∈Vg(x,v)maxu∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) –

minv∈Vg(x¯,v). It means thatx¯is an¯-solution of (RNCP)¯r. Thus, by Lemma .,x¯is an

-solution of (RFP).

Using Remark . and Lemmas . and ., we can obtain the following characterization of an-solution for (RFP).

Theorem .(-Optimality theorem) Let f :Rn_×Rp_{→Rand h}

i:Rn×Rq→R,i= , . . . ,m,be functions such that,for any u∈U,f(·,u),and,for each wi∈Wi,hi(·,wi)are

convex functions.Let g:Rn_×Rp_{→Rbe a function such that,for any v∈V,g(·,v)is a}

concave function.LetU⊂Rp_,V⊂Rp_,andW

i⊂Rq,i= , . . . ,m.Letx¯∈A and let.

Let¯r=max(u,v)∈U×Vf_g(₍x_x¯¯,_,u_v)₎ –.Ifmax(u,v)∈U×Vf_g(₍x¯_x¯,_,vu)₎ <,thenx is an¯ -solution of(RFP).If

max(u,v)∈U×Vgf((x¯¯x,,uv))and

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+

(ii) there existw¯i∈Wiandλ¯i,i= , . . . ,m,,,andi,i= , . . . ,m+  such that ∈∂  max

u∈Uf(·,u)

(x¯) +∂ 

–¯rmin

v∈Vg(·,v)

(x¯)

+ m

i=

∂i _¯

λihi(·,w¯i)

(x¯) +Nm+

C (x¯), ()

max

u∈Uf(x¯,u) –¯rminv∈Vg(x¯,v) =minv∈Vg(x¯,v) and ()

_+_+ m+

i=

i–min v∈Vg(x¯,v) =

m

i= ¯

λihi(x¯,w¯i). ()

Proof [(i) ⇒(ii)] We assume that x¯ is an -solution of (RFP). Then, by Lemma ., x¯

is an ¯-solution of (RNCP)r¯, where r¯=max(u,v)∈U×Vf_g(₍x¯_¯x,_,u_v)₎ – and ¯=minv∈Vg(x¯,v), that is, for anyx∈A,maxu∈Uf(x,u) –r¯minv∈Vg(x,v)maxu∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) –

minv∈Vg(x¯,v). Sincemaxu∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) –minv∈Vg(x¯,v) = , we haveA⊆ {x∈C|maxu∈Uf(x,u) –r¯minv∈Vg(x,v)}. By Lemma .,

(, )∈ epi

max

u∈Uf(·,u)

∗

+epi

–¯rmin

v∈Vg(·,v)

∗

+cl co

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+

.

By assumption,

(, )∈ epi

max

u∈Uf(·,u)

∗

+epi

–¯rmin

v∈Vg(·,v)

∗

+

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+.

So, there existw¯i∈Wiandλ¯i,i= , . . . ,m, such that

(, )∈epi

max

u∈Uf(·,u)

∗

+epi

–¯rmin

v∈Vg(·,v)

∗ +epi _m i= ¯

λihi(·,w¯i)

∗

+epiδ_C∗.

By Proposition ., we obtain

(, )∈

_

ξ_,ξ_,x¯+_–max

u∈Uf(x¯,u)

ξ_∈∂ 

max

u∈Uf(·,u)

(x¯)

+

_

ξ_,ξ_,x¯+_+r¯min

v∈Vg(x¯,v)

ξ_∈∂ 

–r¯min

v∈Vg(·,v)

(x¯)

+

∗

ξ∗,ξ∗,x¯+∗– m

i= ¯

λihi(x¯,w¯i) ξ∗∈∂∗

_m

i= ¯

λihi(·,w¯i)

(x¯)

+

m+

ξm+,ξm+,x¯+m+–δC(x¯)

|ξm+∈∂m+δC(x¯)

So, there existξ¯

∈∂_(maxu∈Uf(·,u))(x¯),ξ¯

∈∂_(–¯rminv∈Vg(·,v))(x¯),ξ¯∗∈∂∗(

m i=λ¯ihi(·, ¯

wi))(x¯),ξ¯m+∈∂m+δC(x¯),,,∗, andm+ such that

 =ξ¯_+ξ¯_+ξ¯∗+ξ¯m+ and

_+_+∗+m+=max

u∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) + m

i= ¯

λihi(x¯,w¯i).

By Proposition ., there existξ¯_∈∂

(maxu∈Uf(·,u))(x¯),ξ¯ 

∈∂_(–r¯minv∈Vg(·,v))(x¯),ξ¯i∈

∂i(λ¯ihi(·,w¯i))(x¯),ξ¯m+∈∂m+δC(x¯),,,i,i= , . . . ,m, andm+ such that

∈∂ 

max

u∈Uf(·,u)

(x¯) +∂ 

–¯rmin

v∈Vg(·,v)

(x¯)

+ m

i=

∂i _¯

λihi(·,w¯i)

(x¯) +Nm+

C (x¯) and

_+_+ m+

i=

i=max

u∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) + m

i= ¯

λihi(x¯,w¯i).

()

Since¯r=max(u,v)∈U×Vfg((x¯x¯,,uv))–,

max

u∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) –minv∈Vg(x¯,v) = . () So, () holds, and so, from () and (), we have

∈∂ 

max

u∈Uf(·,u)

(x¯) +∂ 

–¯rmin

v∈Vg(·,v)

(x¯)

+ m

i=

∂i _¯

λihi(·,w¯i)

(x¯) +Nm+

C (x¯) and

_+_+ m+

i=

i–min v∈Vg(x¯,v) =

m

i= ¯

λihi(x¯,w¯i).

Thus the conditions () and () hold.

[(ii)⇒(i)] Taking account of the converse of the process for proving (i)⇒(ii), we can easily check that the statement (ii)⇒(i) holds.

If for allx∈Rn_{,f(x,·) is concave, and, for allx∈R,g(x,·) is convex, then using} Lem-mas . and ., we can obtain the following characterization of an-solution for (RFP).

Theorem .(-Optimality theorem) Let f :Rn_×Rp_{→Rand h}

i:Rn×Rq→R,i= , . . . ,m,be functions such that,for any u∈U,f(·,u),and,for each wi∈Wi,hi(·,wi)are

convex functions,and,for all x∈Rn_{,f(x,·)is concave function.Let g:R}n_×Rp_{→Rbe a}

function such that,for any v∈V,g(·,v)is a concave function,and,for all x∈R,g(x,·)is a convex function.LetU⊂Rp_,V⊂Rp_,andW

i⊂Rq,i= , . . . ,m.Letx¯∈A and let.

Let¯r=max(u,v)∈U×Vgf((x¯¯x,,uv)) –.Ifmax(u,v)∈U×V f(x¯,u)

max(u,v)∈U×Vgf((x¯¯x,,uv))and

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+

is closed and convex,then the following statements are equivalent: (i) x¯is an-solution of(RFP);

(ii) there existu¯∈U,v¯∈V,w¯i∈Wi,λ¯i,i= , . . . ,m,,,andi,

i= , . . . ,m+ such that

∈∂ 

f(·,u¯)(x¯) +∂ 

–rg¯ (·,v¯)(x¯) + m

i=

∂i _¯

λihi(·,w¯i)

(x¯) +Nm+

C (x¯), ()

max

u∈Uf(x¯,u) –minv∈V¯rg(x¯,v) =minv∈Vg(x¯,v) and ()

_+_+ m+

i=

i–min

v∈Vg(x¯,v) m

i= ¯

λihi(x¯,w¯i). ()

Proof [(i)⇒(ii)] Letx¯be an-solution of (RFP). Then, by Lemma .,x¯is an¯-solution of (RNCP)¯r, wherer¯=max(u,v)∈U×Vf_g(₍¯x_x¯,_,u_v))–and¯=minv∈Vg(x¯,v), that is, for anyx∈A,

maxu∈Uf(x,u) –¯rminv∈Vg(x,v)maxu∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) –minv∈Vg(x¯,v). Since

maxu∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) –minv∈Vg(x¯,v) = , we haveA⊆ {x∈C|maxu∈Uf(x,u) – ¯

rminv∈Vg(x,v)}. By Lemma .,

(, )∈ u∈U

epif(·,u)∗+ v∈V

epi–¯rg(·,v)∗

+cl co

wi∈Wi,λi epi

_m

i=

λihi(·,wi)

∗

+C∗×R+

.

By assumption,

(, )∈ u∈U

epif(·,u)∗+ v∈V

epi–rg¯ (·,v)∗+ wi∈Wi,λi

epi _m

i=

λihi(·,wi)

∗

+C∗×R+.

SinceC∗×R+=epiδ∗C, there existu¯∈U,v¯∈V,w¯i∈Wi, andλ¯i,i= , . . . ,m, such that

(, )∈epif(·,u¯)∗+epi–rg¯ (·,¯v)∗+epi _m

i= ¯

λihi(·,w¯i)

∗

+epiδ∗_C.

By Proposition ., we obtain

(, )∈

_

ξ_,ξ_,x¯+_–f(x¯,u¯)|ξ_∈∂ 

f(·,u¯)(x¯)

+

_

ξ_,ξ_,x¯+_+rg¯ (x¯,v¯)|ξ_∈∂ 

+

∗

ξ∗,ξ∗,x¯+∗– m

i= ¯

λihi(x¯,w¯i) ξ∗∈∂∗

_m

i= ¯

λihi(·,w¯i)

(x¯)

+

m+

ξm+,ξm+,x¯+m+–δC(x¯)

|ξm+∈∂m+δC(x¯)

.

So, there existξ¯_∈∂

(f(·,u¯))(x¯),ξ¯ 

∈∂_(–rg¯ (·,v¯))(x¯),ξ¯∗∈∂∗(

m

i=λ¯ihi(·,w¯i))(x¯),ξ¯m+∈

∂m+δC(x¯),,,∗, andm+ such that

 =ξ¯_+ξ¯_+ξ¯∗+ξ¯m+ and ++∗+m+=f(x¯,u¯) –rg¯ (x¯,v¯) + m

i= ¯

λihi(x¯,w¯i).

By Proposition ., there exist ξ¯_ ∈∂

(f(·,u¯))(x¯), ξ¯ 

 ∈ ∂_(–rg¯ (·,v¯))(x¯), ξ¯i ∈∂i(λ¯ihi(·,

¯

wi))(x¯),ξ¯m+∈∂m+δC(x¯),,,i,i= , . . . ,m, andm+ such that

∈∂ 

f(·,u¯)(x¯) +∂ 

–rg¯ (·,¯v)(x¯)

+ m

i=

∂i _¯

λihi(·,w¯i)

(x¯) +Nm+

C (x¯) and

_+_+ m+

i=

i=f(x¯,u¯) –rg¯ (x¯,v¯) + m

i= ¯

λihi(x¯,w¯i).

()

Sincer¯=max_(u,v)∈U×Vf(x¯,u)

g(¯x,v) –, we havemaxu∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) =minv∈Vg(x¯,v). So, we have

f(x¯,u¯) –rg¯ (x¯,v¯)max

u∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) =minv∈Vg(x¯,v). () So, the condition () holds. Also, from () and (), we have

∈∂ 

max

u∈Uf(·,u)

(x¯) +∂ 

–¯rmin

v∈Vg(·,v)

(x¯)

+ m

i=

∂i _¯

λihi(·,w¯i)

(x¯) +Nm+

C (x¯) and

_+_+ m+

i=

i–min

v∈Vg(x¯,v) m

i= ¯

λihi(x¯,w¯i).

Consequently, the conditions () and () hold.

[(ii)⇒(i)] Taking account of the converse of the process for proving (i)⇒(ii), we can easily check that the statement (ii)⇒(i) holds.

Remark . Assume thatf :Rn_×Rp_→Randg:Rn_×Rp_{→Rare functions such that,}

4

-Duality theorems

Following the approach in [], we formulate a dual problem (RFD) for (RFP) as follows:

(RFD) maxr

s.t. ∈∂ 

max

u∈Uf(·,u)

(x) +∂ 

–rmin

v∈Vg(·,v)

(x)

+ m

i=

∂i

λihi(·,wi)

(x) +Nm+ C (x),

max

u∈Uf(x,u) –rminv∈Vg(x,v)minv∈Vg(x,v),

_+_+ m+

i=

i–min

v∈Vg(x,v) m

i=

λihi(x,wi),

r,wi∈Wi,λi,i= , . . . ,m,

_,_,i,i= , . . . ,m+ .

Clearly,

F:=

(x,w,λ,r)∈∂ 

max

u∈Uf(·,u)

(x) +∂ 

–rmin

v∈Vg(·,v)

(x)

+ m

i=

∂i

λihi(·,wi)

(x) +N

R+(x),max_u∈Uf(x,u) –rmin_v∈Vg(x,v)g(x,v),

_+_+ m+

i=

i–min

v∈Vg(x,v) m

i=

λihi(x,wi),

r,wi∈Wi,λi,,,i,i= , . . . ,m,m+

is the feasible set of (RFD).

Let. Then (x¯,w¯,λ¯,r¯) is called an-solution of (RFD) if, for any (y,w,λ,r)∈F,r¯ r–.

When= ,maxu∈Uf(x,u) =f(x),minv∈Vg(x,v) =g(x), andhi(x,wi) =hi(x),i= , . . . ,m, (RFP) becomes (FP), and (RFD) collapses to the Mond-Weir type dual problem (FD) for (FP) as follows []:

(FD) maxr

s.t. ∈∂f(x) +∂(–rg)(x) + m

i=

∂λihi(x) +NC(x),

f(x) –rg(x),λihi(x),

r,λi,i= , . . . ,m.

Theorem . (-Weak duality theorem) For any feasible x of (RFP) and any feasible

(y,w,λ,r)of(RFD),

max

(u,v)∈U×V

f(x,u)

g(x,v) r–.

Proof Letxand (y,w,λ,r) be feasible solutions of (RFP) and (RFD), respectively. Then there existξ¯_∈∂

(maxu∈Uf(·,u))(y), ξ¯ 

 ∈∂_(–rminv∈Vg(·,v))(y),ξ¯i∈∂i(λihi(·,wi))(y),

¯

ξm+∈NCm+(y),,,i,i= , . . . ,m, andm+ such that

¯

ξ_+ξ¯_+ m+

i= ¯

ξi= , max

u∈Uf(y,u) –rminv∈Vg(y,v)minv∈Vg(y,v) and

_+_+ m+

i=

i–min v∈Vg(y,v)

m

i=

λihi(y,wi).

Thus, we have

max

u∈Uf(x,u) –rminv∈Vg(x,v) +minv∈Vg(x,v)

max

u∈Uf(y,u) –rminv∈Vg(y,v) +

_¯

ξ_+ξ¯_,x–y–_–_+min

v∈Vg(x,v) =max

u∈Uf(y,u) –rminv∈Vg(y,v) –

_m+

i= ¯

ξi,x–y

–_–_+min

v∈Vg(x,v)

max

u∈Uf(y,u) –rminv∈Vg(y,v) + m

i=

λihi(y,wi) – m

i=

λihi(x,wi) ––– m+

i=

i

+min

v∈Vg(x,v)

max

u∈Uf(y,u) –rminv∈Vg(y,v) + m

i=

λihi(y,wi) ––– m+

i=

i

max

u∈Uf(y,u) –rminv∈Vg(y,v) –minv∈Vg(y,v)

.

Hence, we havemax(u,v)∈U×Vf_g(₍x_x,_,u_v)₎r–.

Theorem .(-Strong duality theorem) Suppose that

wi∈Wi,λi epi

_m

i=

λigi(·,wi)

∗

+C∗×R+

is closed.Ifx is an¯ -solution of(RFP)andmax_(u,v)∈U×Vf(x¯,u)

g(x¯,v)–,then there existw¯ ∈R q_,

¯

λ∈Rm₊,and¯r∈R+such that(x¯,w¯,λ¯,r¯)is a-solution of(RFD).

that

∈∂ 

max

u∈Uf(·,u)

(x¯) +∂ 

–¯rmin

v∈Vg(·,v)

(x¯) + m

i=

∂i _¯

λihi(·,w¯i)

(x¯) +Nm+ C (x¯),

max

u∈Uf(x¯,u) –r¯minv∈Vg(x¯,v) =minv∈Vg(x¯,v) and

_+_+ m+

i=

i–min v∈Vg(x¯,v) =

m

i= ¯

λihi(x¯,w¯i).

So, (x¯,w¯,λ¯,r¯) is a feasible solution of (RFD). For any feasible (y,u,v,w,λ,v) of (RFD), it follows from Theorem . (-weak duality theorem) that

¯

r= max

(u,v)∈U×V

f(x¯,u)

g(x¯,v)–r––=r– .

Thus (x¯,w¯,λ¯,r¯) is a -solution of (RFD).

Remark . Using the optimality conditions of Theorem ., robust fractional dual prob-lem (RFD) for a robust fractional probprob-lem (RFP) in the convex constraint functions with uncertainty is formulated. However, when we formulated the dual problem using opti-mality condition in Theorem ., we could not know whether-weak duality theorem is established, or not. It is an open question.

Now we give an example illustrating our duality theorems.

Example . Consider the following fractional programming problem with uncertainty:

(RFP) min max

(u,v)∈U×V

ux+ 

vx+ 

s.t. wx– ,w∈[, ],

x∈R+,

whereU= [, ] andV= [, ].

Now we transform the problem (RFP) into the robust non-fractional convex optimiza-tion problem (RNCP)rwith a parametricr∈R+:

(RNCP)r min max

u∈[,](ux+ ) –rvmin∈[,](vx+ ) s.t. wx– ,w∈[, ],

x∈R+.

Letf(x,u) =ux+ ,g(x,v) =vx+ ,h(x,w) = –wx– , and∈[,_]. ThenA:={x∈

R|x_}is the set of all robust feasible solutions of (RFP) andA¯ :={x∈R|x



–}is the set of all-solutions of (RFP). LetF:={(y,w,λ,r)|∈∂_(maxu∈Uf(·,u))(y) +

∂

(–rminv∈Vg(·,v))(y) + ∂(λh(·,w))(y) + N



minv∈Vg(y,v),

+++–minv∈Vg(y,v)λh(y,w),r,w∈[, ],λ, ,

,,}. Then we formulate a dual problem (RFD) for (RFP) as follows:

(RFD) maxr

s.t. (y,w,λ,r)∈F.

ThenFis the set of all robust feasible solutions of (RFD). Now we calculate the setF=A∪B.

A:=(,w,λ,r)∈∂_

max

u∈Uf(·,u)

()

+∂ 

–rmin

v∈Vg(·,v)

() +∂

λh(·,w)

()

+N

R+(),max_u∈Uf(,u) –rmin_v∈Vg(,v)min_v∈Vg(,v), 

+++–min v∈Vg(,v)

λh(,w),r,u∈[, ],λ,,,,

=(,w,λ,r)|∈ {}+{–r}+{λw}+ (–∞, ],  – r,+++

– –λ,r,w∈[, ],λ,,,,

=

(,w,λ,r)r + λw,r  – 

 , 

+++– –λ,r,

w∈[, ],λ,,,,

,

B:=

(y,w,λ,r)∈∂ 

max

u∈Uf(·,u)

(y) +∂ 

–rmin

v∈Vg(·,v)

(y) +∂

λh(·,w)

(y)

+N

R+(y),max_u∈Uf(y,u) –rmin_v∈Vg(y,v)min_v∈Vg(y,v), 

+++–min v∈Vg(y,v)

λh(y,w),y> ,r,w∈[, ],λ,,,,

=

(y,w,λ,r)∈ {}+{–r}+{λw}+

–

y, 

, y+  –r(y+ )(y+ ),

y> ,_+_++–(y+ )λ(wy– ),r,w∈[, ],λ,

_,_,,

=

(y,w,λ,r)∈

 –r+ λw–



y,  –r+ λw

, y+  –r(y+ )

(y+ ),_+_++–(y+ )λ(wy– ),y> ,r,w∈[, ],

λ,,,,

.

We can check for anyx∈Aand any (y,w,λ,r)∈F,

max

(u,v)∈U×V

f(x,u)