On ɛ-solutions for robust fractional optimization problems

Transcription

1 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 R E S E A R C H Open Access On ɛ-solutions for robust fractional optimization problems Jae Hyoung Lee and Gue Myung Lee * * Correspondence: gmlee@pknu.ac.kr Department of Applied Mathematics, Pukyong National University, 45, Yongso-ro, Nam-Gu, Busan, , Korea Abstract We consider ɛ-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 ɛ-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 ɛ-solution approximate solution, many authors have established ɛ-optimality conditions and ɛ-duality theorems for several kinds of optimization problems [1 7]. Especially, Lee and Lee [8]gave an ɛ-duality theorems for a convex semidefinite optimization problem with conic constraints. Also, they [9] established optimality theorems and duality theoremsfor ɛ-solutions for convex optimization problems with uncertainty data. In [10 15], manyauthorshavetreatedfractionalprogrammingproblemsintheabsence of data uncertainty. Recently, many authors have studied robust optimization problems [9, 16 21]. Very recently, Jeyakumar and Li [22] 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 ɛ-optimality theorems and ɛ-duality theorems in [9] to fractional optimization problems with uncertainty data. Consider the following standard form of fractional programming problem with a geometric constraint set: FP min f x gx s.t. h i x 0, i =1,...,m, x C, where f, h i : R n R, i =1,...,m, are convex functions, C isaclosedconvexconeofr n, and g : R n R is a concave function such that, for any x C, f x 0andgx> Lee and Lee; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 2 of 21 The fractional programming problem FP in the face of data uncertainty in the constraints can be captured by the problem: f x, u UFP min u,v U V gx, v s.t. h i x, w i 0, i =1,...,m, x C, where f : R n R p R, h i : R n R q R, f, u andh i, w i areconvex,andg : R n R p R, g, visconcave,andu R p, v R p,andw i R q are uncertain parameters which belong to the convex and compact uncertainty sets U R p, V R p,andw i R q, i = 1,...,m,respectively. We study ɛ-optimality theorems and ɛ-duality theorems for the uncertain fractional programming model problem UFP by examining its robust worst-case counterpart [18]: f x, u RFP min u,v U V gx, v s.t. h i x, w i 0, w i W i, i =1,...,m, x C. Clearly, A := {x C h i x, w i 0, w i W i, i =1,...,m} is a feasible set of RFP. Let ɛ 0. Then x is called an ɛ-solution of RFP if, for any x A, f x, u u,v U V gx, v u,v U V f x, u g x, v ɛ. Using the parametric approach, we transform the problem RFP into the robust nonfractional convex optimization problem RNCP r with a parametric r R + : RNCP r min f x, u rmin gx, v s.t. h i x, w i 0, w i W i, i =1,...,m, x C. Let ɛ 0. Then x is called an ɛ-solution of RNCP r if, for any x A, f x, u r min gx, v f x, u rmin g x, v ɛ. In this paper, we consider ɛ-solutions for RFP, and we establish optimality theorems and duality theorems for ɛ-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. R n denotes the Euclidean space with dimension n. The nonnegative orthant of R n is denoted by R n + and is defined by Rn + := {x 1,...,x n R n : x i 0}.WesaythesetA is convex

3 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 3 of 21 whenever μa 1 +1 μa 2 A for all μ [0, 1], a 1, a 2 A. A function f : R n R {+ } is said to be convex if, for all μ [0, 1], f 1 μx + μy 1 μf x+μf y for all x, y R n. The function f is said to be concave whenever f is convex. Let g : R n R {+ } be a convex function. The subdifferential of g at a dom g is defined by ga:= { v R n gx ga+ v, x a x dom g }, where, is the inner product on R n and dom g := {x R n : gx<+ }.Letɛ 0. Then the ɛ-subdifferential of g at a dom g isdefinedby ɛ ga:= { v R n gx ga+ v, x a ɛ x dom g }. The function f is said to be proper if f x> for all x R n.wesayf is a lower semicontinuous function if lim inf y x f y f x for all x R n. As usual, for any proper convex function g on R n, its conjugate function g : R n R {+ } is defined, for any x R n, by g x =sup{ x, x gx x R n }. The epigraph of a function g : R n R {+ }, epi g, is defined by epi g = {x, r R n R gx r}.wedenotetheconvexhullofasubseta of R n by co A, and denote the closure of the set A by cl A.LetC beaclosedconvexsetinr n and x C. Then the normal cone N C xtoc at x is defined by N C x= { v R n v, y x 0, for all y C }, and we let ɛ 0, then the ɛ-normal cone NC ɛ xtoc at x is defined by N ɛ C x={ v R n v, y x ɛ, for all y C }. When C isaclosedconvexconeinr n,wedenoten C 0 by C and call it the negative dual cone of C. Proposition 2.1 [23] Let f : R n R be a convex function and let δ C be the indicator function with respect to a closed convex subset C of R n, that is, δ C x=0if x C, and δ C x=+ if x / C. Let ɛ 0. Then ɛ f + δ C x= ɛ 0 0,ɛ 1 0 ɛ 0 +ɛ 1 =ɛ { ɛ0 f x+ ɛ1 δ C x }. Proposition 2.2 [24, 25] If f : R n R {+ } is a proper lower semicontinuous convex function and if a dom f := {x R n f x<+ }, then epi f = ɛ 0{ v, v, a + ɛ f a v ɛ f a }. Proposition 2.3 [26] Let f : R n R be a convex function and g : R n R {+ } be a proper lower semicontinuous convex function. Then epif + g = epi f + epi g.

4 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 4 of 21 Moreover, if f, g : R n R {+ } are proper lower semicontinuous convex functions, and if dom f dom g, then epif + g = cl epi f + epi g. Proposition 2.4 [22, 26] Let h i : R n R {+ }, i I where I is an arbitrary index set, be a proper lower semicontinuous convex function. Suppose that there exists x 0 R n such that sup i I h i x 0 <+. Then epi sup i I h i = cl co epi h i. i I Proposition 2.5 [23] Let h i : R n R {+ }, i =1,...,m, be proper lower semicontinuous convex functions. Let ɛ 0. If m ri dom h i, where ri dom h i is the relative interior of dom h i, then for all x n dom h i, m ɛ h x= { m } m ɛi h i x ɛ i 0, i =1,...,m, ɛ i = ɛ. Proposition 2.6 [9] Let h i : R n R q R, i =1,...,m, be continuous functions such that, for all w i R q, h i, w i is a convex function and let C be a closed convex cone of R n. Suppose that each W i, i =1,...,m, is compact and convex, and there exists x 0 Csuchthat h i x 0, w i <0,for all w i W i, i =1,...,m. Then is closed. m epi λ i h i, w i + C R + Proposition 2.7 [9] Let h i : R n R q R, i =1,...,m, be continuous functions and let C be a closed convex cone of R n. Suppose that each W i R q, i =1,...,m, is convex, for all w i R q, h i, w i is a convex function, and, for each x R n, h i x, is concave on W i. Then m epi λ i h i, w i + C R + is convex. Now we give the following relation between the ɛ-solutions of RFP and RNCP r. f x,u Lemma 2.1 Let x Aandletɛ 0. If u,v U V ɛ 0, then the following statements are g x,v equivalent: i x is an ɛ-solution of RFP; ii x is an ɛ-solution of RNCP r, where r = u,v U V f x,u g x,v ɛ and ɛ = ɛ min g x, v.

5 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 5 of 21 f x,u Proof Let x A be an ɛ-solution of RFP. Then for any x A, u,v U V gx,v f x,u u,v U V g x,v ɛ.put r = u,v U V f x,u g x,v ɛ and ɛ = ɛ min g x, v. Then we have, for any x A, f x, u min rgx, v 0. Since f x, u r min g x, v ɛ min g x, v=0,foranyx A, f x, u r min gx, v = f x, u r min f x, u r min g x, v ɛ. g x, v ɛmin g x, v Hence x is an ɛ-solution of RNCP r. Let x A be an ɛ-solution of RNCP r. Then for any x A, f x, u r min gx, v f x, u r min g x, v ɛ. Since f x, u r min g x, v ɛmin g x, v =0,foranyx A, f x, u r min gx, v 0. So, we have f x,u u,v U V gx,v r.since r = u,v U V f x,u g x,v ɛ, f x, u u,v U V gx, v u,v U V f x, u g x, v ɛ. Hence x is an ɛ-solution of RFP. 3 ɛ-optimality theorems In this section, we establish ɛ-optimality theorems for ɛ-solutions for the robust fractional optimization problem. Now we give the following lemma which is the robust version of Farkas lemma for nonfractional convex functions. Lemma 3.1 Let f : R n R p R and h i : R n R q R, i = 1,...,m, be functions such that, for any u U, f, u and, for each w i W i, h i, w i are convex functions, and, for any x R n, f x, is concave function. Let g : R n R q R be a function such that, for any v V, g, v is a concave function, and, for all x R, gx, isaconvexfunction. Let U R p, V R p, and W i R q, i =1,...,m be convex and compact sets. Let r 0 and let C be a closed convex cone of R n. Assume that A := {x C h i x, w i 0, w i W i, i =1,...,m} =. Then the following statements are equivalent: i {x C h i x, w i 0, w i W i, i =1,...,m} {x R n f x, u rmin gx, v 0}; ii there exist ū U and v V such that { x C hi x, w i 0, w i W i, i =1,...,m } { x R n f x, ū rgx, v 0 } ; iii 0, 0 epi f, u + epi rg, v m + cl co epi λ i h i, w i + C R + ;

6 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 6 of 21 iv 0, 0 epi f, u + epi r min g, v m + cl co epi λ i h i, w i + C R +. Proof Let D := {x R n h i x, w i 0, w i W i, i = 1,...,m}. ThenA = C D. Wewill prove that epi δ A = cl co epi m λ ih i, w i + C R +. For any x R n, δ A x=δ C x+δ D x and δ D x= sup Thus, by Propositions 2.3and2.4,wehave epi δa = epiδ D + δ C = cl epi δd + epi δ C = cl = cl = cl co epi cl co sup w i W i, λ i 0 m λ i h i x, w i. m λ i h i, w i + epi δ m epi λ i h i, w i + epi δ m epi λ i h i, w i + C R +. C C [i iv] Now we assume that the statement iv holds. Then, by Proposition 2.3,the statement iv is equivalent to 0, 0 epi f, u + epi r min g, v + epi δ A = epi f, u r min g, v+δ A Equivalently, by definition of epigraph of f, u r min g, v+δ A, 0 f, u rmin g, v+δ A 0. From the definition of a conjugate function, for any x R n, f, u rmin g, v+δ A x 0.. It is equivalent to the statement that, for any x A, f x, u rmin gx, v 0.

7 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 7 of 21 [ii iii] Now we assume that the statement iii holds. Then the statement iii is equivalent to 0, 0 epi f, u + epi rg, v + epi δ A. It means that there exist ū U and v V such that 0, 0 epi. f, ū rg, v+δ A It is equivalent to the statement that there exist ū U and v V such that f, ū rg, v+δa 0 0. From the definition of a conjugate function, there exist ū U and v V such that, for any x R n, f, ū rg, v+δa x 0. It means that there exist ū U and v V such that, for any x A, f x, ū rgx, v 0. [iii iv]togetthedesiredresult,itsufficestoshowthat epi f, u = epi f, u, 1 epi rg, v = epi r min g, v. 2 By Proposition 2.4, epi f, u = cl co epif, u.letz 1, α 1, z 2, α 2 epif, u and let μ [0, 1]. Then there exist u 1, u 2 U such that z 1, α 1 epif, u 1 and z 2, α 2 epif, u 2,thatis,f, u 1 z 1 α 1 and f, u 2 z 2 α 2. Using the definition of a conjugate function, we have, for all x R n, z 1, x f x, u 1 α 1 and z 2, x f x, u 2 α 2. 3 Since, for all x R n, f x, isconcave,wehavef x, μu 1 +1 μu 2 μf x, u 1 +1 μf x, u 2, i.e., f x, μu 1 +1 μu 2 μf x, u1 1 μf x, u 2. 4 So, from 3and4, we have, for all x R n, μz1 +1 μz 2, x f x, μu 1 +1 μu 2 μα1 +1 μα 2,

8 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 8 of 21 and so f, μu 1 +1 μu 2 μz 1 +1 μz 2 μα 1 +1 μα 2.Hence,wehave μz1 +1 μz 2, μα 1 +1 μα 2 epi f, μu1 +1 μu 2. So, epif, u is convex. Nowweshowthat epif, u is closed. Let z n, α n epi f, u with z n, α n z, α asn. Then there exists u n U such that f, u n z n α n. Since U is compact, we may assume that u n u U as n. So, for each x R n, z n, x f x, u n α n. Since, for all x R n, f x, isconcave,f x, is continuous. Passing to the limit as n, we get, for each x R n, z, x f x, u α.hence,wehave z, α epi f, u. So, epif, u is closed. Thus, 1holds. Moreover, since, for all x R n, gx, isconvexandr 0, for all x R n, rgx, is concave. So, similarly, we can prove that 2holds. Remark 3.1 Using convex-concave mini theorem Corollary in [27], we can prove that the statement i in Lemma 3.1 is equivalent to the statement ii in Lemma 3.1. Remark 3.2 From proving in Lemma 3.1 that the statement i is equivalent to the statement iv, we see that we can prove the equivalent relation without the assumptions that, for all x R n, f x,, and gx, are concave and convex, respectively. From Lemmas 2.1 and 3.1, we can get the following theorem. Theorem 3.1 Let f : R n R p R and h i : R n R q R, i =1,...,m, be functions such that, for any u U, f, u, and, for each w i W i, h i, w i are convex functions, and, for any x R n, f x, is concave function. Let g : R n R p R be a function such that, for any v V, g, v is a concave function, and, for all x R n, gx, is a convex function. Let U R p, V R p, and W i R q f x,u, i =1,...,m. Let x Aandlet r = u,v U V g x,v ɛ. Suppose that epi m λ ih i, w i + C R + is closed and convex. Then the following statements are equivalent: i x is an ɛ-solution of RFP; ii There exist ū U, v V, w i W i, and λ i 0, i =1,...,m such that, for any x C, f x, ū rgx, v+ m λ i h i x, w i 0.

9 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 9 of 21 Proof Let x be an ɛ-solution of RFP. Then, by Lemma 2.1, equivalently, x is an ɛ-solution of RNCP r,where r = u,v U V f x,u g x,v ɛ and ɛ = ɛ min g x, v, that is, for any x A, f x, u r min gx, v f x, u r min g x, v ɛ min g x, v. Since f x, u r min g x, v ɛ min g x, v=0,wehavea {x C f x, u r min gx, v 0}. Then, by Lemma 3.1,wehave 0, 0 + cl co epi f, u + epi rg, v Moreover, by assumption, 0, 0 m epi λ i h i, w i + C R +. epi f, u + epi rg, v + So, there exist ū U, v V, w i W i,and λ i 0, i =1,...,m such that m epi λ i h i, w i +C R +. 0, 0 epi f, ū m + epi rg, v + epi λ i h i, w i + C R +. Then there exist s R n, η 0, t R n, μ 0, z i R n, ρ i 0, i = 1,...,m, c C,and γ R + such that 0, 0 = s, f, ū s+η + t, rg, v t+μ + m zi, λ i h i, w i zi +ρ i + c, γ. So, 0 = s+t + m z i +c and 0 = f, ū s+η + rg, v t+μ+ m λ i h i, w i z i + ρ i +γ.hence,foranyx R n, m z i, x c, x f x, ū rgx, v = s, x + t, x f x, ū rgx, v f, ū s+ rg, v t = η μ m λ i h i, w i zi +ρ i γ. 5 Since η 0, μ 0, ρ i 0, i =1,...,m,andc C,from5, for any x C, m 0 z i, x + c, x + f x, ū+ rgx, v η μ m λ i h i, w 1 m zi λ i ρ i γ

10 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 10 of 21 m m z i, x + f x, ū rgx, v λ i h i, w i zi m f x, ū rgx, v+ λ i h i x, w i. Suppose that there exist ū U, v V, w i W i,and λ i 0, i =1,...,m, suchthat, for any x C, f x, ū rgx, v+ m λ i h i x, w i 0. 6 Since r = u,v U V f x,u g x,v ɛ,wehave f x, u r min g x, v ɛ min g x, v= 0. So, from 6, we have, for any x A, f x, u r min gx, v f x, u r min f x, ū rgx, v+ 0 = f x, u r min gx, v+ m λ i h i x, w i m λ i h i x, w i g x, v ɛmin g x, v. Hence, for any x A, f x, u r min gx, v f x, u r min g x, v ɛ min g x, v. It means that x is an ɛ-solution of RNCP r. Thus, by Lemma 2.1, x is an ɛ-solution of RFP. Using Remark 3.2and Lemmas2.1 and3.1, we canobtain thefollowingcharacterization of an ɛ-solution for RFP. Theorem 3.2 ɛ-optimality theorem Let f : R n R p R and h i : R n R q R, i = 1,...,m, be functions such that, for any u U, f, u, and, for each w i W i, h i, w i are convex functions. Let g : R n R p R be a function such that, for any v V, g, v is a concave function. Let U R p, V R p, and W i R q, i =1,...,m. Let x Aandletɛ 0. f x,u Let r = u,v U V g x,v ɛ. If u,v U V f x,u < ɛ, then x isanɛ-solution of RFP. If g x,v f x,u u,v U V g x,v ɛ and m epi λ i h i, w i + C R + is closed and convex, then the following statements are equivalent: i x is an ɛ-solution of RFP;

11 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 11 of 21 ii there exist w i W i and λ i 0, i =1,...,m, ɛ 1 0 0, ɛ2 0 0, and ɛ i 0, i =1,...,m +1 such that 0 ɛ 1 f, u x+ 0 ɛ 2 r min g, v x 0 m + ɛi λ i h i, w i x+n ɛ C x, 7 f x, u r min g x, v=ɛmin g x, v and 8 ɛ0 1 + ɛ2 0 + ɛ i ɛ min g x, v= m λ i h i x, w i. 9 Proof [i ii] We assume that x is an ɛ-solution of RFP. Then, by Lemma 2.1, x is an ɛ-solution of RNCP r,where r = u,v U V f x,u g x,v ɛ and ɛ = ɛ min g x, v, that is, for any x A, f x, u r min gx, v f x, u r min g x, v ɛ min g x, v. Since f x, u r min g x, v ɛ min g x, v=0,wehavea {x C f x, u r min gx, v 0}. By Lemma 3.1, 0, 0 epi f, u + epi r min g, v m + cl co epi λ i h i, w i + C R +. By assumption, 0, 0 epi f, u + epi r min g, v m + epi λ i h i, w i + C R +. So, there exist w i W i and λ i 0, i =1,...,m,suchthat 0, 0 epi f, u + epi r min g, v m + epi By Proposition 2.2,we obtain ɛ λ i h i, w i + epi δ C. 0, 0 { ξ0 1, } ξ0 1, x + ɛ0 1 f x, u ξ 1 0 ɛ 1 f, u x 0 + { ξ0 2, } ξ0 2, x + ɛ0 2 + r min g x, v ξ 2 0 ɛ 2 r min g, v x 0 ɛ { ξ, ξ, x m m } + ɛ λ i h i x, w i ξ ɛ λ i h i, w i x ɛ 0 + { ξ, ξ, x + ɛ δ C x ξ ɛ δ C x }. ɛ 0

12 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 12 of 21 So, there exist ξ 1 0 ɛ 1 0 f, u x, ξ 2 0 ɛ 2 0 r min g, v x, ξ ɛ m λ i h i, w i x, ξ ɛ δ C x, ɛ 1 0 0, ɛ2 0 0, ɛ 0, and ɛ 0suchthat 0= ξ ξ ξ + ξ and ɛ0 1 + ɛ2 0 + ɛ + ɛ = f x, u r min g x, v+ m λ i h i x, w i. By Proposition 2.5,thereexist ξ 0 1 ɛ0 1 f, u x, ξ 0 2 ɛ0 2 r min g, v x, ξ i ɛi λ i h i, w i x, ξ ɛ δ C x, ɛ0 1 0, ɛ2 0 0, ɛ i 0, i =1,...,m, andɛ 0such that 0 ɛ 1 f, u x+ 0 ɛ 2 r min g, v x 0 m + ɛi λ i h i, w i x+n ɛ C x and ɛ0 1 + ɛ2 0 + ɛ i = f x, u r min g x, v+ m Since r = u,v U V f x,u g x,v ɛ, λ i h i x, w i. 10 f x, u r min g x, v ɛmin g x, v=0. 11 So, 8holds,and so,from10 and11, we have 0 ɛ 1 f, u x+ 0 ɛ 2 r min g, v x 0 m + ɛi λ i h i, w i x+n ɛ C x and ɛ0 1 + ɛ2 0 + ɛ i ɛ min g x, v= m λ i h i x, w i. Thus the conditions 7and9hold. [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 all x R n, f x, is concave, and, for all x R, gx, is convex, then using Lemmas 2.1 and 3.1, we can obtain the following characterization of an ɛ-solution for RFP. Theorem 3.3 ɛ-optimality theorem Let f : R n R p R and h i : R n R q R, i = 1,...,m, be functions such that, for any u U, f, u, and, for each w i W i, h i, w i are convex functions, and, for all x R n, f x, is concave function. Let g : R n R p R be a function such that, for any v V, g, v is a concave function, and, for all x R, gx, is a convex function. Let U R p, V R p, and W i R q, i =1,...,m. Let x Aandletɛ 0. f x,u Let r = u,v U V g x,v ɛ. If u,v U V f x,u < ɛ, then x isanɛ-solution of RFP. If g x,v

13 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 13 of 21 u,v U V f x,u g x,v ɛ and m epi λ i h i, w i + C R + is closed and convex, then the following statements are equivalent: i x is an ɛ-solution of RFP; ii there exist ū U, v V, w i W i, λ i 0, i =1,...,m, ɛ 1 0 0, ɛ2 0 0, and ɛ i 0, i =1,...,m +1such that 0 ɛ 1 0 f, ū x+ ɛ 2 0 rg, v x+ m f x, u min ɛi λ i h i, w i x+n ɛ C x, 12 rg x, v=ɛmin g x, v and 13 ɛ0 1 + ɛ2 0 + ɛ i ɛ min g x, v m λ i h i x, w i. 14 Proof [i ii] Let x be an ɛ-solution of RFP. Then, by Lemma 2.1, x is an ɛ-solution of RNCP r,where r = u,v U V f x,u g x,v ɛ and ɛ = ɛ min g x, v, that is, for any x A, f x, u r min gx, v f x, u r min g x, v ɛ min g x, v. Since f x, u r min g x, v ɛ min g x, v=0,wehavea {x C f x, u r min gx, v 0}. By Lemma 3.1, 0, 0 By assumption, 0, 0 + cl co epi f, u + epi rg, v m epi λ i h i, w i + C R +. epi f, u + epi rg, v + m epi λ i h i, w i +C R +. Since C R + = epi δ C,thereexistū U, v V, w i W i,and λ i 0, i =1,...,m,suchthat 0, 0 epi f, ū + epi rg, v + epi m By Proposition 2.2,we obtain 0, 0 ɛ ɛ λ i h i, w i + epi δ C. { ξ 1 0, ξ0 1, x + ɛ0 1 f x, ū ξ0 1 } ɛ0 1 f, ū x { ξ 2 0, ξ0 2, x + ɛ0 2 + rg x, v ξ0 2 } ɛ0 2 rg, v x

14 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 14 of 21 + { ξ, ξ, x m m } + ɛ λ i h i x, w i ξ ɛ λ i h i, w i x ɛ 0 + { ξ, ξ, x + ɛ δ C x ξ ɛ δ C x }. ɛ 0 So, there exist ξ 1 0 ɛ 1 0 f, ū x, ξ 2 0 ɛ 2 0 rg, v x, ξ ɛ m λ i h i, w i x, ξ ɛ δ C x, ɛ 1 0 0, ɛ2 0 0, ɛ 0, and ɛ 0suchthat 0= ξ ξ ξ + ξ and ɛ ɛ2 0 + ɛ + ɛ = f x, ū rg x, v+ m λ i h i x, w i. By Proposition 2.5, thereexist ξ 0 1 ɛ0 1 f, ū x, ξ 0 2 ɛ0 2 rg, v x, ξ i ɛi λ i h i, w i x, ξ ɛ δ C x, ɛ0 1 0, ɛ2 0 0, ɛ i 0, i =1,...,m,andɛ 0suchthat 0 ɛ 1 0 f, ū x+ ɛ 2 0 rg, v x + m ɛi λ i h i, w i x+n ɛ C x and ɛ0 1 + ɛ2 0 + m ɛ i = f x, ū rg x, v+ λ i h i x, w i. 15 Since r = u,v U V f x,u g x,v ɛ,wehave f x, u r min g x, v=ɛ min g x, v. So, we have f x, ū rg x, v f x, u r min So, the condition 13holds.Also,from 15and16, we have 0 ɛ 1 f, u x+ 0 ɛ 2 r min g, v x 0 m + ɛi λ i h i, w i x+n ɛ C x and ɛ0 1 + ɛ2 0 + ɛ i ɛ min g x, v m λ i h i x, w i. g x, v=ɛmin g x, v. 16 Consequently, the conditions 12and14hold. [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 3.3 Assume that f : R n R p R and g : R n R p R are functions such that, for all x R n, f x,, and gx, are concave and convex, respectively. Then we know that Theorem 3.2is equivalent totheorem 3.3fromLemma 3.1, immediately.

15 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 15 of 21 4 ɛ-duality theorems Following the approach in [13], we formulate a dual problem RFD for RFP as follows: RFD r s.t. 0 ɛ 1 f, u x+ 0 ɛ 2 r min g, v x 0 + m ɛi λi h i, w i x+n ɛ C x, f x, u r min gx, v ɛ min gx, v, ɛ0 1 + ɛ2 0 + ɛ i ɛ min gx, v m λ i h i x, w i, r 0, w i W i, λ i 0, i =1,...,m, ɛ 1 0 0, ɛ2 0 0, ɛ i 0, i =1,...,m +1. Clearly, { F := x, w, λ, r 0 ɛ0 1 f, u x+ ɛ 2 r min g, v x 0 + m ɛi λi h i, w i x+n ɛ 2 R + x, f x, u rmin gx, v ɛgx, v, ɛ0 1 + ɛ2 0 + ɛ i ɛ min gx, v m λ i h i x, w i, r 0, w i W i, λ i 0, ɛ 1 0 0, ɛ2 0 0, ɛ i 0, i =1,...,m, ɛ 0 } is the feasible set of RFD. Let ɛ 0. Then x, w, λ, r is called an ɛ-solution of RFD if, for any y, w, λ, r F, r r ɛ. When ɛ =0, f x, u=f x, min gx, v=gx, and h i x, w i =h i x, i =1,...,m, RFP becomes FP, and RFD collapses to the Mond-Weir type dual problem FD for FP as follows [28]: FD r m s.t. 0 f x+ rgx+ λ i h i x+n C x, f x rgx 0, λ i h i x 0, r 0, λ i 0, i =1,...,m. Now, we prove ɛ-weak and ɛ-strong duality theorems which hold between RFP and RFD.

16 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 16 of 21 Theorem 4.1 ɛ-weak duality theorem For any feasible x of RFP and any feasible y, w, λ, r of RFD, f x, u u,v U V gx, v r ɛ. Proof Let x and y, w, λ, r be feasible solutions of RFP and RFD, respectively. Then there exist ξ 0 1 ɛ0 1 f, uy, ξ 0 2 ɛ0 2 r min g, vy, ξ i ɛi λ i h i, w i y, ξ N ɛ C y, ɛ0 1 0, ɛ2 0 0, ɛ i 0, i =1,...,m,andɛ 0suchthat ξ ξ ξ i =0, f y, u r min ɛ0 1 + ɛ2 0 + ɛ i ɛ min gy, v m λ i h i y, w i. gy, v ɛ min gy, v and Thus, we have f x, u r min gx, v+ɛmin gx, v f y, u rmin gy, v+ ξ ξ 0 2, x y ɛ0 1 ɛ2 0 + ɛ min gx, v = f y, u rmin gy, v ξ i, x y ɛ0 1 ɛ2 0 + ɛ min gx, v f y, u rmin gy, v+ m m λ i h i y, w i λ i h i x, w i ɛ0 1 ɛ2 0 + ɛ min gx, v f y, u rmin gy, v+ m λ i h i y, w i ɛ0 1 ɛ f y, u r min gy, v ɛmin gy, v ɛ i ɛ i Hence, we have u,v U V f x,u gx,v r ɛ. Theorem 4.2 ɛ-strong duality theorem Suppose that m epi λ i g i, w i + C R + is closed. If xisanɛ-solution of RFP and u,v U V f x,u g x,v ɛ 0, then there exist w Rq, λ R m +, and r R + such that x, w, λ, r is a 2ɛ-solution of RFD. Proof Let x A be an ɛ-solution of RFP. Let r = u,v U V f x,u g x,v.then,bytheorem 3.2, thereexist w i W i, λ i 0, ɛ 1 0 0, ɛ2 0 0, ɛ i 0, i = 1,...,m, andɛ such

17 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 17 of 21 that 0 ɛ 1 f, u x+ 0 ɛ 2 r min g, v x+ 0 f x, u r min g x, v=ɛmin g x, v and ɛ0 1 + ɛ2 0 + ɛ i ɛ min g x, v= m λ i h i x, w i. m ɛi λ i h i, w i x+n ɛ C x, So, x, w, λ, r is a feasible solution of RFD. For any feasible y, u, v, w, λ, v ofrfd,it followsfromtheorem 4.1 ɛ-weak duality theorem that r = u,v U V f x, u g x, v ɛ r ɛ ɛ = r 2ɛ. Thus x, w, λ, r isa2ɛ-solutionof RFD. Remark 4.1 Using the optimality conditions of Theorem 3.2, robust fractionaldualproblem RFD for a robust fractional problem RFP in the convex constraint functions with uncertainty is formulated. However, when we formulated the dual problem using optimality condition in Theorem 3.3, we could not know whether ɛ-weak duality theorem is established, or not. It is an open question. Now we give an example illustrating our duality theorems. Example 4.1 Consider the following fractional programming problem with uncertainty: ux +1 RFP min u,v U V vx +2 s.t. 2w 1 x 3 0, w 1 [1, 2], x R +, where U = [1, 2] and V = [1, 2]. Now we transform the problem RFP into the robust non-fractional convex optimization problem RNCP r with a parametric r R + : RNCP r min ux +1 r min u [1,2] v [1,2] s.t. 2w 1 x 3 0, w 1 [1, 2], vx +2 x R +. Let f x, u=ux +1,gx, v=vx +2,h 1 x, w 1 = 2w 1 x 3,andɛ [0, 9 ]. Then A := {x 22 R 0 x 3 } is the set of all robust feasible solutions of RFP and Ā := {x R 0 x 4 4ɛ 3 2ɛ } is the set of all ɛ-solutions of RFP. Let F := {y, w 1, λ 1, r 0 ɛ 1 f, uy+ 0 ɛ 2 r min g, vy + ɛ1 λ 1 h 1, w 1 y +N ɛ 2 0 R + x, f y, u r min gy, v

18 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 18 of 21 ɛ min gy, v, ɛ 1 0 +ɛ2 0 +ɛ 1 +ɛ 2 ɛ min gy, v λ 1 h 1 y, w 1, r 0, w 1 [1, 2], λ 1 0, ɛ 1 0 0, ɛ 2 0 0, ɛ 1 0, ɛ 2 0}. Then we formulate a dual problem RFD for RFP as follows: RFD r s.t. y, w 1, λ 1, r F. Then F is the set of all robust feasible solutions of RFD. Now we calculate the set F = Ã B. Ã := {0, w 1, λ 1, r 0 ɛ 10 f, u 0 + ɛ 2 r min g, v 0 + ɛ1 λ1 h 1, w N ɛ 2 R + 0, f 0, u rmin g0, v ɛ min g0, v, ɛ1 0 + ɛ2 0 + ɛ 1 + ɛ 2 ɛ min g0, v } λ 1 h 1 0, w 1, r 0, u [1, 2], λ 1 0, ɛ0 1 0, ɛ2 0 0, ɛ 1 0, ɛ 2 0 = { 0, w 1, λ 1, r 0 {2} + { r} + {2λ 1 w 1 } +,0],1 2r 2ɛ, ɛ0 1 + ɛ2 0 + ɛ 1 + ɛ 2 2ɛ 3λ 1, r 0, w 1 [1, 2], λ 1 0, ɛ0 1 0, ɛ2 0 0, ɛ 1 0, ɛ 2 0 } { = 0, w 1, λ 1, r r 2+2λ 1 w 1, r 1 2ɛ, ɛ ɛ2 0 + ɛ 1 + ɛ 2 2ɛ 3λ 1, r 0, } w 1 [1, 2], λ 1 0, ɛ0 1 0, ɛ2 0 0, ɛ 1 0, ɛ 2 0, B := {y, w 1, λ 1, r 0 ɛ 10 f, u y+ ɛ 2 r min g, v y+ ɛ1 λ1 h 1, w 1 y 0 + N ɛ 2 R + y, f y, u rmin gy, v ɛ min gy, v, ɛ1 0 + ɛ2 0 + ɛ 1 + ɛ 2 ɛ min gy, v } λ 1 h 1 y, w 1, y >0,r 0, w 1 [1, 2], λ 1 0, ɛ0 1 0, ɛ2 0 0, ɛ 1 0, ɛ 2 0 { = y, w 1, λ 1, r 0 {2} + { r} + {2λ 1 w 1 } + [ ɛ ] 2 y,0,2y +1 ry +2 ɛy +2, y >0,ɛ0 1 + ɛ2 0 + ɛ 1 + ɛ 2 ɛy +2 λ 1 2w 1 y 3,r 0, w 1 [1, 2], λ 1 0, } ɛ0 1 0, ɛ2 0 0, ɛ 1 0, ɛ 2 0 { = y, w 1, λ 1, r 0 [ 2 r +2λ 1 w 1 ɛ ] 2 y,2 r +2λ 1w 1,2y +1 ry +2 ɛy +2,ɛ0 1 + ɛ2 0 + ɛ 1 + ɛ 2 ɛy +2 λ 1 2w 1 y 3,y >0,r 0, w 1 [1, 2], } λ 1 0, ɛ0 1 0, ɛ2 0 0, ɛ 1 0, ɛ 2 0. We can check for any x A and any y, w 1, λ 1, r F, f x, u u,v U V gx, v r ɛ,

19 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 19 of 21 that is, ɛ-weak duality holds. Indeed, let x A and y, w 1, λ 1, r à be fixed. Then f x, u r min gx, v+ɛ min gx, v u [1,2] v [1,2] v [1,2] =2x +1 rx +2+ɛx +2 =2 rx +1 2r + ɛx +2 2λ 1 w 1 x +2ɛ + ɛx +2 3λ 1 +2ɛ + ɛx +2 3λ 1 + ɛ ɛ2 0 + ɛ 1 + ɛ 2 +3λ 1 + ɛx Moreover, let x A and y, u, v, w 1, λ 1, r B be fixed. f x, u r min gx, v+ɛ min gx, v u [1,2] v [1,2] v [1,2] =2x +1 rx +2+ɛx +2 =2y +1 ry +2+2 rx y+ɛx +2. If x y 0, then f x, u r min gx, v+ɛ min gx, v u [1,2] v [1,2] v [1,2] =2y +1 ry +2+2 rx y+ɛx +2 2y +1 ry +2 2λ 1 w 1 x y+ɛx +2 2y +1 ry +2+2λ 1 w 1 y 2λ 1 w 1 x + ɛx +2 ɛy +2+2λ 1 w 1 y 3λ 1 + ɛx +2 ɛ ɛ2 0 + ɛ 1 + ɛ 2 +3λ 1 3λ 1 + ɛx If x y <0,then f x, u r min gx, v+ɛ min gx, v u [1,2] v [1,2] v [1,2] =2y +1 ry +2+2 rx y+ɛx +2 2y +1 ry +2+ 2λ 1 w 1 + ɛ 2 x y+ɛx +2 y 2y +1 ry +2+2λ 1 w 1 y ɛ 2 2λ 1 w 1 x + ɛ 2 x + ɛx +2 y ɛy +2+2λ 1 w 1 y ɛ 2 3λ 1 + ɛ 2 x + ɛx +2 y ɛ0 1 + ɛ2 0 + ɛ 1 + ɛ 2 +3λ 1 ɛ 2 3λ 1 + ɛ 2 x + ɛx +2 y 0.

20 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 20 of 21 Let ɛ = 1 3.Then x Ā := {x R 0 x 4 } is the set of all ɛ-solutions of RFP and r 1 2. If x =0,then r = 1 6.Whenɛ = 1,wecancalculatethesetà as follows: 3 { à := 0, w 1, λ 1, r 0 r 1 6,0 λ 1 2 } 9, w 1 [1, 2]. Let w 1 =2, λ 1 = 1 9. Then 0, 2, 5 8, 1 9 Ã.So,wehave r = u,v U V f x, u g x, v ɛ = 1 6 = 5 2ɛ r 2ɛ. 6 Hence, 0, 2, 1 9, 1 6 isa2ɛ-solution of RFD. So, ɛ-strong duality holds. If 0 < x 4 7,then 1 6 < r 1 2.Whenɛ = 1 3,wecancalculatetheset B as follows: { B := y, w 1, λ 1, r y >0,2+2λ 1 w 1 ɛ 2 5y +1 r y 3y +2, ɛ 2 1 y +2 3 } λ 1 2w 1 y 3,r 0, u [1, 2], v [1, 2], w 1 [1, 2], ɛ 2 0. Let w 1 =2, λ 1 =0,andɛ 2 = x+2 3.Then x,2,0, 5 x+1 3 x+2 B.So,wehave r = u,v U V Hence, x,2,0, f x, u 5 x +1 ɛ = g x, v 3 x +2 r r 2ɛ. 5 x+1 isa2ɛ-solution of RFD. So, ɛ-strong duality holds. 3 x+2 Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally in this paper and they read and approved the final manuscript. Acknowledgements The authors are grateful to the referees and the editor for their valuable comments and constructive suggestions, which have contributed to the final version of the paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education NRF-2013R1A1A Received: 29 April 2014 Accepted: 28 November 2014 Published: 16 Dec 2014 References 1. Govil, MG, Mehra, A: ɛ-optimality for multiobjective programming on a Banach space. Eur. J. Oper. Res. 1571, Gutiérrez, C, Jiménez, B, Novo, V: Multiplier rules and saddle-point theorems for Helbig s approximate solutions in convex Pareto problems. J. Glob. Optim. 323, Hamel, A:An ɛ-lagrange multiplier rule for a mathematical programming problem on Banach spaces. Optimization 491-2, Liu, JC: ɛ-duality theorem of nondifferentiable nonconvex multiobjective programming. J. Optim. Theory Appl. 691, Liu, JC: ɛ-pareto optimality for nondifferentiable multiobjective programming via penalty function. J. Math. Anal. Appl. 1981, Strodiot, JJ, Nguyen, VH, Heukemes, N: ɛ-optimal solutions in nondifferentiable convex programming and some related question. Math. Program. 253, Yokoyama, K: Epsilon approximate solutions for multiobjective programming problems. J. Math. Anal. Appl. 2031, Lee, GM, Lee, JH: ɛ-duality for convex semidefinite optimization problem with conic constraints. J. Inequal. Appl. 2010, Article ID

21 Lee and Lee Journal of Inequalities and Applications 2014, 2014:501 Page 21 of Lee, JH, Lee, GM: On ɛ-solutions for convex optimization problems with uncertainty data. Positivity 163, Barros, A, Frenk, JBG, Schaible, S, Zhang, S: Using duality to solve generalized fractional programming problems. J. Glob. Optim. 82, Chinchuluun, A, Yuan, D, Pardalos, PM: Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity. Ann. Oper. Res. 1541, Frenk, JBG, Schaible, S: Fractional programming. In: Handbook of Generalized Convexity and Monotonicity, pp Springer, Berlin Gupta, P, Mehra, A, Shiraishi, S, Yokoyama, K: ɛ-optimality for mini programming problems. J. Nonlinear Convex Anal. 72, Gupta, P, Shiraishi, S, Yokoyama, K: ɛ-optimality without constraint qualification for multibojective fractional program. J. Nonlinear Convex Anal. 62, Liang, ZA, Huang, HX, Pardalos, PM: Optimality conditions and duality for a class of nonlinear fractional programming problems. J. Optim. Theory Appl. 1103, Beck, A, Ben-Tal, A: Duality in robust optimization: primal worst equals dual best. Oper. Res. Lett. 371, Ben-Tal, A, Ghaoui, LE, Nemirovski, A: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton Ben-Tal, A, Nemirovski, A: A selected topics in robust convex optimization. Math. Program., Ser. B 1121, Bersimas, D, Brown, D: Constructing uncertainty sets for robust linear optimization. Oper. Res. 576, Bersimas, D, Pachamanova, D, Sim, M: Robust linear optimization under general norms. Oper. Res. Lett. 326, Jeyakumar, V, Li, GY: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 206, Jeyakumar, V, Li, GY: Robust duality for fractional programming problems with constraint-wise data uncertainty. J. Optim. Theory Appl. 1512, Hiriart-Urruty, JB, Lemarechal, C: Convex Analysis and Minimization Algorithms, vols. I and II. Springer, Berlin Jeyakumar, V: Asymptotic dual conditions characterizing optimality for convex programs. J. Optim. Theory Appl. 931, Jeyakumar, V, Lee, GM, Dinh, N: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J. Optim. 142, Jeyakumar, V, Lee, GM, Dinh, N: Characterization of solution sets of convex vector minimization problems. Eur. J. Oper. Res. 1743, Rockafellar, RT: Convex Analysis. Princeton University Press, Princeton Liu, JC, Kimura, Y, Tanaka, K: Three types dual model for mini fractional programming. Comput. Math. Appl , / X Cite this article as: Lee and Lee: On ɛ-solutions for robust fractional optimization problems. Journal of Inequalities and Applications 2014, 2014:501