Research Article Robust Linear Programming with Norm Uncertainty

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1 Joural of Applied Mathematics Article ID pages Research Article Robust Liear Programmig with Norm Ucertaity Lei Wag ad Hog Luo School of Ecoomic Mathematics Southwester Uiversity of Fiace ad Ecoomics Chegdu Sichua Chia Correspodece should be addressed to Lei Wag; leiwag@swufe.edu.c Received 27 March 204; Accepted 28 April 204; Published May 204 Academic Editor: Na-Jig Huag Copyright 204 L. Wag ad H. Luo. This is a ope access article distributed uder the Creative Commos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited. We cosider the liear programmig problem with ucertaity set described by (p w)-orm. We suggest that the robust couterpart of this problem is equivalet to a computatioally covex optimizatio problem. We provide probabilistic guaratees o the feasibility of a optimal robust solutio whe the ucertai coefficiets obey idepedet ad idetically distributed ormal distributios.. Itroductio Robust optimizatio is a rapidly developig methodology to address the optimizatio problems uder ucertaity. Compared with sesitivity aalysis ad stochastic programmig the robust optimizatio approach ca hadle cases where fluctuatios of the data may be large ad ca guaratee satisfactio of hard costraits which are required i some practical settigs. The advatage of the robust optimizatio is that it could prevet the optimal solutio agaist ay realizatio of the ucertaity i a give bouded ucertaity set. The robust liear optimizatio problem where the data are ucertai was first itroduced by Soyster []. The basic idea is to assume that the vector of ucertai data ca be ay poit (sceario) i the ucertaity set to fid a solutio that satisfies all the costraits for ay possible sceario from the ucertaity set ad to optimize the worst-case value of the objective fuctio. Be-Tal ad Nemirovski [2 3] ad El Ghaoui et al. [4 5] addressed the overcoservatism of robustsolutiosbyallowigtheucertaitysetsforthedata to be ellipsoids ad proposed some efficiet algorithms to solve covex optimizatio problems uder data ucertaity. Bertsimas et al. [6 7] proposed a differet approach to cotrol the level of coservatism o the solutio that has the advatage that leads to a liear optimizatio model. For more about the robust optimizatio we refer to [8 5]. Cosider the followig liear programmig problem: c T x Axb x P x where x R A R m is a ucertai matrix which belogs to a ucertaity set U c R adp x is a give set. The robust couterpart of problem () is c T x Axb x P x A U. A optimal solutio x is said to be a robust solutio if ad oly if it satisfies all the costraits for ay A U. I this paper we cosider the liear optimizatio problem () with ucertaity set described by (p w)-orm for the reasootolytomakeupthedisadvatagesoftheucertai parameters of all possible values that will give the same weightbutalsotocosidertherobustcostoftherobust optimizatio model which is metioed i [6]. We suggest the robust couterpart of problem () that is a computatioally covex optimizatio problem. We also provide probabilisticguarateesothefeasibilityofaoptimalrobust solutio whe the ucertai coefficiets obey idepedet ad idetically distributed ormal distributios. () (2)

2 2 Joural of Applied Mathematics Here is the structure of this paper. I Sectio 2 we itroduce the (p w)-orm ad its dual orm ad give the compariso with the Euclidea orm. I Sectio 3weshow that the liear optimizatio problem ()withucertaityset described by (p w)-orm is equivalet to a covex programmig. I Sectio 4weprovideprobabilisticguarateesothe feasibility of a optimal robust solutio whe the ucertaity set U is described by the (p w)-orm. S S J S p + S S J S p w j y j w j x j x pw + y pw. (5) 2. The (pw)-norm I this sectio we itroduce the (p w)-orm ad its dual orm. Furthermore we show worst-case bouds o the proximity of the (p w)-orm as opposed to the Euclidea orm cosidered i Be-Tal ad Nemirovski [2 0] ad El Ghaoui et al. [4 5]. 2.. The (pw)-norm ad Its Dual. We cosider the ith costrait of the problem () a i xb i.wedeotebyj i the set of coefficiets a ij ad a ij takesvaluesitheiterval [a ij a ij a ij + a ij ] accordig to a symmetric distributio with mea equal to the omial value a ij.foreveryiweitroduce a parameter p i whichtakesvaluesitheiterval[0 J i ].Itis ulike the case that all of the a ij will chage which is proposed by []. Our goal is to protectthe cases that are up to p i of these coefficiets which are allowed to chage ad take the worst-case values at the same time. Next we itroduce the followig defiitio of (p w)-orm. Defiitio. For a give ozero vector w R with we defie the (p w)-orm as with y R. y pw = S S J S p w j y j (3) Remark 2. Obviously y pw is ideed a orm sice () y pw 0ad y pw =0if ad oly if y=0sice w is ot a zero matric with...; (2) αy pw = S S J S p = α S S J S p w j αy j w j y j = α y pw ; (4) Remark 3. () Supposed that (i) y y 2 y 0; (ii) w =w 2 = =w p = w p =p p w i w p p < i ; the the (p w)-orm degeerates ito D-orm studied by Bertsimas ad Sim [6]; that is y p = S t S J S p t J\S y j + (p p ) y t. (6) (2) If w = (...) T ad p = the(p w)-orm degeerates ito L adwecaget y pw = y e = i= y i i=... (3) If w = (...) T ad p = the(p w)-orm degeerates ito L adwehave y pw = y e = y i i=... Next we derive the dual orm. Propositio 4. The dual orm of the (p w)-orm is s pw = s s/w w p (7) with w=(w w 2...w ) T... Proof. The orm y pw is equivalet to y pw = μ j w j y j μ j p 0μ j (8) (3)... x+y pw = S S J S p w j x j +y j Accordig to liear programmig strog duality we have mi p r + t j

3 Joural of Applied Mathematics 3 r+t j w j y j r 0 t j 0 The y pw if ad oly if p r +... t j r+t j w j y j r 0 t j 0 is feasible. We give the followig dual orm s pw by s pw = From (0)we obtai that s pw = s y p r+ y s y. pw t j w j y j r t j 0 w j y j r t j 0 r 0 t j 0 (9) (0) () (2) ad we obtai that θ s j /w j p (4) s pw = s s/w. (5) w p Remark 5. () Whe the (p w)-orm degeerates ito Dorm we ca get its dual orm: s p = ( s s /p ). (6) (2) Whe the (p w)-orm degeerates ito L wecaget its dual orm: s e = s. (7) (3) Whe the (p w)-orm degeerates ito L weca get its dual orm: s e = s. (8) 2.2. Compariso with the Euclidea Norm. The ucertaity sets i the related literatures have bee described usig the Euclidea orm ad it is of iterest to study the proximity betwee the (p w)-orm ad the Euclidea orm. Propositio 6. For every y R... UsigtheLPdualityagaiwehave mi mi w j w j y pw y wj 2 s pw = mi θ p θ μ j ] j 0 mi k N w 2 k mi k N /w k p (9) θ μ j ] j 0 (μ j ] j )w j =s j (3) y pw y mi. k N wk 2 p k N w k θ 0 μ j 0 ] j 0 Proof. First we will give a lower boud o y pw / y 2 by solvig the followig problem: Thus s pw = mi θ... y 2 j j N y pw = (20) θ s j w j... where N=... Let S =... p ;wecagetthaty j y t j Sad y j y t j N\S. It is easy to see that the objective fuctio

4 4 Joural of Applied Mathematics ca ever decrease if we let y j =y t j N \ S;thewehave that (20) is equivalet to the followig problem: y 2 j + ( p ) y2 t w j y j = y j y t j S y t 0. (2) Our goal is to imize the covex fuctio over a polytope; the there exists a extreme poit optimal solutio for the above problem. We ca get the S + extreme poits: y j = e j w j y S + = j S e w j (22) where e j is the uit vector with the jth elemet equal to oeadtherestisequaltozero.obviouslytheproblemca get the optimum value of /w 2 j /( w j) 2. The the iequality follows as y w j w y pw (23) j by takig the square root. By the same way i order to obtai a upper boud of y pw / y 2 by solvig the followig oliear optimizatio problem: mi y 2 j j N y pw = (24) clearly the objective fuctio ca ever icrease with y j = 0 j N \ S ad we ca show that the above problem is equivalet to the followig problem: mi y 2 j w j y j = y j 0. (25) FirstlyweuseLagragemultipliermethodsreformulatig the problem as L(yμ)= y 2 j +μ( w j y j ). (26) Applyig the KKT coditios for this problem a optimal solutio ca be foud: w j y j = wj 2 j S; (27) otherwise y j =0. It is easy to see that the optimal objective value is / w 2 j. By takig the square root we have that that is Sice So we ca deduce that y wj 2 y pw ; (28) y pw w 2 j y. (29) y y y y. mi y/w k N w k y k N w 2 k y/w y k N w 2 k mi k N w k s pw = s s/w w p with w=(w w 2...w ) T w i >0i=... Thus we have mi k N w 2 k y pw y Therefore the results hold. mi k N /w k p mi. k N wk 2 p k N w k (30) (3) (32) Remark 7. () Whe y pw = y p we obtai the compariso betwee D-orm ad Euclidea orm easily; that is p mi y p y p + (p p ) 2. (33) The compariso betwee the duality of D-orm ad Euclidea orm is mi p y p y p. (34)

5 Joural of Applied Mathematics 5 (2) Whe (p w)-orm degeerates ito L ad L the compariso results are y y (35) y y. 3. Robust Couterpart I this sectio we will show that the robust formulatio of () with the (p w)-orm is equivalet to a liear programmig problem. We cosider the followig robust formulatio of ()with the (p w)-orm: c T x a ij x j + b i S i S i J i S i = p i i=... y j x j y j lxu y j 0... a ij w j y j i (36) If p i is selected as a iteger the protectio fuctio of the ith costrait is β i (x p i )= S i S i J i S i =p i a ij w j x j. (37) i Note that whe p i = 0 β i (x p i ) = 0thecostraitsare equivalet to the omial problem. Ad if p i = J i w j = jwehavethemethodofsoyster[].likewiseifweassume that y y 2 y 0ad w =w 2 = =w p = w p =p p wehavethemethodofbertsimasadsim [6]. Therefore by varyig p i [0 J i ]wehavetheflexibility of adjustig the robustess of the method agaist the level of coservatism of the solutio. We eed the followig propositio to reformulate (36)as a liear programmig problem. Propositio 8. Give a vector x the protectio fuctio of the ith costrait β i (x p i )= a ij w j S i S i J i S x j (38) i = p i i is equivalet to the followig liear programmig problem: β i (x p i )= a ij w j x j z ij z ij p i 0z ij... (39) Proof. A optimal solutio of problem (39) obviously cosists of p i variables at which is equivalet to a subset S i S i J i S i = p i. (40) The objectio fuctio of problem (39)covertsto which is equivalet to problem (38). a ij w j x j (4) i Next we will reformulate problem (36) asaliearprogrammig problem. Theorem 9. Problem (36) is equivalet to the followig liear programmig problem: c T x a ij x j +z i p i + t ij b i z i +t ij a ij w j y j y j x j y j l j x j u j t ij 0 i y j 0 z i 0 w j >0 i=... i j J i i= Proof. First we cosider the dual problem of (39): mi t ij + p i z i z i +t ij a ij w j x j t ij 0 z i 0 i= ij J i (42) (43) Sice problem (39) isfeasibleadboudedforallp i [0 J i ] by strog duality we kow that the dual problem (43) isalsofeasibleadboudedadtheirobjectivevalues

6 6 Joural of Applied Mathematics coicide. By Propositio 8weobtaithatβ i (x p i ) is equivalet to the objective fuctio value of (43). Substitutig i problem (36) we have that problem (36) equalstheliear programmig problem (42). Remark 0. Whe (p w)-orm degeerates ito D-orm we have the followig robust couterpart of problem (36)[6]: c T x a ij x j +z i Γ i + p ij b i i=... z i +p ij a ij y j ij J i y j x j y j l j x j u j p ij 0 i y j 0 z i 0 i= Probabilistic Guaratees (44) I this sectio we will provide probabilistic guaratees o the feasibility of a optimal robust solutio whe the ucertaity set U is described by the (p w)-orm. Propositio. We deote by S i ad t i the set ad the idex respectively which achieve the imum for β i (x p i ) i (38). Assume that x is a optimal solutio of problem (42). The violated probability of the ith costrait satisfies where Pr ( a ij x j >b i) Pr ( γ ij η ij Ξ) (45) j if i γ ij = a ij x j if j J a i \S ir x i (46) r r = arg mi a ir r S x i r. Proof. Let x S i be the solutio of problem (36). The the violated probability of the ith costrait is Pr ( a ij x j >b i) j = Pr ( a ij x j + η ij a ij x j >b i) j Pr ( η ij a ij x j > a ij w j x j ) i = Pr ( η ij a ij x j > a ij x j (w j η ij )) \Si i Pr ( η ij a ij x j > a ir x r (w j η ij )) \Si i Pr ( \S i η ij a ij x p i j + η a ij > w j ) ir x r i p i = Pr ( γ ij η ij > w j ) p i Pr ( γ ij η ij w j ). Let Ξ= p i w j; we get the result. (47) Remark 2. Clearly Ξ is related to p i ad w=(w w 2... w ) T ; the role of the parameter X i or p i (for w is a give vector) is to adjust the robustess of the proposed method agaist the level of coservatism of the solutio. We defie Ξ ad p i as robust cost ad protectio level which cotrol the tradeoff betwee the probability of violatio ad the effect to the objective fuctio of the omial problem. Naturally we wat to boud the probability Pr( j Ji γ ij η ij Ξ). The followig result provides a boud that is idepedet of the solutio x. Theorem 3. Let η ij be idepedet ad symmetrically distributed radom variables i [ ];the we have Pr ( γ ij η ij Ξ)exp ( (Ξ) 2 2 J ). (48) i Proof. Let θ>0. The we obtai that Pr ( γ ij η ij ξ) E[exp (θ γ ij η ij )] = E[exp (θγ ij η ij )] j Ji 2 0 k=0 ((θγ ijη) 2k / (2k)!) df ηij (η) =

7 Joural of Applied Mathematics 7 j Ji k=0 ((θγ ij) 2k / (2k)!) exp (θ 2 γ 2 ij /2) exp ( J i θ2 2 θξ) (49) whereweusethekowledgeofmarkov siequalitythe idepedece ad symmetric distributio ad γ ij. Selectig θ = Ξ/ J i weobtaitheresultoftheorem Coclusios I this paper we itroduce the defiitio of (p w)-orm its dual ad some propositios to show a ew ucertaity set. We suggest that the robust couterpart of liear programmig problem described by (p w)-orm is a computatioally covex optimizatio problem. We provide probabilistic guaratees o the feasibility of a optimal robust solutio whe the ucertai coefficiets obey idepedet ad idetically distributed ormal distributios. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. [8] A. Be-Tal ad A. Nemirovski Robust solutios of liear programmig problems cotamiated with ucertai data Mathematical Programmig vol. 88 o. 3 pp [9] A. Be-Tal ad A. Nemirovski Robust solutios of ucertai liear programs Operatios Research Lettersvol.25o.pp [0] A. Be-Tal ad A. Nemirovski Robust solutios of ucertai liear programs Operatios Research Lettersvol.25o.pp [] C. Gregory K. Darby-Dowma ad G. Mitra Robust optimizatio ad portfolio selectio: the cost of robustess Europea Joural of Operatioal Researchvol.22o.2pp [2] G. N. Iyegar Robust dyamic programmig Mathematics of Operatios Researchvol.30o.2pp [3] E. Erdoǧa ad G. Iyegar Ambiguous chace costraied problems ad robust optimizatio Mathematical Programmigvol.07o.-2pp [4] A.-L. Ya G.-Y. Wag ad N.-H. Xiu Robust solutios of split feasibility problem with ucertai liear operator Joural of Idustrial ad Maagemet Optimizatiovol.3o.4pp [5] H.-M. Zhao X.-Z. Xu ad N.-J. Huag Robust solutios of ucertai exteded weighted steier problems with applicatios Commuicatios o Applied Noliear Aalysis vol.6 o. 4 pp Ackowledgmets This work was supported by the Natioal Natural Sciece Foudatio of Chia ( ) ad the Fudametal Research Fuds for the Cetral Uiversities of Chia (JBK3040). Refereces [] A. Soyster Covex programmig with set-iclusive costraits ad applicatios to i-exact liear programmig Operatio Research vol. 2 o. 5 pp [2] A. Be-Tal ad A. Nemirovski Robust solutios of ucertai liear programs Operatio Research Letters vol. 25 o. pp [3] A. Be-Tal ad A. Nemirovski Robust covex optimizatio Mathematics of Operatios Researchvol.23o.4pp [4] L. El Ghaoui F. Oustry ad H. Lebret Robust solutios to ucertai semidefiite programs SIAM Joural o Optimizatiovol.9o.pp [5] L.ElGhaouiadH.Lebret Robustsolutiostoleast-squares problems with ucertai data SIAM Joural o Matrix Aalysis ad Applicatios vol. 8 o. 4 pp [6] D. Bertsimas ad M. Sim The price of robustess Operatios Researchvol.52o.pp [7] D. Bertsimas D. Pachamaova ad M. Sim Robust liear optimizatio uder geeral orms Operatios Research Lettersvol.32o.6pp

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