Dantzig-Wolfe and Lagrangian Decompositions in Integer Linear Programming

Transcription

1 In. J. Mahemacs n Operaonal Research, Vol. 4, No. 3, Danzg-Wolfe and Lagrangan Decomposons n Ineger Lnear Programmng L. Léocar* LIPN, UMR 7030 CNRS, Insu Gallée Unversé Pars 13, 99 avenue Jean-Bapse Clémen, Vlleaneuse, France E-mal: lucas.leocar@lpn.unv-pars13.fr *Correspondng auhor A. Nagh LITA, Unversé Paul Verlane, Ile du Saulcy Mez Cedex 1, France E-mal: anass.nagh@unv-mez.fr N. Toua-Moungla LIX, École polyechnque, Palaseau Cedex, France E-mal: oua@lx.polyechnque.fr Absrac: We propose n hs paper a new Danzg-Wolfe maser model based on Lagrangan Decomposon (LD). We esablsh he relaonshp wh classcal Danzg-Wolfe decomposon maser problem and propose an alernave proof of he domnance of LD on Lagrangan Relaxaon (LR) dual bound. As llusraon, we gve he correspondng models and numercal resuls for wo sandard mahemacal programs: he 0-1 bdmensonal knapsack problem and he generalsed assgnmen problem. Keywords: Danzg-Wolfe decomposon; column generaon; LR; Lagrangan relaxaon; Lagrangan decomposon; 0-1 bdmensonal knapsack problem; generalsed assgnmen problem. Reference o hs paper should be made as follows: Léocar, L., Nagh, A. and Toua-Moungla, N. (2012) Danzg-Wolfe and Lagrangan Decomposons n Ineger Lnear Programmng, In. J. Mahemacs n Operaonal Research, Vol. 4, No. 3, pp Bographcal noes: Lucas Léocar s Assocae Professor of Operaons Research and Combnaoral Opmsaon a he Unversy Pars 13, France. He s member of he CNRS laboraory LIPN. He receved hs MSc n Compuer Scence and Operaons Research n 1999 from he Unversy Pars 6 and hs PhD n Compuer Scence n 2002 from he Conservaore Naonal des Ars e Méers. Hs research areas nclude relaxaon, decomposon and heursc approaches for solvng Copyrgh 2012 Inderscence Enerprses Ld.

2 248 L. Léocar e al. large NP-hard combnaoral opmsaon problems, neger lnear and quadrac programmng as well as complexy and graph heory. In parcular, he s neresed n applcaons o vehcle roung, logsc and bology. Anass Nagh s Full Professor of Compuer Scence a he Unversy Paul Verlane, Mez, France. He s a member of Algorhmc and Opmsaon research eam of he laboraory of heorecal and appled compuer scence (LITA). He receved hs MSc n appled mahemacs n 1992 from he unversy Pars Sud and hs PhD and Hablaon boh n compuer scence respecvely n 1996 and 2004 from he Unversy Pars 13. Hs research areas nclude reopmzaon, decomposon and lagrangan approaches for neger lnear, quadrac and fraconal programmng, effcen solvng of large NP-Hard combnaoral opmsaon problems and here applcaons n logsc and ranspor. Nora Toua-Moungla s currenly workng as a Posdocoral Researcher a LIX (Ecole Polyechnque, Palaseau, France) whn he Opmsaon for Susanable Developmen (OSD) char. She receved her PhD n Compuer Scence n 2008 from he LIPN laboraory (Vlleaneuse, France). Her research neress are focused on decomposon and heursc approaches for solvng large NP-hard combnaoral opmsaon problems. She s currenly applyng hem o opmsaon problems lnked o he very mporan and passonang area of susanable developmen. Ths paper s a revsed and expanded verson of a paper enled Décomposon lagrangenne e généraon de colonnes presened a Journées Polyèdres e Opmsaon Combnaore, June, 2009, Bordeaux, France. 1 Inroducon An neger lnear program where consrans are paroned n wo subses can be formulaed as follows: (P ) c x Ax = a Bx = b x X, where c R n, A s a m n marx, B s a p n marx, a R m, b R p and X N n. These problems are generally NP-hard and bounds are needed o solve hem n generc branch and bound lke schemes. To mprove he bound based on he connuous relaxaon of (P ), Lagrangan mehods, lke Lagrangan

3 Danzg-Wolfe and Lagrangan Decomposons 249 Relaxaon (LR) (Geoffron, 1974), Lagrangan Decomposon (LD) (Gugnard and Km, 1987a, 1987b, Mchelon, 1991, Nagh and Plaeau, 2000a, 2000b), Lagrangan subsuon (Renoso and Maculan, 1992) and Surrogae Relaxaon (SR) (Glover, 1965), are well-known echnques for obanng bounds n Ineger Lnear Programmng (ILP). Ths work recalls he exsng lnk beween LR and classcal Danzg-Wolfe Decomposon (DWD) (Danzg and Wolfe, 1960) and esablshes he relaonshp beween LD and DWD o derve a new DW maser model. The equvalence beween DWD and LR s well known (Lemaréchal, 2003). Solvng a lnear program by Column Generaon (CG), usng DWD, s he same as solvng he Lagrangan dual by Kelley s cung plane mehod (Kelley, 1960). Ths work recalls he prevous resul and exends o LD, whch can be vewed as a specfc DWD, o prove he superory of he new bound obaned. The paper s organsed as follows. Secon 2 deals wh LR, LD and DWD prncples. Secon 3 shows he relaonshp beween LD and DWD, and gves a new proof on he LD bound domnance over he LR one. In Secon 4 we llusrae wh wo DW maser models on he 0-1 B-dmensonal Knapsack Problem (0-1_BKP) and he Generalsed Assgnmen Problem (GAP). In Secon 5 we presen some compuaonal resuls on he wo prevous problems. 2 Lagrangan duals and Danzg-Wolfe decomposon These approaches can be used n he pre-reamen phase of an exac or heursc mehod n order o compue beer bounds han lnear relaxaon. In hs secon, we recall he prncple of Lagrangan dualy and s lnk wh DWD and CG. 2.1 Dual Lagrangan relaxaon LR consss n omng some complcang consrans (Ax = a) and n ncorporang hem n he objecve funcon usng a Lagrangan mulpler π R m. We oban he followng relaxaon: c x + π (a Ax) (LR(π)) Bx = b x X. For any π R m, he value of (LR(π)) s an upper bound on v(p ). The bes one s gven by he LR dual: (LRD) mn π R m(lr(π)) mn π R m {x X,Bx=b} c x + π (a Ax).

4 250 L. Léocar e al. Le be X B = {x X Bx = b} and Conv(X B ) s convex hull (boundary of he convex polygon), supposed bounded. We denoed by x (k),k {1,...,K} he exreme pons of Conv(X B ). Hence, (LRD) can be reformulaed as follows: (LRD) mn π R m,...,k c x (k) + π (a Ax (k) ) mn z s.. z + π (Ax (k) a) c x (k), k =1,...,K π R m, z R. Ths new formulaon poenally conans an exponenal number of consrans, equal o K. Kelley s cung plans mehod (Kelley, 1960) consders a reduced se of hese consrans ha handle a resrced problem. Cus (consrans) are added a each eraon unl he opmum reached. 2.2 Lagrangan Decomposon dual I s well-known ha he effcency of branch and bound lke scheme depends on he qualy of he bounds. To mprove hose provded by LR, Gugnard and Km (1987a, 1987b) have proposed o use LD. In such an approach, copy consrans are added o he formulaon (P ) o buld an equvalen problem: c x Ax = a By = b x = y x X, y Y, wh Y X where he copy varables perms o spl he nal problem n wo ndependen sub-problems afer applyng LR on he copy consrans x = y: c x + w (y x) Ay = a (LD(w)) Bx = b x X, y Y, where w R n are dual varables assocaed o he copy consrans. We oban he wo followng ndependen sub-problems: (LD y (w)) w y Ay = a y Y and (LD x (w)) (c w) x Bx = b x X

5 Danzg-Wolfe and Lagrangan Decomposons 251 The LD dual s gven by where wh (LDD) mn w R n v(ld(w)) v(ld(w)) = {w y y Y A } + {(c w) x x X B } Y A = {y Ay = a, y Y } X B = {x Bx = b, x X}. Ths dual can be rewren as : { mn (c w) x + w y (LDD) w R n x X B y Y A. We assume ha he convex hull of he ses Y A and X B are bounded. We denoe by x (k),k {1,...,K} he exreme pons of X B and by y (l),l {1,...,L} hose of Y A. We oban he followng formulaon: (LDD) { mn (c w) x (k) + w y (l) w R n k =1,...,K l =1,...,L whch can be expressed n hs equvalen lnear form: mn z 1 + z 2 (LDD) z 1 (c w) x (k), k =1,...,K z 2 w y (l), l =1,...,L w R n, z 1,z 2 R. The followng heorem gve he well-known domnance relaonshp beween (P ), (LRD), (LDD) and (LP) whch s he lnear relaxaon of (P ). Theorem 1 (Gugnard and Km, 1987a, 1987b): v(p ) v(ldd) v(lrd) v(lp ). 2.3 Danzg-Wolfe decomposon and column generaon The key dea of DWD (Danzg and Wolfe, 1960) s o reformulae he problem by subsung he orgnal varables wh a convex combnaon of he exreme pons of he polyhedron correspondng o a subsrucure of he formulaon. We know ha x Conv(X B ), x = λ k x (k)

6 252 L. Léocar e al. wh K λ k =1, λ k 0, k 1,...,K. By subsung n (P ) we oban he maser problem of DWD: (MP) (c x (k) )λ k (Ax (k) )λ k = a λ k =1 λ k 0, k =1,...,K. (MP) conans m +1consrans and (poenally) a huge number of varables (.e., he number K of exreme pons of Conv(X B )). Remark 1: Due o he fac ha (LRD) s a dual of (MP), v(lrd) = v(mp) (Lemaréchal, 2003). CG consss n generang eravely a subse of he exreme pons of X B deermne an opmal soluon of (MP) by solvng alernavely: o a Resrced Maser Problem of DWD on a subse K of {1,...,K}: k K(c x (k) )λ k (RMP) a prcng problem: (Ax (k) )λ k = a k K λ k =1 k K λ k 0, c x π Ax π 0 (SP) Bx = b x X k K where (π, π 0 ) R m R are he dual varables provded by he resoluon of (RMP). The soluon of (SP) s ncorporaed (as a column) n (RMP) f s value s non negave. Ths process ends when here s no more varables n {1,...,K}\K wh a posve reduced cos.

7 Danzg-Wolfe and Lagrangan Decomposons Lagrangan and Danzg-Wolfe decomposons Ths secon s dedcaed o LD dualy. We esablsh he relaonshp beween LD, DWD and CG. We consder he followng DW maser problem : (cx (k) )λ k (MPD) x (k) λ k λ k =1 L γ l =1 L y (l) γ l =0 λ k 0, k =1,...,K, γ l 0, l =1,...,L, where x (k),k {1,...,K} are he exreme pons of X B hose of Y A. and y (l),l {1,...,L} Lemma 1: The value of hs maser problem (MPD) provdes a beer upper bound on v(p ) han he value of he classcal DWD (MP). Proof: v(mpd) = (cx (k) )λ k x (k) λ k λ k =1 L γ l =1 L y (l) γ l =0 λ k 0, k =1,...,K, γ l 0, l =1,...,L. By dualy mn z 1 + z 2 z 1 + w x (k) cx k, k =1,...,K(1) v(mpd) = z 2 w y (l) 0, l =1,...,L(2) w R n, z 1,z 2 R

8 254 L. Léocar e al. If we consder only a subse of he mulplers w R n such ha w = π A, where π s a vecor of R m, and subsue n equaons (1) and (2) we oban he followng problem: mn z 1 + z 2 z 1 + π Ax (k) cx k, k =1,...,K z 2 π Ay (l) 0, l =1,...,L w R n, z 1,z 2 R for whch he dual s: (cx (k) )λ k Ax (k) λ k λ k =1 L γ l =1 L Ay (l) γ l =0 λ k 0, k =1,...,K, γ l 0, l =1,...,L. As y (l),l {1,...,L} are he exreme pons of Y A, we have Ay (l) = a, and by l γ l =1, we oban he problem (MP). Thus v(mpd) v(mp). Remark 2: If n>m, he se {π A, π R m } R n and hen v(mpd) can be srcly beer han v(mp). Remark 3: sae ha As (LDD) (resp. (LRD)) s he dual of (MPD) (resp. (MP)), we can v(mpd) = v(ldd) = mn v(ld(w)) mn w Rn π R v(ld(π A)) m and mn π R v(ld(π A)) = mn v(lr(π)) = v(lrd) = v(mp). m π Rm Ths approach supply an alernave proof o he domnance of LD over LR. 4 Decomposon models Ths secon s devoed o an llusraon of hs new DWD model on wo classcal combnaoral opmsaon problems : he 0-1 b-dmensonal knapsack problem and he generalsed assgnmen problem.

9 Danzg-Wolfe and Lagrangan Decomposons The 0-1 b-dmensonal knapsack problem Ths problem consss n selecng a subse of gven objecs (or ems) n such a way ha he oal prof of he seleced objecs s msed whle wo knapsack consrans are sasfed. The formulaon of hs problem s gven by : (0-1_BKP) n c x n a x A n b x B x {0, 1}, =1,...,n where n s he number of objecs (or ems), he coeffcens a ( =1,...,n), b ( = 1,...,n) and c ( =1,...,n) are posve negers and A and B are negers such ha {a : =1,...,n} A<,...,n a and {b : =1,...,n} B<,...,n b. The classcal Danzg-Wolfe maser problem s gven by: ( n ) c x (k) λ k ( n ) a x (k) λ k A λ k =1 λ k 0, k =1,...,K. where x (k),k =1,...,K, are he exreme pons of Conv({x {0, 1} n b x B, =1,...,n}); and he prcng problem s: mn n (c πa )x πa n b x B x {0, 1}, =1,...,n.

10 256 L. Léocar e al. The maser problem assocaed o LD decomposon s gven by: ( n ) c x (k) λ k ( n ) x (k) λ k λ k =1 L γ l =1 L ( n ) y (l) γ l =0 λ k 0, k =1,...,K, γ l 0, l =1,...,L where x (k),k =1,...,K (resp. y (l),l =1,...,L), are he exreme pons of Conv({x {0, 1},,...,n n b x B, =1,...,n}) (resp. Conv({y {0, 1},,...,n n a y A})); and he prcng problems are: mn n u y n a y A y {0, 1}, =1,...,n and mn n (c u )x n b x B x {0, 1}, =1,...,n. where x,,...,n and y,,...,n are equal o 1 f objec s flled n he knapsack. 4.2 The generalsed assgnmen problem I consss of fndng a mum prof assgnmen of T jobs o I agens such ha each job s assgned o precsely one agen subjec o capacy resrcons on he

11 Danzg-Wolfe and Lagrangan Decomposons 257 agens (Marello and Toh, 1992). The sandard neger programmng formulaon s he followng: c x x =1, =1,...,T r x b, =1,...,I x {0, 1}, =1,...,I, =1,...,T. Two classcal Danzg-Wolfe decomposons can be made, by relaxng he assgnmen consrans or he capacy consrans. The frs classcal Danzg-Wolfe maser problem s gven by: ( ) c x (k) λ k ( x (k) K λ k =1 λ k 0, k =1,...,K ) λ k =1, =1,...,T where x (k), k =1,...,K, are he exreme pons of Conv({x {0, 1} r x b,,...,i}); and he assocaed prcng problem s: mn (c π )x π r x b, =1,...,I x {0, 1}, =1,...,I, =1,...,T. The second classcal Danzg-Wolfe maser problem s gven by: L ( ) c y (l) γ l L ( L γ l =1 r y (l) γ l 0, l =1,...,L ) γ l b, =1,...,I

12 258 L. Léocar e al. where y (l),l =1,...,Lare he exreme pons of Conv({y {0, 1} y =1,= 1,...,T}); and he assocaed prcng problem s: mn (c π )y y =1, =1,...,T y {0, 1}, =1,...,I, =1,...,T. π The maser problem assocaed o LD s gven by: ( ) c x (k) λ k ( ) x (k) λ k λ k =1 L γ l =1 L ( ) y (l) γ l =0 λ k 0, k =1,...,K, γ l 0, l =1,...,L where x (k),k =1,...,K (resp. y (l),l =1,...,L), are he exreme pons of Conv({x {0, 1} r x b,,...,i}) (resp. Conv({y {0, 1} y = 1,=1,...,T})); and he prcng problems are: and mn mn u y y =1, =1,...,T y {0, 1}, =1,...,I, =1,...,T (c u )x r x b, =1,...,I x {0, 1}, =1,...,I, =1,...,T where x,,...,i,=1,...,t and y,,...,i,=1,...,t are equal o 1 f job s assgned o agen.

13 Danzg-Wolfe and Lagrangan Decomposons Numercal expermens Ths secon s devoed o an expermenal comparave sudy beween LD and LR when solved by he CG algorhm. We consder he wo opmsaon problems defned n he prevous secon : he 0-1 bdmensonal knapsack problem and he generalsed assgnmen problem. We consder n our ess 6 nsances of he 0-1 b-dmensonal knapsack problem from he OR-Lbrary. Table 1 presens a comparave sudy beween CG resoluon of LD and LR formulaons (denoed CG_LD and CG_LR respecvely). The maser and prcng problems are solved by CPLEX11.2 solver. CG_LR and CG_LD opmaly are reached for all nsances. As expeced, LD gves beer upper bounds hen LR. On average on nsances WEING, =1,...,6, %ve assocaed o LD (resp. RL) s 0.02 (resp. 0.78), bu we observe ha he average resoluon me of CG_LR (0.07 s) s very small compared o CG_LD compuaon me (10.54 s), hs s due o he fac ha he compuaonal effor of each CG_LD eraon s greaer han he CG_LR one and o he slow convergence of CG_LD compared o CG_LR. We consder also n our ess 6 nsances of he GAP from he OR-Lbrary. All nsances gap, =1,...,6 have he same sse, 5 agens and 15 jobs. The maser and prcng problems are solved by CPLEX11.2 solver. Table 2 shows a comparson beween LR and LD algorhms performances, when we apply for LR he second classcal Danzg-Wolfe decomposon, by relaxng he capacy consrans (cf. Secon 4.2). As before, CG_LR and CG_LD opmaly are reached for all nsances. LD gves beer upper bounds hen LR. On average on nsances gap, =1,...,6, %ve assocaed o LD (resp. RL) s 0.13 (resp. 2.85), bu we observe ha he average resoluon me of CG_LR (0.24 s) s sll very small compared o CG_LD compuaon me ( s). The frs classcal Danzg-Wolfe decomposon for LR, by relaxng he assgnmen consrans (cf. Secon 4.2), has been also esed on he same nsances, he resuls show ha he bounds are gher (bu hey are no beer hen hose obaned by LD) and he CG algorhm akes more eraons and me o converge. 6 Concluson Ths paper focused on Danzg-Wolfe Decomposon prncple. We propose a new Danzg-Wolfe maser problem for ILP, whch allows o propose an alernave domnance proof of LD bound over LR bound. As llusraon, we have gven he wo Danzg-Wolfe decomposon models for he 0-1 B-dmensonal Knapsack Problem and he Generalsed Assgnmen Problem. The obaned expermenal resuls demonsrae he superory of he LD bound, bu he gan on bound qualy mpose an addonal compuaon effor. In fac, a each eraon of he CG algorhm for he LD, wo prcng problems (generally neger problems) have o be solved. Through hs expermenal sudy, we conclude ha column generaon resoluon of LD can be useful f we wan o oban a good nal bound, as for example a he roo node of a branch and bound or a branch and prce scheme.

14 260 L. Léocar e al. Table 1 Lagrangan Relaxaon and LD for (0-1_BKP)

15 Danzg-Wolfe and Lagrangan Decomposons 261 Table 2 Lagrangan Relaxaon and LD for (GAP)

16 262 L. Léocar e al. References Danzg, GB. and Wolfe, P. (1960) Decomposon prncple for lnear programs, Operaons Research, Vol. 8, pp Desrosers, J., Dumas, Y., Solomon, M.M. and Soums, F. (1995) Tme consraned roung and schedulng, n Ball, M.O., Magnan, T.L. and Nemhauser, G.L. (Eds.): Handbooks n Operaons Research and Managemen Scence, Vol. 8: Nework Roung, Amserdam, Norh-Holland, The Neherlands. Geoffron, A.M. (1974) Lagrangan relaxaon for neger programmng, Mahemacal Programmng Sud., Vol. 2, pp Glover, F. (1965) A mulphase dual algorhm for he 0-1 neger programmng problem, Operaons Research, Vol. 13, No. 6, pp Gugnard, M. and Km, S. (1987) Lagrangan Decomposon: a model yeldng sronger langrangan bounds, Mahemacal Programmng, Vol. 32, pp Gugnard, M. and Km, S. (1987) Lagrangan decomposon for neger programmng: heory and applcaons, R.A.I.R.O, Vol. 21, pp Kelley, J.E. (1960) The cung-plane mehod for solvng convex programs, SIAM Journal on Opmsaon, Vol. 8, pp Lasdon, L.S. (1972) Opmsaon Theory for Large Sysems, Macmllan Seres n Operaons Research. Lemaréchal, C. (2003) The omnpresence of Lagrange, 4OR, Vol. 1, No. 1, pp Marello, S. and Toh, P. (1992) Generalzed assgnmen problems, Lecure Noes n Compuer Scence, Vol. 660, pp Mchelon, P. (1991) Méhodes lagrangennes pour la programmaon lnéare avec varables enères, Invesgacón Operava, Vol. 2, No. 2, pp Nagh, A. and Plaeau, G. (2000) A Lagrangan Decomposon for 0-1 hyperbolc programmng problems, Inernaonal Journal of Mahemacal Algorhms, Vol. 14, pp Nagh, A. and Plaeau, G. (2000) Dualé lagrangenne en programmaon fraconnare concave-convexe en varables 0-1, CRAS : Compes Rendus de l Académe des Scences de Pars, ome 331, sére I, pp Renoso, H. and Maculan, N. (1992) Lagrangan decomposon for neger programmng: a new scheme, INFOR, Vol. 52, No. 2, pp.1 5.