A Facet Generaton Procedure for solvng 0/ nteger programs by Gyana R. Parja IBM Corporaton, Poughkeepse, NY 260 Radu Gaddov Emery Worldwde Arlnes, Vandala, Oho 45377 and Wlbert E. Wlhelm Teas A&M Unversty, College Staton 77843-33 Abstract Ths paper presents the Facet Generaton Procedure (FGP) for solvng 0/ nteger programs. The FGP seeks to dentfy a hyperplane that represents a facet of an underlyng polytope to cut off the fractonal soluton to the lnear programmng relaaton of the nteger programmng problem. A set of standard problems s used to provde nsght nto the computatonal characterstcs of the procedure.
Polyhedral cuttng plane methods have generated consderable nterest due to ther success n solvng a varety of dffcult nteger programmng problems (e.g., Crowder, Johnson and Padberg (983); Balas, Cera and Cornujelos (993); Balas, Cera, Cornuejols and Natraj (995); Boyd (99), (993b) (994); Savelsbergh, Gu and Nemhauser (994)). Most of these studes dealt wth the followng separaton problem: Gven a full dmensonal polytope R n R m and an m-dmensonal vector f, f? R, fnd a hyperplane H separatng f from R. The purpose of ths paper s to present the Facet Generaton Procedure (FGP) (Parja (994), Gaddov (998)) an algorthm that dentfes a hyperplane that represents a facet of an underlyng polytope and cuts off the fractonal soluton to the lnear programmng relaaton of the nteger programmng problem. As a general reference for the defnton of varous terms used n ths paper see Nemhauser and Wolsey (998). The FGP reles upon the followng four assumptons: (A) the underlyng polytope R s full-dmensonal and 0? R (A2) the pont f to be separated s not n R (A3) there ests a set E of m lnearly ndependent etreme ponts of R such that f belongs to the conve cone generated by E (A4) an oracle ests to mamze a lnear functon over the underlyng polytope. For the convenence of the reader we recall (e.g. Nemhauser and Wolsey (988) pp. 86) that f X = {,..., k } s a set of vectors n R m, then the conve cone C generated by X s the set of all nonnegatve lnear combnatons of vectors n X,.e. C = { k α : α 0, =,..., k }. = A hyperplane H separatng f from R can be obtaned by solvng the followng lnear programmng problem usng column generaton: (P) mn z = α et R st 2
α = f et R a? 0,? et R n whch et R denotes the set of etreme ponts of the poytope R. Reason From A3, problem (P) s feasble. Let z be the optmal value to problem (P) and B = {,..., m } be the assocated optmal bass. If n R m s such that n T =, =,,m, then the hyperplane H generated by B can be wrtten as H = {? R m : n T = }. Snce B s a feasble bass for (P) and f s n the cone generated by B, there est nonnegatve numbers a,, a m such that () m α = and (2) = f m α = = z Frst we note that from A and A2 t follows that z = n T f >. To see ths we note that from () we have m m f = ( α = α + z ) 0. = = If z =, then f belongs to R snce t can be epressed as a conve combnaton of ponts n R. Ths contradcts A2, so z >. Moreover, puttng together () and (2) we get n T f = so m = α m n T = α z =, = (3) z = n T f >. 3
On the other hand, t s easy to see that the reduced cost c of any etreme pont of R can be epressed as c = -n T. Snce z s the optmal value to problem (P), t follows that for all the etreme ponts of R, n T =, so all the ponts of R le on the same sde of hyperplane H. From (3) t follows that f les on the other sde of hyperplane H, so H separates f from R. We now descrbe the FGP. Step 0 Set k =. Step Fnd n k? R m such that n T k = for all n E k. Use the oracle to solve w k = ma { n T k :? R }. If w k =, STOP. The nequalty n T k = represents a facet of R. Else, GO TO Step 2. Step 2 Set k = k+ The oracle n Step generated a new column, whch wll enter the bass E k. Update E k and GO TO Step. Computatonal evaluaton A typcal applcaton of the Facet Generaton Procedure s to ndvdual 0/ knapsack constrants. We used a subset of sparse bnary problems from MIPLIB to benchmark the FGP. The reason for selectng them s that they are already "classc", havng been studed by several researchers. They are avalable electroncally as descrbed by Bby, Boyd, and Indovna (992) and Boyd (993a).Table I descrbes the test problems, gvng the number of constrants and number of varables for each, along wth the optmal values for the ntal LP relaaton, Z LP, and the optmal bnary soluton, Z BP, respectvely. We ran all tests on an IBM RISC 6000 model 550. To clarfy our computatonal procedure we note that we appled our algorthm to generate facets of ndvdual knapsack constrants and the cuts generated were added durng the supernode branch and bound routne of OSL release 3. In all these tests the ntal set E n A3 was chosen to be the set of unt vectors n the underlyng vector space. As a further means of reducng computatonal requrements the classcal varable fng procedure (e.g. Balas and Martn 980) was nvoked effcently, so that n the problems tested we needed to solve the knapsack n Step only 42% of the tme. As oracle n Step we used software from Martello and Toth (989). Table II gves the results of these tests. Column 4
"Cuts" lsts the number of cuts generated by FGP, column "Nodes" gves the number of nodes requred by OSL supernode branch and bound subroutne to fnd and verfy an optmal soluton, and column "CPU" gves the run tme n seconds for solvng the problem. The last two columns demonstrate the results of usng only OSL to solve each problem and report the number of nodes to fnd and verfy the optmal soluton and the CPU run tme, respectvely. These CPU run tmes do not nclude I/O. In all test problems, the FGP mproved on the run tme of OSL, or the number of nodes eplored, or both. In partcular, consderng sets of problems for whch OSL requred 0-00, 00-,000 or,000-7,000 CPU seconds, the FGP requred 30%, 35%, and 48% less run tme, respectvely. These computatonal benchmarks suggest that the FGP s able to facltate solutons, especally for larger nstances. Acknowledgements Ths materal s based upon work supported by the Natonal Scence Foundaton on Grants DDM-94396 and DMI-95002. We are grateful that Professor P. Toth asssted us n usng software from Martello and Toth (989). We are ndebted to two anonymous referees, an Assocate Edtor and the Area Edtor whose comments allowed us to mprove an earler verson of ths paper. 5
Table I: Test Problems Problem Rows Varables Z LP Z BP bm23 20 27 20.5 34.0 cap6000 276 6000-245537.3-245336.0 Engma 2 00 0.0 0.0 l52lav 97 989 4656.3 4722.0 lp4l 85 086 2942.5 2967.0 lseu 28 89 834.7 20.0 mtre 2054 0724 4740.5 585.0 mod008 6 39 290.9 307.0 mod00 46 2655 6532.0 6548.0 p0033 5 33 2520.5 3089.0 p020 33 20 6875.0 765.0 p028224 24 282 76867.5 2584.0 p029 252 29 705. 5223.7 p0548 76 548 35.3 869.0 p2756 755 2756 2688.7 324.0 ppe 25 48 773.7 788.2 Table II: Test Results Problem FGP OSL Cuts Nodes CPU Nodes CPU bm23 55 30 9 35 7 cap6000 24 68 9623 04 688 engma 372 688 770 587 3888 l52lav 82 506 4404 2399 7934 lp4l 22 6 29 7 30 lseu 43 6 4 73 20 mtre 370 0 660 0 887 mod008 62 2 4 5 63 mod00 2 57 227 20 570 p0033 76 2 3 p020 87 76 29 27 58 p0282 49 6 8 4 30 p029 7 3 4 4 4 p0548 42 0 9 0 0 p2756 57 6 8 8 32 ppe 83 3 2 8 2 All tmes are n seconds 6
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