0/1 INTEGER PROGRAMMING AND SEMIDEFINTE PROGRAMMING

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1 CONVEX OPIMIZAION AND INERIOR POIN MEHODS FINAL PROJEC / INEGER PROGRAMMING AND SEMIDEFINE PROGRAMMING b Luca Buch ad Natala Vktorova

2 CONENS:.Itroducto.Formulato.Applcato to Kapsack Problem 4.Cuttg Plaes ad Lft-ad-Project Methods 4. Balas-Cera-Coruéjols Costructo ad Eample 4. Lovász-Schrjver Costructo 4. Sheral-Adams Costructo 5.Worst Case Eamples 6.Refereces

3 . INRODUCION he defto of a / teger lear programmg model s the followg: (.) Before gog a further, the cove hull has to be defed as well: (.) Usuall, the problems represeted ths wa are NP-Hard, whch makes them ver dffcult to solve. It s a commo approach to use LP modelg stated below: However, f K does ot represet the cove hull of the feasble rego, we wll ot get a feasble / soluto. he soluto that we obta s the called LP Relaato. If the poltope K s the same as the cove hull, we wll get the optmal soluto just b solvg the correspodg relaato. I order to fd K that wll be the same as P, oe of the most commo methods s the cuttg plaes method wch wll be descrbed ths project. Before that, we wll troduce a alteratve formulato for the / Programmg Model, usd semdefte programmg.. FORMULAION Let the matr Y be defed as followg: (.) We wll defe the matr X, as a submatr of Y. he matr X s: (.) { } ma subject to,, c A b { } b A K R : + { } { } ( ) ma subject to, : cov, c P P A b [ ] Y X

4 I order to model the Objectve Fucto, we wll use the operator, (.) whch the commo defto represets: (.4) he dfferece both objectve fuctos s the square varable. However, ths do ot pla a mportat role due to the fact that s bar. I order to model the / costrat, we have two possble alteratves. Alteratve Frst of all, we use the fact that Y s postve semdefte, so we state: (.5) hs s true alwas due to the -rak matres behavor. he followg costrat s totall ecessar: (.6) It makes the / costrat: Alteratve I ths alteratve, we defe dfferetl the matr Y. I other words, we plug the costrat (.6) to the Y matr obtag the followg costrat: (.7) ( ) ( ) X c dag ma c ma Y Y Y,...,, Y Y ' Y

5 here are alteratves to represet the A ol oe equalt a β from A b b sstem of equaltes. Let s cosder Alteratve. ( dag( a) * X ) β ma a (.8) Alteratve. Alteratve. ( aa * X ) β for a, β (( aa β dag( a) ) X ) for a, β ( a ) β ( a ) a β (.9) (.) All these alteratves are equvalet.. NUMERICAL EXAMPLE I ths secto we wll appl the costructos provded the prevous secto, order to formulate a small stace of the Kapsack problem as a semdefte programmg. he stace s: ma s. t. 5, {, } he semdefte formulato wll be the followg: ma 4

6 Subject to: ad or or 4. CUING PLANES AND LIF-AND-PROJEC MEHODS he / lear program that we are gog to use ths secto looks lke he reformulato of ths problem s (.). As eplaed the troducto of ths project, the ma dea behd the relaato methods as to get as close as possble to the cove hull represetato of the feasble rego, e.g. we wat to get a lear descrpto or at least good lear relaatos of P. If the orgal relaato does ot have that propert, we ca costruct some cuttg plaes order to elmate the o-teger etreme pots of the relaato. he techque that ths project presets for the costructo of such cuttg plaes s the Lft-ad-Project method. I geeral, those methods seek to fd a poltope Q, whose projecto P would represet the cove hull we look for. he ma advatage of these methods s the fact that the Q ca have less facets that the projecto tself. hs allows us to get a { } ma subject to,,. c A b

7 polomal poltope Q, eve f the cove hull P s ot. If we kow that Q has a polomal umber of facets ad ts dmeso s polomal the dmeso of P, the our lear optmzato problem over P s solvable polomal tme. hs s whe lft-ad-project method come to pla. he easest lear relaato whch we are gog to start wth s K { [,] A b} ad we wat to add costrats such that we ed up P cov (K {,} ). I ths secto we are gog to preset three lft-ad-project methods that start wth the polhedro K ad allow us to get the cove hull P of our problem (.) whch meas that the optmal soluto of ma c subject to Ab s attaed for Boolea values of the compoets of. hese three methods are the Balas-Cera-Coruéjols method (BCC), also kow as lft-ad-project for BCC, the Lovász-Schrjver method (LS), also called matr-cuts for LS, ad the Sheral-Adams method (SA), also called Reformulato-Learzato echque for SA. hese costructos proceed the followg wa: Multpl each row of the sstem Ab b varous products of ad, the substtute the products j, j, for a ew varable j ad substtute the squares j for j. he resultg polhedro les a hgher dmesoal space tha the orgal problem. herefore, we project the polhedro oto R, because the projecto s stll cotaed K ad cotas P. After at most teratos the projecto s equal to the cove hull P of problem (.). We are gog to assume wthout loss of geeralt that the equaltes for Є {,, } are cluded A b. 4. Balas-Cera-Coruéjols Costructo ad Eample he costructo of the BCC projectos goes as follows: Step : F a de j Є {,, }. Step : Acheve the olear sstem (A b) * j (A b) * ( j ) b multplg the orgal sstem A b b j ad respectvel j. Step : Substtute for j (,, j) ad j for j order to receve the learzed sstem A b j A( ) b( j ) Step 4:Project the resultg poltope oto R b elmatg the learzed sstem. he projecto P j (K) satsfes P Pj ( K ) K. Step 5: Defe Ph, j ( K) : Ph ( Pj ( K)) ad terate from step o wth de h j utl ever elemet of {,, } has occurred eactl oce.

8 he ma purpose of ths algorthm s to gve the equaltes that descrbe the cove hull of the set of feasble solutos problem (.) ad we reach the cove hull wth teratos. We have Pj ( K ) cov( K { j {,}}) P (K) : P (P (K)) cov (K { {,}, {,}}) h,j h j h j P,,... (K) cov (K {,} ) ad therefore we obta the clusos Eample: Oe smple eample to llustrate the algorthm s oe kapsack problem varables. ma ma s. t. 5 wth ts lear relaato K s. t for {,,},,, { } he frst terato goes as followg: Step : Choose j Є {,,}, WLOG j. Step : Multpl the costrats b ad to obta the sstem of olear equatos Step : Substtute for ad for. K P j(k) P h,j(k)... P,,... (K) P ( ) ( ) ( ) ( ) ( ) ( ) - - ( ) ( ) ( ) ( ) - - ( ) ( ) ( ) ( ) - - ( ) ( ) for {,,}

9 Step 4: Project the sstem from step oto R b elmatg, Є {,,}\{}{,}. WLOG start the elmato b elmatg. he equaltes cotag are (5 6 ) ( ) + m{ (5 6 ),, } ma{ ( ), +, } Now we are gog to wrte dow all the possble equaltes that are mplcated ma{ } m{ } Elmate the remag -varable the ew sstem that we receved b the ma{ } m{ } combatos lke doe above ad addg sstem from step wthout the equaltes cotag

10 leads to the polhedro of the frst projecto P (K) that s descrbed b for j {,... } for j {,... } Lovász-Schrjver Costructo he costructo of the LS method goes as follows: Step : For ever de j Є {,, } acheve the olear sstem (A b) * j (A b) * ( j ) b multplg the orgal sstem A b b j ad respectvel j. Step : Substtute j for j (,, j) ad j for j to receve a learzed sstem of costrats. Use the fact that j j to have j j. Step : Project the resultg poltope oto R b elmatg the learzed sstem. he projecto N(K) satsfes P N( K ) K. Step 4: Defe N (K) : N(K) t t- N (K) : N(N (K)) for t ad terate. We kow that N( K) cov( K { j {,}}), j,.... Aga, after at most teratos we receve the cove hull N (K) P Istead of havg a lear relaato N t (K) we ca add the restrcto that the matr Y( j ) has to be postve semdefte order to receve a semdefte relaato.

11 4. Sheral-Adams Costructo he costructo of the SA method goes as follows: Step : Let t Є {,, }. Step : Multplg the orgal sstem A b b products of the form ( j Є J j )*( j Є J ( j )) where J ad J dsjot subsets of {, } such that J J t. Step : Learze the sstem b substtutg j for j ad J for j Є J j ( J {,, }) Step 4: Project the resultg poltope oto R b elmatg the learzed sstem ad deote the projecto S t (K). he clusos K S (K) S (K)... S (K) P hold ad after at most teratos the cove hull s acheved. o sum t up, the relatos amog the lft-ad-project methods (for the lear relaatos) are as follows: t S t (K) N (K) P j,j,...j (K) t ad we reach the cove hull P at most teratos. hs boud s tght as we wll demostrate b eamples the et secto. 5. WORS CASE EXAMPLES For the followg two eamples, the SA lft-ad-project method has the worst (bggest) umber of teratos, whch s. Eample : K : [, ] Eample : K : [, ] + ( ) I I

12 6. REFERENCES Moque Lauret, Fraz Redl: Semdefte Programmg ad Iteger Programmg, Aprl, Ego Balas, Sabastá Cera, Gérard Coruéjols: A lft-ad-project cuttg plae algorthm for med - programs, Joural Mathematcal Programmg Volume 58, Numbers -, p. 95-4, Sprger Berl / Hedelberg, Jauar, 99, ISSN: 5-56 (Prt) (Ole) Ego Balas, Sabastá Cera, Gérard Coruéjols: Solvg med - programs b a lft-ad-project method, Smposum o Dscrete Algorthms, Proceedgs of the fourth aual ACM-SIAM Smposum o Dscrete algorthms, p. 4, Aust, eas, USA, 99, ISBN: