A Modified Centered Climbing Algorithm for Linear Programming

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1 Applied Matheatics, 0, 3, Published Olie October 0 ( A Modified Cetered Clibig Algorith for Liear Prograig Mig-Fag Dig, Yaqu Liu, Joh Athoy Gear School of Matheatical & Geospatial Scieces, RMIT Uiversity, Melboure, Australia Eail: igfag_dig@yahoo.c Received July 9, 0; revised August 9, 0; accepted August 6, 0 ABSTRACT I this paper, we propose a odified cetered clibig algorith (MCCA) for liear progras, which iproves the cetered clibig algorith (CCA) developed for liear progras recetly. MCCA ipleets a specific clibig schee where a violated costrait is probed by eas of the cetered vector used by CCA. Coputatioal copariso is ade with CCA ad the siple ethod. Nuerical tests show that, o average CPU tie, MCCA rus faster tha both CCA ad the siple ethod i ters of tested probles. I additio, a siple iitializatio techique is itroduced. Keywords: Liear Prograig; Ladder Method; Clibig Rules; Siple Method. Itroductio I the last decades a variety of algoriths have bee proposed for solvig liear prograig (LP) probles (see, e.g., [-5]). However, so far there has t bee a available sigle best LP algorith which is suitable for solvig all types of LP probles. Cosiderable research is still uder way to fid a faster LP algorith. Recetly, a ladder ethod was developed i [5] for solvig geeral LP probles. I this ethod, the iclusive oral coe is updated at each iteratio by clibig i the associated iclusive regio (ladder) util the proble is solved. A clibig rule used to update the iclusive oral coe is of crucial iportace, directly deteriig the perforace of a ladder algorith. The clibig rule ivolves picig up a violated costrait ad droppig a costrait fro the curret iclusive coe. Ladder algoriths applyig various clibig rules were proposed i [5,6]. I this paper, we preset a ew ladder algorith called the Modified Cetered Clibig Algorith (MCCA), where a specific clibig rule is eployed by eas of the cetered vector used by CCA [5]. At each iteratio, a violated costrait is selected whose associated outer oral vector fors the iiu agle with the cetered vector. Coputatioal results show that, the proposed ladder algorith has surprisig superiority to CCA as well as the siple ethod i ters of average CPU tie for radoly geerated test probles. I additio, the sigle artificial costrait techique is preseted to iitialize the ladder ethod for a certai class of LP probles. The paper is orgaized as follows. I the reaiig of this sectio, for the sae of readability, we itroduce cocepts used i the ladder ethod. A ew ladder algorith is proposed i Sectio, followed by a iitializatio techique of the ladder ethod i Sectio 3. I Sectio 4, a specific eaple is provided to illustrate effectiveess of the ew algorith. We report coputatioal results i Sectio 5, followed by a brief coclusio i Sectio 6. Cosider the followig liear prograig proble: (P): i T c s.t. A b where R is decisio variables, A R with, c = [c ; c ; ; c ] R (c 0), ad b = [b ; b ; ; b ] R. Here, square bracets with etries separated by sei-colos idicate colu vectors. Throughout the paper, we assue that ra(a) =. We deote the costrait ide set {,,, } by J. Let J be a ordered subset of J. Deote by J(i ) the ordered subset with i-th etry of J replaced by J J. The -th row of A is deoted by a. We deote by A(J) the subatri with its -th row as the i -th row of A. Also, deote by b(j) the -vector with -th eleet of b(j) as the i -th eleet of b. I additio, deotes the Euclidea or. Before proceedig, we preset the followig defiitios developed i [5], which will be used i this paper. Defiitio [5] Cosider proble (P). Let Copyright 0 SciRes.

2 44 M.-F. DING ET AL. J =,,, J be a ordered subset. A cove coe geerated by liearly idepedet vectors T T T a, a,, a, where a l l are rows of A, is said to be a iclusive oral coe geerated by J if it cotais the vector c, where c is the obective vector. The geerated coe is deoted by N(J). If J geerates a iclusive coe, the set defied by = :, L J R a b for J is called the iclusive regio or the ladder associated with J. The correspodig ordered ide set J is called the geerator of L(J), ad the uique solutio of A(J) = b(j), deoted by J, is called the base poit of the ladder L(J). Accordig to Theore. i [5], proble (P) has a optial solutio (if a optial solutio eists) if ad oly if it has a feasible base poit. A feasible base poit is eactly a optial solutio. Defiitio [5] A ladder L(J) of proble (P) is said to be degeerate if at least oe of its edges is oral to the obective vector c. Proble (P) is said to be odegeerate if it does ot have a degeerate ladder. Throughout the paper, we assue that proble (P) is o-degeerate sice it is show i [5] that the degeerate case ca be readily treated by iposig a appropriate perturbatio o the obective fuctio of the origial proble without affectig the optial solutio of the origial proble. O the basis of the above assuptio, we give the followig ladder algorith.. The Modified Cetered Clibig Algorith (MCCA) Step 0 Iitializatio. Start with a ow ladder geerator, which is deoted by J0 =,,, J (Refer to [5] or the subsequet sectio for how to fid such a geerator if it is ot iediately available). Deote by 0 = J the base 0 poit associated with J 0. Calculate the iitial base poit. 0 = A J0 b J0 Set = 0 ad D =. Step Chec optiality. Let V = J \ J D : a > b. If V =, eit with the proble attais optiality. Otherwise, go to Step. Step Updatig the ladder.. Picig up a ew ide as a pic. Let v AJ where = ;; ; =, J R, ad v J is the ceter vector of the curret ladder L(J ) [5]. Select a pic such that where t p t =a t V, p V as av J =. () a. Try to fid a ide d J as a drop such that = J J d p is a ladder geerator ad the associated base poit L J (See Procedure i [5] for how to idetify). If such a ide does ot eist, eit with the proble is ifeasible. Otherwise, go to et step..3 Let = J J d p ad D = d. Calculate the updated base poit. = A J b J Set :=. Go to Step. Note that eistece of a iitial ladder (geerator) iplies that the case of uboudedess ca ot occur (Refer to Theore.5 (d) i [5] for details). Step. costitutes the ai differece with respect to the previous ladder algoriths. At each iteratio, a violated costrait is selected as a pic whose associated outer oral vector fors the iiu agle with the cetered vector. Before uerically eaiig its efficiecy, we would lie to itroduce the sigle artificial costrait techique to costruct a iitial ladder for a certai class of LP probles. 3. Fidig a Iitial Ladder for LP Probles with Bouded Variables A iitial ladder is required to get the ladder algorith started. To fid a ladder L(J) is to fid the associated geerator J =,,, J, equivaletly, to fid idepedet outward orals a ( =,,, ) such that there eist costats 0 ( =,,, ) satisfyig T c = a. Various approaches were developed i [5] to obtai a iitial ladder. I the followig, we preset a iitializatio techique for LP probles ivolvig bouded variables. 3.. Variables with Upper Bouds I this subsectio, we cosider the case where variables of proble (P) have upper bouds. Teporarily, we assue that at least oe copoet of c is positive. For Copyright 0 SciRes.

3 M.-F. DING ET AL. 45 coveiece of discussio, write the proble as below: (P) i c c cdd c s.t. a a a b a a a b a a a b b b d b d b With this assuptio that c cotais at least oe positive copoet, it is easily see that the ide set { +, +,, + d,, } is ot a ladder geerator. I order to obtai a ladder for the above proble, we add a artificial costrait i i M, where M is a sufficietly large uber. For clarity, display the proble with the additioal costrait as below: i c c cdd c s.t. a a a b a a a b a a a b b b d b d b M Eecutig the followig siple procedure, we ca readily fid a iitial ladder for the above proble. At start, tae J = { +, +,, + d,, }. Let c d = a{c i } > 0 ( d ). Tae = + d as a drop (the associated costrait is d b + d ) ad p = + a pic (the associated costrait is M). It is easy to verify that J( p) is a ladder geerator of the above proble. Ideed, fro c c cd d = c d d cd d c we have = c c 0 i d, = c > 0 i d i d d which iplies J( p) is a ladder geerator of the above proble. 3.. Variables with Lower Bouds I this subsectio, we cosider the case where variables of proble (P) have lower bouds. For coveiece of discussio, we rewrite the proble i the followig for: c c c c (P) i d d s.t. a a a b a a a b a a a b b b d b d b Note that here we use the sae otatios i proble (P) as i (P) for coveiece. I this subsectio, we teporarily assue that c cotais at least oe egative copoet. With this assuptio, it is easy to be see that the ide set { +, +,, + d,, } is ot a ladder geerator. Addig a artificial costrait i i M, we get the followig syste: i c c c c d d s.t. a a a b a a a b a a a b b Copyright 0 SciRes.

4 46 M.-F. DING ET AL. b d b d b M Perforig the siilar procedure as the above subsectio, we ca easily obtai a iitial ladder for the above proble. Iitially, tae J = { +, +,, + d,, }. Let c d = i{c i } < 0 ( d ). Tae = + d as a drop (the associated costrait is d b +d ) ad p = + as a pic (the associated costrait is M). It is easy to verify that J( p) is a ladder geerator of the above proble. I fact, fro c c cd d = cd d cd d c we have = c c 0 i d, = c > 0 i d i d d which iplies J( p) is a ladder geerator of the above proble. If variables i a LP proble are bouded fro both below ad above, that is, a LP proble cotais costraits taig the for of l i i u i ( i ), where l i ad u i deote the lower ad upper bouds of i ad l i < u i, the after rewritig the above costraits as two costraits i l i ad i u i we ca follow the procedure i either Subsectio 3. or Subsectio 3. to obtai a iitial ladder for the proble. 4. A Specific Eaple To illustrate the efficiecy of the above ladder algorith, we use both the siple ethod ad MCCA to solve a Klee-Mity proble [7,8]. Eaple Cosider the followig Klee-Mity proble with = 3 i 00 0 s.t , ,, 3 0. O the oe had, we use the siple ethod to solve the above proble. Itroducig the slac variables s, s, s 3 0, write the above proble as the stadard for i s.t. s = 0 s = s = 0, ,, 3, s, s, s3 0 The tableau i Table shows that the siple ethod with the ost egative rule requires = 3 = 7 iteratios to attai the optiality. O the other had, we solve the sae proble usig MCCA. Firstly we rewrite all costraits as type: i s.t , To fid a iitial ladder, we add a artificial costrait M. For clarity, write the proble with the additioal costrait as below. i s.t M It is easy to verify that the ide set {7, 5, 6} is a iitial ladder geerator (see Subsectio 3.). With the ow ladder geerator at had, it taes oly two iteratios to reach a optial solutio for MCCA. For solutio detail, see Table. Clearly MCCA is uch ore efficiet for solvig the above Klee-Mity proble. Firstly, our algorith requires o additioal variables (slac, surplus ad artificial variables). Secodly, the uber of iteratios is reduced greatly. I additio, we would lie to stress that Copyright 0 SciRes.

5 M.-F. DING ET AL. 47 Table. The tableau obtaied fro siple for Eaple. Iteratio 3 s s s 3 rhs 0 z s s s ,000 z s s z s z s s z s z z s z ,000 s s ,000 Table. The table obtaied fro MCCA for Eaple. Iteratio Ladder geerator Base poit Optial value 0 {7,5,6} [M; 0; 0] {7,5,3} 0,000 M 0,000 00M ;0; {4,5,3} [0; 0; 0,000] 0,000 Copyright 0 SciRes.

6 48 M.-F. DING ET AL. although here we use a eaple with o-egativity variables to illustrate the efficiecy of our algorith, there is o o-egativity requireet for variables i our proble for. Thus, our algorith is suitable for a wider rage of LP probles. 5. Nuerical Tests I this sectio, we ae coputatioal tests to deostrate the efficiecy of MCCA. The ladder algoriths were coded i MATLAB Test probles are radoly geerated i the sae way as i [5], which is preseted as below. Eaple [5] (Radoly geerated feasible proble) Geerate a liear prograig proble by specifyig A R, c = c; c; ; c R, ad b = b; b ; ; b R i the followig ethod.. Radoly geerate c R ad a vector R such that copoets of c tae values betwee 5 ad 5, ad copoets of betwee 0 ad 0.. Geerate A ad b by two steps. (a) For, the -th row a of A is T a = c sigce, where e is the -th row of the idetity atri. The, b is defied by b = a, where γ is a rado uber i (0, ). (b) For +, radoly geerate a row vector R ad a uber R such that β ad all the copoets of α are betwee 5 ad 5. If, the the -th row a of A ad the -th eleet of b are defied by a = α, b = β. Otherwise, they are defied by a = α, b = β. Tests were ru o a des-top coputer (HP Itel(R) Core(TM), i7-600 CPU@3.40GHz, 3.39GHz, 3.4GB of R) uder Microsoft Widows XP operatig syste. For copariso, the cetered clibig algorith (CCA) [5] ad the liprog solver i MATLAB optiizatio toolbo (Versio 5., (R00b)) were used for solvig the sae test probles. The ediu-scale siple algorith (SP) was ipleeted. Tables 3 ad 4 preset coputatioal results for 0 test probles with various diesios. The average CPU tie is reported i secods. Sice our algorith ad the siple ethod actually wor o probles with differet fors ad diesios, the uber of iteratios does ot provide uch helpful iforatio. Therefore, here we do ot tae the copariso of iteratio ubers ito accout. Tables 3 ad 4 reveal that, MCCA has the absolute advatage over CCA as well as the siple ethod for tested probles. We would lie to poit out that i the preset code we adopt the traditioal techique of the iverse of atri to calculate base poits. If the advaced uerical techique was icorporated ito the curret code, the coputatioal perforace would proise further iproveet. Table 3. Average CPU tie (i secods) for test probles i Eaple ( = ). Size Algoriths (, ) MCCA CCA SP (40. 0) (80, 40) (0, 60) (60, 80) (00, 00) (40, 0) (80, 40) (30, 60) (360, 80) (400, 00) Table 4. Average CPU tie (i secods) for test probles i Eaple ( = 00). Size Algoriths (, ) MCCA CCA SP (600, 500) (650, 550) (700, 600) (750, 650) (800, 700) (850, 750) (900, 800) (950, 850) (000, 900) Coclusio A ew ladder algorith, tered the Modified Cetered Clibig Algorith, was proposed i this paper. Coputatioal results deostrated that the ladder algorith outperfors CCA as well as the siple algorith i ters of average CPU tie for radoly geerated test probles. I additio, the sigle artificial costrait techique was preseted to iitialize the ladder ethod for LP probles with bouded variables. A illustratio showed that this iitializatio techique is ituitive ad siple. REFERENCES [] G. B. Datzig, Liear Prograig ad Etesios, Priceto Uiversity Press, Priceto, 963. [] N. Kararar, A New Polyoial-Tie Algorith for Copyright 0 SciRes.

7 M.-F. DING ET AL. 49 Liear Prograig, Cobiatorica, Vol. 4, No. 4, 984, pp doi:0.45/ [3] K. G. Murty ad Y. Fathi, A Feasible Directio Method for Liear Prograig, Operatios Research Letters, Vol. 3, No. 3, 984, pp. -7. doi:0.06/ (84) [4] L. G. Khachia, A Polyoial Algorith i Liear Prograig, Soviet Matheatics Dolady, Vol. 0, 979, pp [5] Y. Liu, A Eterior Poit Liear Prograig Method Based o Iclusive Noral Coes, Joural of Idustrial ad Maageet Optiizatio, Vol. 6, No. 4, 00, pp doi:0.3934/io [6] M.-F. Dig, Y. Liu ad J. A. Gear, A Iproved Targeted Clibig Algorith for Liear Progras, Nuerical Algebra, Cotrol ad Optiizatio (NACO), Vol., No. 3, 0, pp doi:0.3934/aco [7] R. J. Vaderbei, Liear Prograig: Foudatios ad Etesios, 3rd Editio, Spriger, New Yor, 008. [8] V. Klee ad G. J. Mity, How Good Is the Siple Algorith? I: O. Shisha, Ed., Iequalities III, Acadeic Press, New Yor, 97, pp Copyright 0 SciRes.