Systematic Construction of examples for cycling in the simplex method

Transcription

1 Systemtc Constructon of emples for cyclng n the smple method Prof. r. Peter Zörng Prof. Assocdo Unversdde de Brsíl UnB eprtmento de Esttístc e-ml: peter@un.r

2 ) Lner progrmmng prolem cnoncl form: stndrd form: s.t. s.t. Felds of pplcton: Sles nd producton plnnng Blendng prolems Cuttng stock prolems Agrculture Fnncl plnnng Soluton methods: m z = c A T T m z = c A + Iu =, u. n c, R m A R n m, R, I R m m, 9: Smple method of ntzg (nonpolynoml) 979: Ellpsod method of Khchn (polynoml) 98: Interor pont method of Krmkr (polynoml) u R m

3 ) Emple for the smple method nd cyclng. m s.t. z = + + +, z = 8 M (). 8. ( ) ( ) () BV z - z z - - / -/ 8 - / -/ / 6 / / / 8 Pvo selecton rule of ntzg: Column ν such tht: Row µ such tht: > zν z. ~ ~ ~ > : / ~ / ~ wth ~ > µ, ν µ µ, ν, ν, ν.

4 A tleu s clled degenerte, f t lest one vlue of the rght-hnd sde s zero. When every tleu s nondegenerte, the smple lgorthm solves lner prolem n fnte numer of tertons. Otherwse cyclng my occur,.e. fter some tertons the lgorthm returns to prevously generted tleu. Cyclng emple: m , K,

5 BV BV The net terton results n the ntl tleu! In the followng we present condtons whch re necessry nd suffcent for cyclng nd use them to construct cyclng emples.

6 ) Hstory of smple cyclng urng two yers fter the pulcton of ntzg s smple method n 9 t ws n open prolem whether smple cyclng s possle. In more thn yers only few cyclng emples hve een pulshed. Collecton of Gss/Vnmur (): 9 Hoffmn 9 Bele 96 Yudn / Gol shten 969 Mrshll / Suurlle 98 Chvátl 98 Solow 99 Nerng / Tucker 996 Serksm 997 Kuhn (n: Blnsk / Tucker) The constructon of cyclng emples ws knd of mentl sport, occsonlly lmost mystfed. There dd not est ny systemtc constructon procedure

7 Some cttons: Hoffmn 9: Cyclng s certnly not completely understood. Bele 9: lner progrmmers re stll ntrgued y cyclng nd seek n understndng of the sc resons underlyng ts occurrence Lee 997: None of those emples s s mysterous s Hoffmn s Guerrero-Grcí/Sntos-Plomo 7: Sumsson of the pper: On Hoffmn s celerted cyclng emple 6

8 ) Bsc de of the constructon of cyclng emples Here we restrct ourselves to the esest cse wth n = nd m =,. e. the lner progrm hs the form s.t. m z = c + K+ c + K+ + K+, K, Wthout loss of generlty we my ssume tht the rght-hnd sde zero. Let the smple cycle e C = ({,}, {,}, {,}, {,}, {,6}, {,6}, {,} ),. e. the frst ss conssts of columns nd, the second of columns nd, etc. 7

9 The tleu must hve the followng propertes (rght-hnd sde elmnted): VB BV [+] 6 Bsc ndces 6 {, 6} [ ] c c c 6 [+] [ ] {, 6} [+] [+] [ ] M [ ] M : element must e postve : element must e the most negtve M {, } 8

10 In order to otn necessry nd suffcent condtons for cyclng, every tleu must e epressed n terms of. c,, The tleu ssocted wth the sc vrles nd s of the form T = T 9 For emple, pvot step from to s possíle, ff hs the followng propertes:, T T, T, K K K K,, / K K >,,,,,, / / K K /,, >, : = ν ν ν ν c c c := where ( ) v <, 6,,.

11 After some smplfctons we otn the followng condtons whch re necessry nd suffcent for the smple lgorthm to run through the cycle C = ({,},{,},{,},{,},{,6},{,6},{,} ) :, > >,,,,,, > >,,,,,, > >,,,,6,, > >,,6,,,,6 > >,6,,6,,,6 > >,6,,6,,,,,,6,,,,,6,,6, () () () () () (6), 6, (6) () () C, 6, () () (),,

12 Note tht the vrles n the ove system re the elements of the ntl tleu. For emple,,, < mens c c c A specfc soluton of the ove system s: < c =, 9 c =, c =, c =, =, =,, = =, 9 =, =, =, = whch corresponds to the llustrted cyclng emple., Oservton: etermnng cyclng emple s equvlent to solvng system of determnntl nequltes of the ove type! Thus, nonlner progrmmng softwre cn e ppled to construct cyclng emples! e c

13 Crucl de for the chrcterzton of the cyclng property: Epress ll tleu n terms of the ntl tleu, usng Crmer s Rule: If the prolem s m z = c + + c s.t ,,.e. the ntl tleu s c c c c, the tleu T, hvng sc vrles nd cn e epressed s

14 T = where =, ν = ν ν ν c c c. For emple, T = = (for the numercl emple).

15 Thus, the sequence of tleu s T 6 = T 6 = T = 6 6 6

16 nd we get the condtons for cyclng ν >, > for ν =,..., ν >, > for ν =,,..., >, > ν for ν =,..., whch cn e smplfed to 6 >, > 6 >, > >, > >, > >, > 6 6 >, > 6 6, 6,, 6 6,, 6, 6

17 ) Cyclng emples for dverse pvot rules ) ntzg s rule wth the Lest-nde te-rekng rule : (When severl sc vrles stsfy the crteron for levng the ss, choose the vrle wth the lest nde): m =, n = m : m =, n = : (.) (.) m , K, Soluton: unounded Soluton: K = =, K,

18 Emple wth nonzero rght-hnd sde: m =, n = m : (.) , K, Soluton:,,, ) (,, 7, ) ( =

19 Emple wth hgher dmensons: m =, n = C = 6 : ({,}, {,}, {,}, {,}, {,6}, { 6,7}, { 7,8}, {,8}, {,} ) m (.), K, 6 Soluton: unounded

20 ) Lrgest-coeffcent te-rekng rule In prctcl clcultons vrous prolems rse f the pvot element s too smll (ll-condtoned ss mtrces). Therefore common prctcl te-rkng rule for the levng sc vrle conssts n selectng the lrgest (postve) element of the enterng column s the pvot element. Emples: m =, n = : 6 m (.) Soluton: unounded, K,

21 6 m =, n = m + 7 : (.6) , K, 7 Soluton: unounded

22 7 c) Steepest-edge column selecton crteron Most of the lner progrmmng solvers offer the steepest-edge column selecton crteron s n lterntve for the most negtve reduced cost rule. Here the enterng vrle s selected on the ss of the most negtve rto of reduced cost to the length of the vector, correspondng to unt chnge n the nonsc vrle,. e. ~, ν ~ m+, ν + K+ ~ ~ m+, ν m, ν + < ~, ~ m+, + K+ ~ m, + for ll =, K, m + n (the levng sc vrle s determned y the Lrgest-coeffcent te-rekng rule ).

23 8 m =, n = 6 : + m (.7) , K, 6 Soluton: unounded

24 M M M 9 6) Cyclng emples wth permutton structure The cyclng emple of Hoffmn (9) hs the followng form. After two pvot steps the tleu s column permutton of the ntl tleu: BV z c z c z c9 c 9 9 K K c c c K K c c + c + c c K K K K c + c c 8 K 8 K 8

25 Such permutton structure occurs ff the ntl tleu hs the followng form, see Zörng (8, formul (.7)): 6 K n+ n+ c T c T B B ( I + B) K c T B B n ( I + B + B + K+ ) I B n B K B where the mtr B R stsfes B n = I, nd the tleu stsfes some determnntl nequltes. + Oserve tht two pvot steps, susttutng y nd y to premultplcton of the tleu y c B T T B c B B. = B (6.) correspond

26 Usng the ove theory nd some mtr theory (,. e. B s nvolutry) one cn construct cyclng emples wth permutton structure. B n = I + Emple: In (6.) choose n = 8 nd = = c c B c T B

27 7) Prctcl relevnce/concludng remrks The lrge numer of ntcyclng rules, pulshed over the decdes demonstrte the prctcl sgnfcnce of cyclng. The occurrence of ths phenomenon s not restrcted to the orgnl verson of the smple lgorthm. Almost ll mprovements nd vrnts of the smple method, s well s mny of the smple type lgorthms n (nonlner) mthemtcl progrmmng nvolve the posslty of cyclng or stllng, for emple: steepest edge smple lgorthm prml-dul smple lgorthm eteror pont smple lgorthm trnsportton prolems network prolems qudrtc prolems lner complementrty prolems ottleneck progrmmng pecewse-lner progrmmng lnerly constrned optmzton ntegrl smple method for comntorl optmzton

28 Occurrence of cyclng n prctce: (see Zörng 6: pge 8) scusson of cyclng n the Internet: Brn s gest, see u/dgest.html Lner Progrmmng FAQ s, see

29 Types of cyclng It s ndspensle to dstngush etween clsscl cyclng nd computer cyclng. Clsscl cyclng: rses when the prolem dt my e epressed s rtonl frctons nd computtons re performed wthout round-off errors,.e. the dt re lwys trnsformed from rtonl frctons to rtonl frctons. Computer cyclng: cused y round-off errors. All emples ove re of the frst type! They need not cuse cyclng when professonl softwre s ppled.

30 Prctcl use of the results The results nswer the clsscl queston, under whch condtons (clsscl) cyclng my rse. The theory permts the constructon of cyclng emples wth hgher dmensons (ll emples n the lterture re smll). The posslty of constructon s only lmted y the cpcty of the softwre used to solve the systems of determnntl nequltes. There s lrge numer of generl lner progrmmng test prolems vlle, ut only few emples for clsscl cyclng. For emple, the TOMLAB OPERA Toolo, developed t Mälrdlen Unversty n Sweden offers only three (!) cyclng emples mong ther test prolems. A gret collecton of constructed cyclng emples could e useful to evlute the prctcl performnce of (ntcyclng) procedures or new vrnts of smple type methods