Post-optimality Analysis in Bounded Variables Problem
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7 nd ch th n f th h cpndng cn vct nt th Stp 6 t th nw ttd d p t f d ptt Rpt th thd nt th n pt f tn h n tnd n fnt n f tp th n ndctn f nn-tnc f p f tn R : If pp nd f th dcn v fnt thn p p ndd If nt thn th d p nwn t f th p p cnnt nndd w dt th h gth dcd h w tnt wth tht tf th th ptt ctn p nft ctn Sntvt An nd fwng th chng n dt n Dtn f v Dtn f cntnt hng n th cffcnt t A Dtn f V : Dtn f nn-c v Whn nn-c v thn nd dppd w nt chng n nd w ndg chng tn w t w dppd h tw pt th : thn nd In th c ptt ntnd t ft hpd If thn nw U tn pt w f If nt f thn pp d p gth nd pcd : P t c : Dtn f c v Dtn f c v ffct th ptt w ft F dtn f w nn-c gv t hgh n nztn c P nd ngtv ct n pt t f th p chng t nd ct vd v f nd [ [ And ] ] w v n tfc v Wh ng th chng n ptt cn hp If ptt hpd thn pp p gth nd fnd pt tn In th pt t chc whth c nnc If nn-c thn t w t t w nd nd nw dt t cn n chng n ctv fnctn v nd c v If c nd nt pc thn th p w nf [8hpt p-] thw pc t nd pcd dcd v Dtn f cntnt Dtn f nctv cntnt An nctv cntnt tht n whch tfd tct nqt S t cpndng c p v wd c nd t nn-z v Spp w wnt t dt th cntnt whch nctv hn dt th w nd cn f th c/p v cpndng t th
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10 ng c v Snc ] [ ] [ 7 It cn vfd tht f thn P / / / / P P S nn-ng ff P 8 Ung 8 nd 9 F tn t f nd F tn t f hd ptv nd tf Ung 7 nd nd
11 [ng 7] ] [ Inqt p tht > < nd < > Ft pt f nqt p tht > n < γ γ γ γ wh γ 6 And cnd pt f nqt p tht < n > η η η η 7 wh η 8 nd 7 tgth gv th ng f tht f tn f nw p t tht f f thn f fndng th ng f f tn t pt w d nt cnd th nqt 7 cpndng t tht Rng f f t pt tn ng 8 w n pt tn f nw p f nd ] [ < n > α α α α 9 wh α F α p tht
12 α < α n α α > Rtn 8 nd tgth gv th qd ng f f tn t pt tn f nw p And 8 7 gv th ng f f tn t pt w f tn f nw p R : Stn f th p whn ndg chng F pct t th c v n th pt c vct f P Spp w hv t df ft npt-tpt cffcnt n th cn vct f t h dfd cn vct f w t ndct th v f th ctvt cpndng t th nw cn vct h pv cn vct n ng pt f th p nd t hd ntd Phc pc n th pt t h ddtn f v nd dtn f nth ntct nw p gntng th pnt gn t wth nw v wth t cn vct nd nd ct cffcnt A chng th ct cffcnt f t n nztn c wh v g ptv n h d t th p whch w c P In th nw p h hgh ngtv ct v n tfc v W cn t t t w nd/ppnd nd chng n nd ctv fnctn v dcd dng th ddtn f nw ctvt Wh vng th p th th pt On ptt hpd On ft hpd Optt w ft th hpd cn dcd t c chngng th v f ddtn f v dcd dng th In th c pp p gth nd pcd If ttn t w nd whch z nw A p P f nd nn-c wth t n pt c f tn OFS f P hn n w pt c f * tn f th qd p P t pt ctv fnctn v f P nd hn pt ctv fnctn v f th * qd p nc If t c nd cn pcd thn pc t nd dt t cn f th pt t Othw th qd p w nf t 976 App d p gth t d ptt p nft ctn tfd If P f thn pcd n c thw qd p nf ncdng R Oth chng chng n ct vct chng n qnt vct ddtn f v nd ddtn f cntnt cn dcd n th n f th th c In th pv dcn w hv cndd chng n n n nt f t A t th thd dcd v cn d t v th p whn w cnd thn n chng n n ctvt vct S hng n w vct thn n nt t t qvnt t ddtn f n cntnt nd dtn f nth S chng n cn vct thn n nt t t qvnt t ddtn f n v nd dtn f nth
13 c Ittn P z ct t 9 wh 7 Stn: Opt f tn f th p P / nd 7/ Opt t hwng th tn : * / / / / / / / / / / 7/ / / / / / / / / 6/ A / > 7 t dpt nd nt th Appng d p gth ptd pt tn f ptd p gvn * dnt th v f th v n th tn Dtn f v Dtn f nn-c v t n t dppd thn / / / / / 7 6 Dtn f c v t dtd H w cnd t v n tfc v tht A chng n nd ccdng w F t ng hv fwng t H ptt hpd / > A nn-c n dtng th cn n t w hv
14 - / -/ -/ - / / / / -/- -/- / 6 / A < ndg chng Appng p gth - -/ -/ - / / / -/- -/- -/ -8 Agn < ndg chng Appng p gth gn w hv S th tn n v t pt nd n nn-c nd t w nd n dtng pt tn f th ptd p 8 nd Dtn f cntnt Dtn f n nctv cntnt A / > ft cntnt nctv S t fnd th pt tn f th ptd p w dt th cn nd ft w f t nd th w n chng n nd Dtn f n ctv cntnt In p P cnd cntnt ctv t nctv chng nd cct n / -/ -/ / / / / -/ / /- 7/ A < ndg chng Appng p gth ptd w hv
15 - - - t tht > cnd cntnt nctv On dtng nd cnd w n th v t nd ng chng n nd w hv th fwng t h tn n v t f t nt pt ppng p gth ptd; th pt tn f th ptd p gvn 6 / / / -/ -/ / chng thn ndg chng dcd / / nd n Rng f f ntnng ptt gvn > < Snc nw tn w pt f v f Rng f f ntnng ft gvn < > < n > > F w hv / nd f tn f f v f h ng f f ntnng ptt w ft / t thn ft w hpd hng n A t ndg chng nd H Whn / / / / / 7/
16 / / < nd > -/ -/ -/ / / / / -/ / / 7 / A < t dpt Appng d p thd OFS f ptd p w gvn 9/6 -/6 / -/6 /6 /6 -/6 / / /6 7/6 / H t ndg chng nd ndg chng t Whn Rng f f tn t pt tn f th ptd p f / > 7/ > 7 F γ > < 7 < t tht nqt 7 nt d h 7 nng nd ng f f tn 7 t f Rng f t pt: α { } { } α α 8 nd α
17 α α α n α h w pt f α > α < α < α > F nd ng f f tn t OFS f th ptd p 7 t F vng th p nd ntdc nw v wth nd 7 St thn / nd / 6 / 6 / / 9/ / / / 7 F t 9/ -/ -/ - / / / / / -/ / / / / h tn pt w f w w dt On chngng nd cct n v t - 9/ -/ -/ - - / / / / / -/- -/- -/ /- -/ A < ndg chng Appng p gth ptd w gt th fwng t
18 - 9/ / / -/ / / / / /6 -/ / -/ /6 / w nn- nd t w nd dt h w n chng n nd S th pt tn f ptd p w / nd / Acnwdgnt h th cnwdg th fnnc ppt pvdd Unvt Gnt n Gvnnt f Ind Rfnc [] P Jn- On th % f ntvt n n n pgng ptng nd ntcshngh ct t n pt Sc 76 Spng n 997 [] V vt n Pgng Fn nd 98 [] G Dntzg n Pgng nd Etnn Pnctn Unvt P Pnctn J 96 [] G Dntzg t n n pgng: Pt VIII I- Upp nd cnd cntnt n c tngt n n pgng h RAD ptn ch nd R-67 Oct 9 Phd n Ecntc Ap n 7-8 [] Dg dd nd H Wgn n pgng wth tv ndd v" ngnt Sc /7 [6] G Fn A dfd ndd v gth" Pcdng f th nth nt nfnc n c thtc nd ptngunv nt Wnnpg n [7] n ng H A vd p gth f n pgng wth ndd v" Ofd J ngnt Sc [8] d Hn K P An tntv thd f vng PP wth ndd v" P App Sc Sct E th Stt 9 n - [9] K Swp PK Gpt n hn Optn Rch Stn hnd & Sn [] KG t n nd nt pgng Jhn W & Sn I w ndn Sdn nt976 [] I A gnd d ph- p gth" Epn Jn f Optn Rch 9-6 [] Fnt nvc A dfctn f th dcptn gth f vng n pgng wth pp ndd v" En-t Oz [] H Wgn h d p gth f ndd v" v R gt Qt [] J E Wd Rchd E Wnd Appch t ntvt n n n pgng Ann Op R 7 n [] E Aph d ch J dd Sntvt n n n pgng nd dfnt pgng ng nt-pnt thd th Pg 9 n S A 9-6
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