Application of the Two Nonzero Component Lemma in Resource Allocation

Transcription

1 Joral of Aled Mathematcs ad Physcs, 04,, Pblshed Ole Je 04 ScRes. htt:// htt://dx.do.org/0.436/am Alcato of the Two Nozero Comoet Lemma Resorce Allocato Morteza Seddgh Mathematcs Deartmet, Idaa Uversty East, Rchmod, IN, USA Emal: Receved 6 Arl 04; revsed 6 May 04; acceted May 04 Coyrght 04 by athor ad Scetfc Research Pblshg Ic. Ths work s lcesed der the Creatve Commos Attrbto Iteratoal Lcese (CC BY. htt://creatvecommos.org/lceses/by/4.0/ Abstract I ths aer we wll geeralze the athor's two ozero comoet lemma to geeral self-redcg fctos ad tlze t to fd closed from aswers for some resorce allocato roblems. Keywords Two Nozero Comoet Lemma, Resorce Allocato, The Dstrbto of the Search Effort. Itrodcto The techqe we wll se ths aer was frst aled by ths athor to roblems matrx eqaltes ad matrx otmzato. Hstorcally, may researchers have establshed matrx eqaltes by varatoal methods. I a varatoal aroach oe dfferetates the fctoal volved to arrve at a Eler eqato ad the solves the Eler Eqato to obta the mmzg or maxmzg vectors of the fctoal. The same techqe s also ofte sed matrx otmzato. Solvg the Eler eqatos obtaed are tedos ad geerally rovde lttle formato. Others have establshed eqaltes for matrces ad oerators by gog throgh a two-ste rocess whch cossts of frst comtg er bods for stable fctos o tervals cotag the sectrm of the matrx or oerator ad the alyg the stadard oeratoal calcls to that matrx. Ths method, whch we refer to as the oeratoal calcls method, has the followg two lmtatos: Frst,t does ot rovde ay formato abot vectors for whch the establshed eqaltes become eqaltes (a matrx otmzato roblem. Secod, the oeratoal calcls method s ftle extedg Katorovch-tye eqaltes to oerators o fte dmesoal Hlbert saces. See [] for examles sg each of the two methods metoed above. I hs stdy of matrx eqaltes ad matrx otmzato, the athor has dscovered ad roved a lemma called the Two Nozero Comoet Lemma or TNCL for short. I ths aer we wll state a exteso of the athor s Two Nozero Comoet Lemma ad tlze t to solve some resorce allocato roblems. Whle resorce allocato roblems are ot geerally formlated terms of matrces, as we wll see, How to cte ths aer: Seddgh, M. (04 Alcato of the Two Nozero Comoet Lemma Resorce Allocato. Joral of Aled Mathematcs ad Physcs,, htt://dx.do.org/0.436/am

2 M. Seddgh there are some smlartes betwee matrx otmzato roblems ad resorce allocato roblems Let s frst state the TNCL as t was sed athor's revos aers o matrx eqaltes ad matrx otmzato.. The Two Nozero Comoet Lemma It was hs vestgato o roblems of ategevale theory that the athor dscovered ad roved the Two Nozero Comoet Lemma (see []-[4]. Althogh ths lemma s tlzed effectvely by the athor matrx theory, t s by atre a dmeso redcg otmzato lemma whch has otetal alcatos all areas of mathematcs ad hyscs. Whle TNCL was mlctly sed all of the aers st cted, t was ot tl 009 that the athor stated a formal descrto of the lemma hs aer ttled, Ategevale Techqes Statstcs (see [4]. Below s the statemet of the lemma. For the roof of the lemma lease see the athor s work cted above. Lemma (The Two Nozero Comoet Lemma Let l be the set of all seqeces wth oegatve terms the Baach Sace l. That s, let l = t = ( t, 0. εl t ( Let be a fcto from mmzg vectors for the fcto { } ( x F x, x, m ( m k R to R. Assme gk ( ct o the covex set C = {( t : l t = } k t for ( c l = ( ( t, ( t, m ( t, t l, ad k m. The the F g g g (3 have at most two ozero comoets. What make the lemma ossble are the followg two facts: Frst, the fact that the set C = {( t : l t } = (4 s covex. Secod, a secal roerty that the fctos F g ( t, g ( t, gm ( t (5 volved ossess. If we set the all restrctos of ( (,, 3, = ( ( t, ( t, 3( t, D t t t F g g g (6 (,,,,0,, D t t t t (7 (,, 3, t eqal to zero, have the same algebrac form as D( t, t,, D t t t (8 obtaed by settg oe comoet t3 tself. We call fctos wth ths roerty self-redcg fctos. Please ote that TNCL s vald for both fte ad fte varable cases. Let s look at a examle of a self-redcg fcto where there are oly a fte mber of varables t, t, t3 t volved. Cosdered the fcto β t βt βt (9 λ t λ t λ t 3 3 where β, β, β3, β are real mbers ad λ, λ, λ are comlex mbers (ths fcto aeared []. Let β t βt βt D( t, t,, t = (0 λ t λ t λ t

3 M. Seddgh the we have whch has the same algebrac form as β t ( 0,,, = D t t β t λ t λ3 t3 β t β t β t D t, t,,. ( t = λ t λ t λ3 t3 Ideed, for ay, < ; all restrctos of the fcto β t β t β t (,,, t = D t t λ t λ t λ3 t3 obtaed by settg a arbtrary set of comoets of D( t, t,, t brac form as D( t, t,, t. Obvosly, ot all fctos have ths roerty. For stace, for the fcto G( t, t =t tt G( t,0 = t, whch does ot have the same algebrac form as G( t, t. Note that fctos g ( ( ( (3 eqal to zeros have the same alge-, we have t aearg the statemet of TNCL above are fte or fte lear combatos of t, t, t 3, The lemma was g t s orgally stated ths way becase whe we deal wth a matrx or oerator otmzato roblem each ( ether a fte or fte lear combato of varables t, t, t3,. Examle I Theorem of [4] we sed TNCL to fd the mmm of a Raylegh qotet. A Raylegh qotet of a ostve oerator C over ostve oerators A ad B s a qotet of the form The t vectors f for whch the mmm f Af 0, Bf 0 ( Cf, f ( Af, f ( Bf, f ( Cf, f ( Af, f ( Bf, f s attaed are called statoary vales for (4. I Theorem of [4] the mmm of (4 was fod by covertg the roblem to the roblem of fdg the mmm of = αt βt = = λ t Sbect to t =. = (4 (5 (6 (7 I ths case g ( t = λ t, g ( t = t, ad g ( t = β t The sets { λ }, { α }, { } = α = set of egevales of C, A, ad B resectvely. 3. A Geeralzato of TNCL (GTNCL 3 = β are the I ths secto we wll show how a geeralzato of TNCL ca be formlated. I the roof of the TNCL [] ad [3] we took advatage of the facts that the set s a covex set ad the fcto {( : } C = t l t = 655

4 M. Seddgh s a self-redcg fcto. A fcto ( ( t, ( t, m ( t F g g g ( t F t, t, k ca be a self-redcg fcto wthot beg comosed of lear combatos of the form gk( t = ct. Examle 3 Cosder the fcto t t t3 t ( = ( ( ( ( F t, t, t e e e e. Ths fcto s self-redcg bt s ot a comosto of lear combatos. A close look at or argmets [3] shows that the oly roerty sed was the fact that the fcto to be mmzed was self-redcg ad the set t = = was covex. Therefore, we ca state the followg lemma whch s a geeralzato of TNCL. We state the lemma for the case that the mber of varables fte (a case whch occrs may aled roblems bt the argmets sed [3] show that the lemma s also vald the case where the mber of varables s fte. Lemma 4 If the fcto s a ostve self-redcg fcto o the covex set the the mmzg vectors ( t F t, t, (8 {( : } C = t l t = (9 ( t t t t =,, have at most two ozero comoets. We call the lemma stated above the Geeral Two Nozero Comoet Lemma or GTNCL for short. Remark 5 We ca also se TNCL ad GTNCL to fd the maxmm of a ostve self-redcg fcto o (9. To see ths lease ote that f s a ostve self-redcg fcto so s ( t F t, t, (0 F t, t, ( t ad maxmm of (0 o (9 s the recrocal of the mmm of ( o (9. A geeral resorce allocato roblem s stated as ( mmze f x, x,, x ( whch ca be coverted to sbect to x = N, x 0, =,,,. = ( t mmze g t, t,, sbect to t =, t 0, =,,,. = I the followg sectos we wll se GTNCL to comte a closed form aswer for the dstrbto of the search effort roblem. 4. The Dstrbto of the Search Effort Ths roblem s formlated as 656

5 M. Seddgh where α s a ostve mber, = α x ( mmze e = sbect to x = N, x 0, =,,,. α x s the robablty of a obect beg at osto ad ( e the codtoal robablty of detectg the obect at osto. If we defe x = N the the dstrbto of the search effort roblem wll be trasformed to Theorem 6 The mmm of sbect to s ether for some or = ( mmze e = sbect to =, x 0, =,,,. = for a a ar of ad, ad. Proof. Sce x ( e s ( = =, 0, =,,, (3 ( e (4 β e ( e (5 (6 = s a self -redcg fcto, the GTNCL ca be sed to fd the mmm of ths fcto sbect to =, 0, =,,, (7 = Sce α s ostve so s β. Sose 0, =,,, are comoets of a mmzg vector o the feasble set (7. By GTNCL ether there s a, so that = ad = 0 for ad or there s a ar of ad,, sch that o, o ad k = o for k, k, k. I the frst case the mmm of (6 o (7 s obvosly (4. I the secod case the mmm of (6 o (7 s the same as the mmm of sbect to ( e ( e (8 = (9 657

6 M. Seddgh Exresso (8 ca be smlfed to Sbstttg = (30 we have If we dfferetate (3 wth resect to If we solve (3 for Sbstttg, we obta from (33 (9 gves s If we sbsttte (33 ad (34 (30 we have The last exresso s eqvalet to Please ote that the dervatve of the fcto wth resect to s e e (30 ( e e (3 ad t the dervatve eqal to zero we have β ( e β βe = 0 (3 = ( β l l = β l β β = β l β l β β l (33 (34 e e. (35 β e. (36 β ( βe βe (37 β ( β e (38 β e whch s ostve for, 0. Therefore, by the secod dervatve test (36 s a mmm vale ot a maxmm vale for the obectve fcto the resorce allocato roblem. Althogh the GTNCL states two comoets ad are ozero bt geeral we do ot kow exactly whch ar of ad exresses (36. Whe alyg TNCL to roblems of matrx otmzato, the athor was able to determe exactly whch comoet or whch ar of comoets of the otmzg vectors are ozero (see [5] der certa codtos. The same ca be doe here. Theorem 7 Sose the robabltes, are ordered sch that The the mmm of sbect to s = 3 = ( e =, 0, =,,, ( e 658

7 M. Seddgh as the Proof. Assme Sce β >0 (39 mles that Frthermore, (5 assme < Note that k >0. If we defe Sce x the 3. (39 {( } = ( m e e. Let x = ad = k =e β x x k x x x x f ( x = x k x x x 3 ( x df = 3kx k x 3 dx x 3. Obvosly x. Now (5 ca be wrtte (40 (4 (4 3kx k x > 3k k = k > 0 (43 Ths shows that f ( x s a creasg fcto o [,. Hece o the fte set the fcto,, x f ( x = x k x x x has ts mmm at. Therefore, f two comoets ad ad ths case the mmm of the obectve fcto s for some,. Next otce that β ( β e are ozero, we mst have = e < < e β for. Hece the mmm of the obectve fcto s ( e Sce both TNCL ad GTNCL are vald for fte mber of varables, these techqes ca be sed to solve resorce allocato roblems volvg a fte mber of varables as well. For examle, the dstrbto of the search effort roblem we ca assme the search s for a obect o the lae that ca be otetally. 659

8 M. Seddgh detected at a fte set of locatos (sch as ots wth teger x ad y coordates. 5. Otmal Portfolo Selecto There are other resorce allocato roblems that we are able to tackle wth GTNCL, Oe of these roblems s the roblem of otmal ortfolo selecto. Oe model for ths geeral roblem s formlated as fdg the maxmm of Rx = = σ xx, sbect to x =, x 0, =,,,. (see [6]. Here R s exected retr o secrty ad σ s the covarace betwee secrtes ad. If the correlato coeffcets betwee ad are costat ρ, wth 0< ρ <, the σ = σ, σ = ρσ σ ad from Karsh-Kh-Tcker codtos the roblem s redced to ( ρ maxmze R σ x σ x (44 = sbect to x =, x 0, =,,,. = Notce that (44 s a self-redcg fcto ad oe ca aga aly the GTNCL to fd a maxmm vale for t. The roblems of dstrbto of search effort ad otmal ortfolo selecto are both examles of searable resorce allocato roblems. A searable resorce allocato roblem s a roblem where we wat to mmze or maxmze = f = ( x sbect to x = N, x 0, =,,,. where each f s cotosly dfferetable over a terval cldg [ ] sch a roblem f f ( 0 = 0, for each, 0, N. The GTNCL ca be aled to. Ths codto s ot satsfed for a mber of resorce allocato roblems cldg otmal samle allocato stratfed samlg, ad rodcto lag (see [5]. Remark 8 I a broader sese, each Katorovch-tye matrx otmzato roblem sch as the oe Examle ca be regarded as a resorce allocato roblem where or resorce s st the set of re mbers o the terval [ 0, ]. For stace Examle the roblem s redced to fdg mmm of = αt βt = = λ t Sbect to t = Each t = z, where each z s a comoet of a mmzg vector f of orm for the oerator otmzato roblem (5. Ideed ozero comoets of a mmzg vector f are mortat alca- = 660

9 M. Seddgh tos. Hstorcally, the otmzato roblem (5 was frst dscssed by R. Camero ad B. Kovartaks a effort to mmze the orm of ott feedback cotrollers sed ole lacemet (see [7]. Remark 9. The athor s ot aware of ay other theorem that rovdes closed form aswers for resorce allocato roblems. The reslts we obta mght be of terest for stace sgal aalyss where oe eeds to mmze the resorce set fdg a sgal that s robable to detected at a certa locato. Comter models are sed for solvg sch roblems ad t s terestg to vestgate how cosstet the reslts of comter models are wth or reslts here. Also, may theores ortfolo selecto sggest that dversfcato maxmzes the roft. At the frst glace ths mght sod cosstet wth the reslts oe mght obta sg the two ozero theorem. However, we have to remember that the over theory also esres dversfcato creases roft. If the roft s maxmzed for oe or two secrtes, the the more the mber of secrtes, the more ars of secrtes we have. Refereces [] Gstafso, K. ad Rao, D. (997 Nmercal Rage. Srger, New York. htt://dx.do.org/0.007/ [] Gstafso, K. ad Seddgh, M. (989 Ategevale Bods. Joral of Mathematcal Aalyss ad Alcatos, 43, htt://dx.do.org/0.06/00-47x( [3] Seddgh, M. (00 Ategevales ad Total Ategevales of Normal Oerators. Joral of Mathematcal Aalyss ad Alcatos, 74, htt://dx.do.org/0.06/s00-47x( [4] Seddgh, M. (009 Ategevale Techqes Statstcs. Lear Algebra ad Its Alcatos, 430, htt://dx.do.org/0.06/.laa [5] Seddgh, M. ad Gstafso, K. (005 O the Egevales whch Exress Ategevales. Iteratoal Joral of Mathematcs ad Mathematcal Sceces, 005, htt://dx.do.org/0.55/ijmms [6] Ibarak, T. ad Katoh, N. (988 Resorce Allocato Problems. The MIT Press, Cambrdge. [7] Camero, R. ad Kovartaks, B. (980 Mmzg the Norm of Ott Feedback Cotrollers Used Pole Placemet: A Dyadc Aroach. Iteratoal Joral of Cotrol, 3, htt://dx.do.org/0.080/

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