Discrete Simultaneous Perturbation Stochastic Approximation on Loss Function with Noisy Measurements
|
|
- Bruno Rodgers
- 5 years ago
- Views:
Transcription
1 0 Amercan Control Conference on O'Farrell Street San Francco CA USA June 9 - July 0 0 Dcrete Smultaneou Perturbaton Stochatc Approxmaton on Lo Functon wth Noy Meaurement Q Wang and Jame C Spall Abtract Conder the tochatc optmzaton of a lo functon defned on p-dmenonal grd of pont n Eucldean pace We ntroduce the mddle pont dcrete multaneou perturbaton tochatc approxmaton () algorthm for uch dcrete problem and how that convergence to the mnmum acheved Content wth other tochatc approxmaton method th method formally accommodate noy meaurement of the lo functon Keyword Stochatc optmzaton; recurve etmaton; SPSA; noy data; dcrete optmzaton I INRODUCION HE optmzaton of real-world tochatc ytem typcally nvolve the ue of a mathematcal algorthm that teratvely ee out the oluton It often the cae that the doman of optmzaton dcrete Reource allocaton for ntance nvolve the dtrbuton of dcrete amount of ome reource to a fnte number of uer n the face of uncertanty; other problem of nteret wthn th framewor nclude weapon agnment plant locaton networ reource and expermental degn h paper ntroduce a method for tochatc dcrete optmzaton that baed on tochatc approxmaton technque cutomarly ued n contnuou optmzaton problem Many method have been propoed to deal wth dcrete optmzaton problem hee method nclude random earch [] mulated annealng [] tochatc comparon [7] ordnal optmzaton [] neted partton [8] Recently Hannah and Powell [8] propoe an algorthm for one-tage tochatc combnatoral optmzaton problem baed on evolutonary polcy teraton L et al [3] ntroduce a method baed on random earch n the mot promng area propoed n [] And Slenar [9] conder an exhautve local earch method whch degned explctly for noy lo he am here to preent an alternatve method that can fully ue the nformaton of the tructure of objectve functon (eg gradent ) and potentally nvolve fewer functon meaurement he multaneou perturbaton tochatc approxmaton (SPSA) algorthm [0 ] wa Q Wang wth the Department of Appled Mathematc and Stattc of the John Hopn Unverty Baltmore MD 8 USA (e-mal: qwang9@ jhuedu) Jame C Spall wth he John Hopn Unverty Appled Phyc Laboratory Laurel MD USA and wth the Department of Appled Mathematc and Stattc of the John Hopn Unverty Baltmore MD 8 USA (e-mal: jamepall@ jhuapledu) developed for contnuou optmzaton problem of hgh dmenon and where the lo functon expenve to evaluate SPSA a popular algorthm that create gradenttype nformaton from only two noy functon meaurement n each teraton he ncreae n effcency over the fnte dfference tochatc approxmaton method for example ha been hown to be a factor equal to the dmenon of the problem [0] Spall [0] ha condered the convergence of SPSA for three tme dfferentable functon wherea He et al [9] have analyzed the convergence for nondfferentable but contnuou optmzaton Alo Youefan et al [3] have dcued a local randomzed moothng technque for convex nondfferentable contnuou tochatc optmzaton We want to ue a mlar dea of SPSA for the dcrete cae Becaue the uual noton of a gradent doe not apply n dcrete problem t not obvou that the convergence properte demontrated for SPSA hold for the dcrete cae Hll et al [0] conder a dcrete form of SPSA and develop prelmnary reult aocated wth convergence for a eparable dcrete lo functon under pecal condton However th algorthm can be hown to not converge to the optmal oluton n mple example We ntroduce a dfferent form of dcrete algorthm that apple to a broader range of problem whle potentally retanng the eental effcency advantage of tandard SPSA In partcular we ntroduce a mddle pont dcrete multaneou perturbaton tochatc approxmaton () algorthm that apple n a cla of dcrete problem A n conventonal SPSA the method need only two noy meaurement of the lo functon at each teraton Although a full convergence and convergence rate analy ha not yet been conducted we how condton for almot ure convergence of the algorthm to the true parameter value he paper organzed a the follow In Secton II we motvate the general approach by conderng the cae of one dmenonal and decrbe the bac algorthm for general p In Secton III we how that the algorthm converge to the optmal oluton for ome cla of functon under ome condton In Secton IV we how how th algorthm compare wth the localzed random earch method n two example In Secton V we conclude wth a dcuon //$600 0 AACC 450
2 II PROBLEM FORMULAION A Motvaton: One Dmenon Cae Let u frt conder one dmenonal dcrete functon L: where denote the et of nteger { 0 } We want to fnd the mnmal oluton of the lo functon L Let the noy meaurement of the lo functon be y where y L and ndcate the noe Fg how an example of a dcrete functon n one dmenon wth a lne connectng the neghborng nteger pont he lne L can be regarded a a contnuou extenon of L but L a nondfferentable functon at the nteger pont For a pont \ the gradent where g( ) L L and L ( ) L ( ) L ( ) L ( ) the floor functon the celng functon ( ) the mddle pont between and and a Bernoull random varable tang the value Actually ( ) n and n an odd number o ( ) mut be nteger We can ee that g( ) alo well defned at nteger pont and t a ubgradent (a vector γ a ubgradent of L() at f L()L() γ () for all p ) at (() now the mddle pont between and +) hen the etmated gradent for noy functon y( ) y( ) gˆ( ) Ref [9] ha hown that SPSA method tll converge for nondfferentable functon when the functon are contnuou and convex and the doman are convex and compact et L( ) Fg Example of trctly dcrete convex functon and L a contnuou extenon L B Bac Algorthm of Motvated by the pecal example hown above we wll conder the cae when θ p-dmenonal p 3 We have the general bac algorthm a below for functon y L + where L: p and ε noe he bac algorthm : Step0: Pc an ntal gue ˆθ 0 Step: Generate Δ [ p ] where the are ndependent Bernoull random varable tang the value wth probablty Step: ˆ ˆ p π ( θ ) θ where a p-dmenonal p vector wth all component equal unty and ˆ ˆ ˆ θ p Step3: Evaluate y at π( θˆ ) Δ and π( θˆ ) Δ form the etmate of g ˆ ( ˆ θ) ˆ ˆ ˆ ˆ g ( θ ) y( θ ) y ( ) θ where Δ p Step4: Update the etmate accordng to the recuron θ ˆ ˆ ˆ θ a g ˆ ( θ ) In the theoretcal analy below we mae ue of the followng mean gradent-le expreon centered at π( θ ): g ( π( θ)) E L π( θ) Δ L ( ) π θ Δ Δ θ where Δ p-dmenonal vector that ha the ame defnton a Δ mentoned above and may be a random varable n ome cae If each drecton choen equally then g ( π( θ)) ( ) ( ) p L L π θ Δ π θ Δ Δ where ndcate the ummaton over all poble drecton Note that Bernoull cae; we ue Δ Δ and Δ Δ n the Δ to accommodate future extenon to perturbaton dtrbuton other than Bernoull III CONVERGENCE PROPERIES We now preent an almot ure (a) convergence reult for θ ˆ Frt we ntroduce ome defnton that are ued n 45
3 the proof to follow For any pont θ we denote the et of mddle pont of all unt hypercube contanng θ a If θ le trctly nde one unt hypercube contan one pont But f θ le on the boundary contan multple pont For any pont m n we have m t for p where [t t t p ] and m the th component of m Furthermore let ˆ ˆ ˆ { θ0 θ θ } heorem Aume L a bounded functon on p and t ha unque mnmal pont θ Aume alo () a 0 lm a 0 0a and 0a ; () the component of Δ are ndependently Bernoull dtrbuted; () For all [( ) Δ ] 0 a and E the varance of unformly bounded; (v) up ˆ θ a; and (v) gm ( ) ( θ θ ) 0 for all 0 m and all p \{ } hen Proof By the algorthm we have θˆ ˆ ˆ θ a gˆ ( θ ) ˆ θ θ a ˆ ˆ ˆ θ a y ( θ ) y ( ) θ ˆ ˆ ˆ θ a L ( θ ) L ( ) ε ε θ By condton () () (v) and boundedne of L we have ˆ ˆ lm a L( ( θ ) ) L( ( θ ) ) 0 a () Alo uppoe the varance of are Chebyhev nequalty and () () ( ) hen by ( ) lm P a for ome m lm a 0 m m m mplyng by [4 heorem 4] that lm a { } Δ 0 a (3) hrough () we have the relatonhp that ( ) ( ) θ ˆ ˆ ˆ ˆ θ a L L θ θ ε ε and by the reult of () and (3) we get θ ˆ ˆ θ 0 a Hence there ext uch that θ ˆ ˆ ( ) θ ( ) 0 and P( ) By condton (v) { θ ˆ ( )} a bounded equence for any hen there ext a ubequence { θˆ ( )} and pont θ ( ) uch that { θˆ ( )} θ ( ) In addton we can rewrte () a θˆ θˆ a g( ( θˆ )) ˆ ˆ ˆ a L L g( ( θ )) ( θ ) ( θ ) a ε ε By the defnton of g () we have ˆ ˆ ˆ g ( π( θ )) π( θ ) Δ ( ) π θ Δ Δ E L L ˆ ˆ L ( ) L ( ) p π θ Δ π θ Δ Δ Let ˆ ˆ b L ( ) L ( ) π θ Δ π θ Δ then for all <j we have ˆ ˆ E ( π( θ )) b Δ ( π( θ j )) bjδ j g g ˆ ˆ ( π( θ )) Δ ( π( θ )) Δ E E g b g j bj j j ˆ ˆ ( π( θ )) Δ ( π( θ )) Δ E b E b due to condton () () and (v) for any we have g g j j j j 0 hen ˆ ˆ Δ ˆ Δ g( π( θ )) π( θ ) π( θ ) Δ E a L L a E L L ˆ ˆ Δ ˆ Δ g ( π( θ )) π( θ ) π( θ ) Δ Smlarly due to condton () () and () we have E a ( ) Δ <(4) a E ( ) Δ < (5) Snce for all n ˆ ˆ ˆ a ( π( θ )) L π( θ ) Δ L π( θ ) Δ Δ g and n a ( ) Δ are martngale then by (4) (5) and [3 n heorem 355] we now for all ˆ ˆ ˆ ( ( )) ( ) ( ) a L L g π θ π θ Δ π θ Δ Δ ext and Δ ext Let ( ) a M Δ and a ( ) N ( ( ˆ )) ( ˆ ) ( ˆ ) a L L g π θ π θ Δ π θ Δ Δ then M and N are revere martngale ([ p47]) and by [ heorem 358] there ext random varable M and N uch that M M a and N N a Furthermore n 45
4 due to (4) and (5) we have lm E N lm E M 0 Alo EM 0 and lm 0 and lm EN 0 hen M0 a whch ndcate M 0 a and N0 a whch ndcate N 0 a hen there ext and 3 uch that P( ) P( 3 ) and uch that for any a ( ( ) ( )) Δ 0 and for any 3 ( ( ˆ (ω))) ( ˆ (ω)) ( ˆ (ω)) a L L g π θ π θ Δ π θ Δ Δ 0 Let 4 3 wth P( 4 ) hen for any 4 we have θ(ω) θˆ (ω) a g( π( θˆ (ω))) ˆ ˆ ˆ a L L g( π( θ (ω))) π( θ (ω)) Δ π( θ (ω)) Δ Δ Δ a ε (ω) ε (ω) mplyng ε (ω) ε (ω) a ˆ ˆ ˆ a g L L 0 and π θ π θ Δ π θ Δ Δ 0 a In addton we now{ θ ˆ ( )} θ ( ) ( ( (ω))) ( (ω)) ( (ω)) ndcatng that a g ( π( θˆ (ω))) 0 a (6) Becaue θˆ (ω) θ (ω) then for any>0 there ext S>0 uch that when >S θˆ (ω) θ(ω) < hu there exts when >S all π( θˆ (ω)) We now how θ(ω) the optmal pont By way of contradcton uppoe θ (ω) not the optmal oluton hen by condton (v) we have gm ( ) ( θ( ) θ ) 0 for all m whch a contradcton of g ( π( θˆ (ω))) 0 when >S hen a for all 4 the lmtng pont of the equence { θ ˆ ( )} unque whch equal to θ hu ˆ θ θ a Comment : he nner product condton (v) a natural extenon of the tandard nner product condton for contnuou problem (eg [ p06]) whch nclude convex functon a a pecal cae Comment : Actually ome people have condered the dcrete convexty Mller [4] a forerunner n the early 970 n the area of dcrete convex functon Ref [4] ha ntroduced the defnton of dcrete convex functon and howed that the local optmal pont for dcrete convex functon are alo global optmal oluton here are other defnton of dcrete convex functon [5][5][6][6] but [7] how that Mller dcrete convexty contan the other clae of dcrete convexty Note that Mller defnton doe not nclude all functon atfyng condton (v) and condton (v) doe not nclude all functon atfy Mller defnton of dcrete convexty However for p dcrete convex functon atfyng Mller defnton alo atfy (v) he corollare below gve two common functon atfyng condton (v) Even though we decrbe the functon n contnuou form for we only ue ther value at multvarate nteger pont Strctly convex eparable functon mentoned n corollary are dcued n [0] Corollary Strctly convex eparable functon wth mnmal value at multvarate nteger pont atfy the condton (v) n heorem Proof A eparable functon can be wrtten a p L( θ ) L ( ) t where θ [ t t p] And L a dcrete functon ha ame value wth L at multvarate nteger pont Suppoe the unque mnmal pont of L and a multvarate nteger pont wth t t p θ [ ] hen alo the optmal pont of L Becaue t trctly convex then L ( t ) we have for all p \{ } and any ubgradent L ( t )( t t ) 0 for p Moreover for any m gm ( ) p L L m Δ m Δ Δ p L ( m ) L ( m ) p Δ Δ p L ( m ) L ( m ) p Δ Δ p L ( m ) L ( m ) e hen we have p gm ( ) ( θ θ ) L ( m ) L ( m ) ( t t ) Becaue the mnmal pont a multvarate nteger pont then L ( m ) L ( m ) ha the ame gn wth one of the ubgradent of L at t ndcatng that L ( m ) L ( m ) ( t t ) 0 for all p 453
5 hu gm ( ) ( θ θ ) 0 for all m and all p \{ } QED Corollary L a trctly convex pecewe lnear functon wth mnmal value at a multvarate nteger pont and t lnear n each unt hypercube then L atfy the condton (v) n heorem Proof L a dcrete functon that ha ame value wth L at multvarate nteger pont Snce L trctly convex functon then for all p \{ } and for any ubgradent L( θ ) we have L( θ) ( θ θ ) 0 Furthermore for any m g( m) p L L m Δ m Δ Δ p L L m Δ m Δ Δ ( ) p L m Δ Δ where the notaton of gradent hu gm ( ) ( θ θ) L( ) ( ) p m Δ Δ θ θ L( m) ( ) p ΔΔ θ θ L( m ) ( ) θ θ In addton for any m there wll be one ubgradent L( θ ) at pont θ uch that L( m ) L( θ) hen gm ( ) ( θ θ) L( θ) ( θ θ ) < 0 whch ndcate gm ( ) ( θ θ ) 0 for all m and all p \{ } QED IV COMPARISION WIH LOCALIZED RANDOM SEARCH MEHOD Let u now compare the performance of and the localzed random earch method for two lo functon he frt functon condered here a eparable functon p t he econd one a ewed quartc lo functon whch mentoned n [ Ex 66]: L( θ ) B B p 3 p 4 0 ( B ) 00 ( B ) θ θ θ θ where pb an upper trangular matrx of Even though the ewed quartc lo functon doe not atfy condton (v) we wll ee that tll wor for th lo functon We conder the hgh-dmenonal cae for both functon where p 00 and the meaurement noe ε d N(0) Snce the localzed random earch method more effcent n noe-free cae than n noy cae then we wll conder both the noe-free tuaton and noy tuaton he localzed random earch method decrbed n [ Secton 3] whch conder both noe-free lo functon and noy lo meaurement where a threhold parameter τ nvolved We wll retrct the random earch to the cloet neghbor pont and all thee pont are choen wth equal probablty Here for let a a ( A) a 006 (for eparable); a 00 (for ewed quartc) A For the localzed random earch method we chooe for the noy cae after everal tunng he ntal gue et to be 000 n all run Fg and 3 how the performance of both method under noe-free and noy tuaton for eparable functon And Fg 4 and 5 how the performance of both method for a ewed quartc functon We can ee that doe better than the random earch method for thee two example ˆ θ θ ˆ θ0 θ Number of Meaurement Fg Performance of localzed random earch method and under noe-free tuaton for eparable functon ˆ θ θ ˆ θ0 θ Number of Meaurement Fg 3 Performance of localzed random earch method and wth noy meaurement for eparable functon 3 09 ˆ 08 θ θ 07 ˆ θ0 θ Number of Meaurement Fg 4 Performance of localzed random earch method and under noe-free tuaton for ewed quartc functon 454
6 ˆ θ θ ˆ θ0 θ Fg 5 Performance of localzed random earch method and wth noy meaurement for ewed quartc functon V CONCLUSION Number of Meaurement In th paper we ntroduced a dcrete SPSA algorthm and preented ome prelmnary convergence analy A prelmnary numercal tudy how that wor well on hgh-dmenonal problem wth or wthout noe n the lo meaurement A part of future wor we plan to formally tudy the convergence rate of the and conder non-bernoull random varable for the perturbaton vector Alo we ntend to compare wth other popular dcrete optmzaton algorthm ncludng thoe degned explctly for handlng noy lo meaurement (eg [7][3][9]) wo mportant practcal problem of nteret that nvolve tochatc dcrete optmzaton are reource allocaton where a fnte amount of a valuable commodty mut be optmally allocated and expermental degn where t neceary to chooe the bet ubet of nput combnaton from a large number of poble nput combnaton n a full-factoral degn (eg [4]) We ntend to explore the applcaton of to thee or other problem ACKNOWLEDGMEN h wor wa upported n part by the JHU/APL IRAD Program REFERENCES [] M H Alrefae S Andradóttr A Smulated Annealng Algorthm wth Contant emperature for Dcrete Stochatc Optmzaton Management Sc vol 45 No5 May 999 pp [] S Andradóttr A Method for Dcrete Stochatc Optmzaton Management Sc vol 4 No December 995 pp [3] P Bllngley Probablty and Meaure Wley-Intercence hrd Edton 995 [4] K L Chung A Coure n Probablty heory Academc Pre hrd Edton 00 [5] P Favat F ardella Convexty n Nonlnear Integer Programmng Rcerca Operatva vol pp 3 44 [6] S Fujhge K Murota Note on L-/M-convex Functon and the Separaton heorem Mathematcal Programmng vol pp 9 46 [7] W B Gong Y C Ho W Zha Stochatc Comparon Algorthm for Dcrete Optmzaton wth Etmaton SIAM J Optm vol 0 No 000 pp [8] L A Hannah and WB Powell Evolutonary Polcy Iteraton Under a Samplng Regme for Stochatc Combnatoral Optmzaton IEEE ranacton on Automatc Control vol 55 No5 May 00 pp [9] Y He M C Fu and S I Marcu Convergence of Smultaneou Perturbaton Stochatc Approxmaton for Nondfferentable Optmzaton IEEE ranacton on Automatc Control vol 48 No8 Augut 003 pp [0] S D Hll L Gerencér and Z Vágó Stochatc Approxmaton on Dcrete Set Ung Smultaneou Dfference Approxmaton Proceedng of the 004 Amercan Control Conference Boton MA June 30July 004 pp [] Y C Ho Q C Zhao and Q S Ja Ordnal Optmzaton: Soft Optmzaton for Hard Problem Sprnger New Yor NY 007 [] L J Hong and B L Nelon Dcrete Optmzaton va Smulaton Ung COMPASS Oper Re vol 54 No 006 pp 5 9 [3] J L A Sava and X Xe Smulaton-Baed Dcrete Optmzaton of Stochatc Dcrete Event Sytem Subject to Non Cloed-Form Contrant IEEE ranacton on Automatc Control vol 54 No December 009 pp [4] B L Mller On Mnmzng Noneparable Functon Defned on the Integer wth an Inventory Applcaton SIAM Journal on Appled Mathematc vol No July 97 pp 6685 [5] K Murota Dcrete Convex Analy Mathematcal Programmng vol pp [6] K Murota A Shoura M-convex Functon on Generalzed Polymatrod Mathematc of Operaton Reearch vol pp [7] K Murota A Shoura Relatonhp of M-/L- Convex Functon wth Dcrete Convex Functon by Mller and Favat-ardella Dcrete Appled Mathematc vol 5 00 pp 5 76 [8] L Sh and S Olafon Neted Partton Method for Global Optmzaton Oper Re vol 48 No3 000 pp [9] J Slenar P Popela Integer Smulaton Baed Optmzaton by Local Search Proceda Computer Scence vol 00 pp [0] J C Spall Multvarate Stochatc Approxmaton Ung a Smultaneou Perturbaton Gradent Approxmaton IEEE ranacton on Automatc Control vol 37 No3 March 99 pp [] J C Spall An Overvew of the Smultaneou Perturbaton Method for Effcent Optmzaton John Hopn APL echncal Dget vol 9 No4 998 pp [] J C Spall Introducton to Stochatc Search and Optmzaton: Etmaton Smulaton and Control Wley Hoboen NJ 003 [3] F Youefan A Nedć and U V Shanbhag Convex Nondfferentable Stochatc Optmzaton: A Local Randomzed Smootng echnque Proceedng of the Amercan Control Conference Baltmore MD June 30 July 00 pp [4] J C Spall Factoral Degn for Choong Input Value n Expermentaton: Generatng Informatve Data for Sytem Identfcaton IEEE Control Sytem Magazne vol 30 no 5 October 00 pp
Additional File 1 - Detailed explanation of the expression level CPD
Addtonal Fle - Detaled explanaton of the expreon level CPD A mentoned n the man text, the man CPD for the uterng model cont of two ndvdual factor: P( level gen P( level gen P ( level gen 2 (.).. CPD factor
More informationTeam. Outline. Statistics and Art: Sampling, Response Error, Mixed Models, Missing Data, and Inference
Team Stattc and Art: Samplng, Repone Error, Mxed Model, Mng Data, and nference Ed Stanek Unverty of Maachuett- Amhert, USA 9/5/8 9/5/8 Outlne. Example: Doe-repone Model n Toxcology. ow to Predct Realzed
More informationImprovements on Waring s Problem
Improvement on Warng Problem L An-Png Bejng, PR Chna apl@nacom Abtract By a new recurve algorthm for the auxlary equaton, n th paper, we wll gve ome mprovement for Warng problem Keyword: Warng Problem,
More informationStart Point and Trajectory Analysis for the Minimal Time System Design Algorithm
Start Pont and Trajectory Analy for the Mnmal Tme Sytem Degn Algorthm ALEXANDER ZEMLIAK, PEDRO MIRANDA Department of Phyc and Mathematc Puebla Autonomou Unverty Av San Claudo /n, Puebla, 757 MEXICO Abtract:
More informationHarmonic oscillator approximation
armonc ocllator approxmaton armonc ocllator approxmaton Euaton to be olved We are fndng a mnmum of the functon under the retrcton where W P, P,..., P, Q, Q,..., Q P, P,..., P, Q, Q,..., Q lnwgner functon
More informationSpecification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction
ECONOMICS 35* -- NOTE ECON 35* -- NOTE Specfcaton -- Aumpton of the Smple Clacal Lnear Regreon Model (CLRM). Introducton CLRM tand for the Clacal Lnear Regreon Model. The CLRM alo known a the tandard lnear
More informationThe multivariate Gaussian probability density function for random vector X (X 1,,X ) T. diagonal term of, denoted
Appendx Proof of heorem he multvarate Gauan probablty denty functon for random vector X (X,,X ) px exp / / x x mean and varance equal to the th dagonal term of, denoted he margnal dtrbuton of X Gauan wth
More informationChapter 11. Supplemental Text Material. The method of steepest ascent can be derived as follows. Suppose that we have fit a firstorder
S-. The Method of Steepet cent Chapter. Supplemental Text Materal The method of teepet acent can be derved a follow. Suppoe that we have ft a frtorder model y = β + β x and we wh to ue th model to determne
More informationEstimation of Finite Population Total under PPS Sampling in Presence of Extra Auxiliary Information
Internatonal Journal of Stattc and Analy. ISSN 2248-9959 Volume 6, Number 1 (2016), pp. 9-16 Reearch Inda Publcaton http://www.rpublcaton.com Etmaton of Fnte Populaton Total under PPS Samplng n Preence
More informationImprovements on Waring s Problem
Imrovement on Warng Problem L An-Png Bejng 85, PR Chna al@nacom Abtract By a new recurve algorthm for the auxlary equaton, n th aer, we wll gve ome mrovement for Warng roblem Keyword: Warng Problem, Hardy-Lttlewood
More informationChapter 6 The Effect of the GPS Systematic Errors on Deformation Parameters
Chapter 6 The Effect of the GPS Sytematc Error on Deformaton Parameter 6.. General Beutler et al., (988) dd the frt comprehenve tudy on the GPS ytematc error. Baed on a geometrc approach and aumng a unform
More informationMethod Of Fundamental Solutions For Modeling Electromagnetic Wave Scattering Problems
Internatonal Workhop on MehFree Method 003 1 Method Of Fundamental Soluton For Modelng lectromagnetc Wave Scatterng Problem Der-Lang Young (1) and Jhh-We Ruan (1) Abtract: In th paper we attempt to contruct
More informationSeparation Axioms of Fuzzy Bitopological Spaces
IJCSNS Internatonal Journal of Computer Scence and Network Securty VOL3 No October 3 Separaton Axom of Fuzzy Btopologcal Space Hong Wang College of Scence Southwet Unverty of Scence and Technology Manyang
More informationVariable Structure Control ~ Basics
Varable Structure Control ~ Bac Harry G. Kwatny Department of Mechancal Engneerng & Mechanc Drexel Unverty Outlne A prelmnary example VS ytem, ldng mode, reachng Bac of dcontnuou ytem Example: underea
More informationA METHOD TO REPRESENT THE SEMANTIC DESCRIPTION OF A WEB SERVICE BASED ON COMPLEXITY FUNCTIONS
UPB Sc Bull, Sere A, Vol 77, I, 5 ISSN 3-77 A METHOD TO REPRESENT THE SEMANTIC DESCRIPTION OF A WEB SERVICE BASED ON COMPLEXITY FUNCTIONS Andre-Hora MOGOS, Adna Magda FLOREA Semantc web ervce repreent
More informationTwo Approaches to Proving. Goldbach s Conjecture
Two Approache to Provng Goldbach Conecture By Bernard Farley Adved By Charle Parry May 3 rd 5 A Bref Introducton to Goldbach Conecture In 74 Goldbach made h mot famou contrbuton n mathematc wth the conecture
More informationAdaptive Centering with Random Effects in Studies of Time-Varying Treatments. by Stephen W. Raudenbush University of Chicago.
Adaptve Centerng wth Random Effect n Stde of Tme-Varyng Treatment by Stephen W. Radenbh Unverty of Chcago Abtract Of wdepread nteret n ocal cence are obervatonal tde n whch entte (peron chool tate contre
More informationSolution Methods for Time-indexed MIP Models for Chemical Production Scheduling
Ian Davd Lockhart Bogle and Mchael Farweather (Edtor), Proceedng of the 22nd European Sympoum on Computer Aded Proce Engneerng, 17-2 June 212, London. 212 Elever B.V. All rght reerved. Soluton Method for
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationm = 4 n = 9 W 1 N 1 x 1 R D 4 s x i
GREEDY WIRE-SIZING IS LINEAR TIME Chr C. N. Chu D. F. Wong cnchu@c.utexa.edu wong@c.utexa.edu Department of Computer Scence, Unverty of Texa at Autn, Autn, T 787. ABSTRACT In nterconnect optmzaton by wre-zng,
More informationConfidence intervals for the difference and the ratio of Lognormal means with bounded parameters
Songklanakarn J. Sc. Technol. 37 () 3-40 Mar.-Apr. 05 http://www.jt.pu.ac.th Orgnal Artcle Confdence nterval for the dfference and the rato of Lognormal mean wth bounded parameter Sa-aat Nwtpong* Department
More informationIntroduction to Interfacial Segregation. Xiaozhe Zhang 10/02/2015
Introducton to Interfacal Segregaton Xaozhe Zhang 10/02/2015 Interfacal egregaton Segregaton n materal refer to the enrchment of a materal conttuent at a free urface or an nternal nterface of a materal.
More informationarxiv: v1 [cs.gt] 15 Jan 2019
Model and algorthm for tme-content rk-aware Markov game Wenje Huang, Pham Vet Ha and Wllam B. Hakell January 16, 2019 arxv:1901.04882v1 [c.gt] 15 Jan 2019 Abtract In th paper, we propoe a model for non-cooperatve
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More information1 Convex Optimization
Convex Optmzaton We wll consder convex optmzaton problems. Namely, mnmzaton problems where the objectve s convex (we assume no constrants for now). Such problems often arse n machne learnng. For example,
More informationPythagorean triples. Leen Noordzij.
Pythagorean trple. Leen Noordz Dr.l.noordz@leennoordz.nl www.leennoordz.me Content A Roadmap for generatng Pythagorean Trple.... Pythagorean Trple.... 3 Dcuon Concluon.... 5 A Roadmap for generatng Pythagorean
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationMarkov Chain Monte Carlo Lecture 6
where (x 1,..., x N ) X N, N s called the populaton sze, f(x) f (x) for at least one {1, 2,..., N}, and those dfferent from f(x) are called the tral dstrbutons n terms of mportance samplng. Dfferent ways
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationa new crytoytem baed on the dea of Shmuley and roved t rovably ecure baed on ntractablty of factorng [Mc88] After that n 999 El Bham, Dan Boneh and Om
Weak Comote Dffe-Hellman not Weaker than Factorng Koohar Azman, azman@ceharfedu Javad Mohajer mohajer@harfedu Mahmoud Salmazadeh alma@harfedu Electronc Reearch Centre, Sharf Unverty of Technology Deartment
More information18.1 Introduction and Recap
CS787: Advanced Algorthms Scrbe: Pryananda Shenoy and Shjn Kong Lecturer: Shuch Chawla Topc: Streamng Algorthmscontnued) Date: 0/26/2007 We contnue talng about streamng algorthms n ths lecture, ncludng
More informationLecture 20: Lift and Project, SDP Duality. Today we will study the Lift and Project method. Then we will prove the SDP duality theorem.
prnceton u. sp 02 cos 598B: algorthms and complexty Lecture 20: Lft and Project, SDP Dualty Lecturer: Sanjeev Arora Scrbe:Yury Makarychev Today we wll study the Lft and Project method. Then we wll prove
More informationLecture 10: May 6, 2013
TTIC/CMSC 31150 Mathematcal Toolkt Sprng 013 Madhur Tulsan Lecture 10: May 6, 013 Scrbe: Wenje Luo In today s lecture, we manly talked about random walk on graphs and ntroduce the concept of graph expander,
More informationIV. Performance Optimization
IV. Performance Optmzaton A. Steepest descent algorthm defnton how to set up bounds on learnng rate mnmzaton n a lne (varyng learnng rate) momentum learnng examples B. Newton s method defnton Gauss-Newton
More informationand decompose in cycles of length two
Permutaton of Proceedng of the Natona Conference On Undergraduate Reearch (NCUR) 006 Domncan Unverty of Caforna San Rafae, Caforna Apr - 4, 007 that are gven by bnoma and decompoe n cyce of ength two Yeena
More informationSmall signal analysis
Small gnal analy. ntroducton Let u conder the crcut hown n Fg., where the nonlnear retor decrbed by the equaton g v havng graphcal repreentaton hown n Fg.. ( G (t G v(t v Fg. Fg. a D current ource wherea
More informationSome modelling aspects for the Matlab implementation of MMA
Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton
More informationAPPROXIMATE FUZZY REASONING BASED ON INTERPOLATION IN THE VAGUE ENVIRONMENT OF THE FUZZY RULEBASE AS A PRACTICAL ALTERNATIVE OF THE CLASSICAL CRI
Kovác, Sz., Kóczy, L.T.: Approxmate Fuzzy Reaonng Baed on Interpolaton n the Vague Envronment of the Fuzzy Rulebae a a Practcal Alternatve of the Clacal CRI, Proceedng of the 7 th Internatonal Fuzzy Sytem
More information728. Mechanical and electrical elements in reduction of vibrations
78. Mechancal and electrcal element n reducton of vbraton Katarzyna BIAŁAS The Slean Unverty of Technology, Faculty of Mechancal Engneerng Inttute of Engneerng Procee Automaton and Integrated Manufacturng
More informationLecture 4: Constant Time SVD Approximation
Spectral Algorthms and Representatons eb. 17, Mar. 3 and 8, 005 Lecture 4: Constant Tme SVD Approxmaton Lecturer: Santosh Vempala Scrbe: Jangzhuo Chen Ths topc conssts of three lectures 0/17, 03/03, 03/08),
More informationExtended Prigogine Theorem: Method for Universal Characterization of Complex System Evolution
Extended Prgogne Theorem: Method for Unveral Characterzaton of Complex Sytem Evoluton Sergey amenhchkov* Mocow State Unverty of M.V. Lomonoov, Phycal department, Rua, Mocow, Lennke Gory, 1/, 119991 Publhed
More informationA Computational Method for Solving Two Point Boundary Value Problems of Order Four
Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 A Computatonal Metod for Solvng Two Pont Boundary Value Problem of Order Four Yoge Gupta Department of Matematc Unted College of Engg and
More informationRoot Locus Techniques
Root Locu Technque ELEC 32 Cloed-Loop Control The control nput u t ynthezed baed on the a pror knowledge of the ytem plant, the reference nput r t, and the error gnal, e t The control ytem meaure the output,
More informationWeak McCoy Ore Extensions
Internatonal Mathematcal Forum, Vol. 6, 2, no. 2, 75-86 Weak McCoy Ore Extenon R. Mohammad, A. Mouav and M. Zahr Department of Pure Mathematc, Faculty of Mathematcal Scence Tarbat Modare Unverty, P.O.
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationOn the SO 2 Problem in Thermal Power Plants. 2.Two-steps chemical absorption modeling
Internatonal Journal of Engneerng Reearch ISSN:39-689)(onlne),347-53(prnt) Volume No4, Iue No, pp : 557-56 Oct 5 On the SO Problem n Thermal Power Plant Two-tep chemcal aborpton modelng hr Boyadjev, P
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationA MODIFIED METHOD FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS
Journal of Mathematcs and Statstcs 9 (1): 4-8, 1 ISSN 1549-644 1 Scence Publcatons do:1.844/jmssp.1.4.8 Publshed Onlne 9 (1) 1 (http://www.thescpub.com/jmss.toc) A MODIFIED METHOD FOR SOLVING SYSTEM OF
More informationEstimation: Part 2. Chapter GREG estimation
Chapter 9 Estmaton: Part 2 9. GREG estmaton In Chapter 8, we have seen that the regresson estmator s an effcent estmator when there s a lnear relatonshp between y and x. In ths chapter, we generalzed the
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationStatistical Properties of the OLS Coefficient Estimators. 1. Introduction
ECOOMICS 35* -- OTE 4 ECO 35* -- OTE 4 Stattcal Properte of the OLS Coeffcent Etmator Introducton We derved n ote the OLS (Ordnary Leat Square etmator ˆβ j (j, of the regreon coeffcent βj (j, n the mple
More informationOn the Multicriteria Integer Network Flow Problem
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of
More informationPower law and dimension of the maximum value for belief distribution with the max Deng entropy
Power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy Bngy Kang a, a College of Informaton Engneerng, Northwest A&F Unversty, Yanglng, Shaanx, 712100, Chna. Abstract Deng
More informationGeneral viscosity iterative method for a sequence of quasi-nonexpansive mappings
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,
More informationAP Statistics Ch 3 Examining Relationships
Introducton To tud relatonhp between varable, we mut meaure the varable on the ame group of ndvdual. If we thnk a varable ma eplan or even caue change n another varable, then the eplanator varable and
More informationFinding Dense Subgraphs in G(n, 1/2)
Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng
More informationThe Essential Dynamics Algorithm: Essential Results
@ MIT maachuett nttute of technology artfcal ntellgence laboratory The Eental Dynamc Algorthm: Eental Reult Martn C. Martn AI Memo 003-014 May 003 003 maachuett nttute of technology, cambrdge, ma 0139
More informationEstimation of a proportion under a certain two-stage sampling design
Etmaton of a roorton under a certan two-tage amng degn Danutė Kraavcatė nttute of athematc and nformatc Lthuana Stattc Lthuana Lthuana e-ma: raav@tmt Abtract The am of th aer to demontrate wth exame that
More informationENTROPY BOUNDS USING ARITHMETIC- GEOMETRIC-HARMONIC MEAN INEQUALITY. Guru Nanak Dev University Amritsar, , INDIA
Internatonal Journal of Pure and Appled Mathematc Volume 89 No. 5 2013, 719-730 ISSN: 1311-8080 prnted veron; ISSN: 1314-3395 on-lne veron url: http://.jpam.eu do: http://dx.do.org/10.12732/jpam.v895.8
More informationThe Price of Anarchy in a Network Pricing Game
The Prce of Anarchy n a Network Prcng Game John Muaccho and Shuang Wu Abtract We analyze a game theoretc model of competng network ervce provder that trategcally prce ther ervce n the preence of elatc
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationInformation Acquisition in Global Games of Regime Change (Online Appendix)
Informaton Acquton n Global Game of Regme Change (Onlne Appendx) Mchal Szkup and Iabel Trevno Augut 4, 05 Introducton Th appendx contan the proof of all the ntermedate reult that have been omtted from
More informationResonant FCS Predictive Control of Power Converter in Stationary Reference Frame
Preprnt of the 9th World Congre The Internatonal Federaton of Automatc Control Cape Town, South Afrca. Augut -9, Reonant FCS Predctve Control of Power Converter n Statonary Reference Frame Lupng Wang K
More informationOPTIMAL COMPUTING BUDGET ALLOCATION FOR MULTI-OBJECTIVE SIMULATION MODELS. David Goldsman
Proceedng of the 004 Wnter Smulaton Conference R.G. Ingall, M. D. Roett, J. S. Smth, and B. A. Peter, ed. OPTIMAL COMPUTING BUDGET ALLOCATION FOR MULTI-OBJECTIVE SIMULATION MODELS Loo Hay Lee Ek Peng Chew
More informationVector Norms. Chapter 7 Iterative Techniques in Matrix Algebra. Cauchy-Bunyakovsky-Schwarz Inequality for Sums. Distances. Convergence.
Vector Norms Chapter 7 Iteratve Technques n Matrx Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematcs Unversty of Calforna, Berkeley Math 128B Numercal Analyss Defnton A vector norm
More informationMULTIPLE REGRESSION ANALYSIS For the Case of Two Regressors
MULTIPLE REGRESSION ANALYSIS For the Cae of Two Regreor In the followng note, leat-quare etmaton developed for multple regreon problem wth two eplanator varable, here called regreor (uch a n the Fat Food
More informationThis appendix presents the derivations and proofs omitted from the main text.
Onlne Appendx A Appendx: Omtted Dervaton and Proof Th appendx preent the dervaton and proof omtted from the man text A Omtted dervaton n Secton Mot of the analy provded n the man text Here, we formally
More informationKinetic-Energy Density-Functional Theory on a Lattice
h an open acce artcle publhed under an ACS AuthorChoce Lcene, whch permt copyng and redtrbuton of the artcle or any adaptaton for non-commercal purpoe. Artcle Cte h: J. Chem. heory Comput. 08, 4, 407 4087
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationMultiple-objective risk-sensitive control and its small noise limit
Avalable onlne at www.cencedrect.com Automatca 39 (2003) 533 541 www.elever.com/locate/automatca Bref Paper Multple-objectve rk-entve control and t mall noe lmt Andrew E.B. Lm a, Xun Yu Zhou b; ;1, John
More informationA NUMERICAL MODELING OF MAGNETIC FIELD PERTURBATED BY THE PRESENCE OF SCHIP S HULL
A NUMERCAL MODELNG OF MAGNETC FELD PERTURBATED BY THE PRESENCE OF SCHP S HULL M. Dennah* Z. Abd** * Laboratory Electromagnetc Sytem EMP BP b Ben-Aknoun 606 Alger Algera ** Electronc nttute USTHB Alger
More informationPrimer on High-Order Moment Estimators
Prmer on Hgh-Order Moment Estmators Ton M. Whted July 2007 The Errors-n-Varables Model We wll start wth the classcal EIV for one msmeasured regressor. The general case s n Erckson and Whted Econometrc
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationDesign of Recursive Digital Filters IIR
Degn of Recurve Dgtal Flter IIR The outut from a recurve dgtal flter deend on one or more revou outut value, a well a on nut t nvolve feedbac. A recurve flter ha an nfnte mule reone (IIR). The mulve reone
More informationOn a direct solver for linear least squares problems
ISSN 2066-6594 Ann. Acad. Rom. Sc. Ser. Math. Appl. Vol. 8, No. 2/2016 On a drect solver for lnear least squares problems Constantn Popa Abstract The Null Space (NS) algorthm s a drect solver for lnear
More informationA A Non-Constructible Equilibrium 1
A A Non-Contructbe Equbrum 1 The eampe depct a eparabe contet wth three payer and one prze of common vaue 1 (o v ( ) =1 c ( )). I contruct an equbrum (C, G, G) of the contet, n whch payer 1 bet-repone
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationAdditional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty
Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationA New Inverse Reliability Analysis Method Using MPP-Based Dimension Reduction Method (DRM)
roceedng of the ASME 007 Internatonal Degn Engneerng Techncal Conference & Computer and Informaton n Engneerng Conference IDETC/CIE 007 September 4-7, 007, La Vega, eada, USA DETC007-35098 A ew Inere Relablty
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
More informationBatch RL Via Least Squares Policy Iteration
Batch RL Va Leat Square Polcy Iteraton Alan Fern * Baed n part on lde by Ronald Parr Overvew Motvaton LSPI Dervaton from LSTD Expermental reult Onlne veru Batch RL Onlne RL: ntegrate data collecton and
More informationApproximate D-optimal designs of experiments on the convex hull of a finite set of information matrices
Approxmate D-optmal desgns of experments on the convex hull of a fnte set of nformaton matrces Radoslav Harman, Mára Trnovská Department of Appled Mathematcs and Statstcs Faculty of Mathematcs, Physcs
More informationThe Study of Teaching-learning-based Optimization Algorithm
Advanced Scence and Technology Letters Vol. (AST 06), pp.05- http://dx.do.org/0.57/astl.06. The Study of Teachng-learnng-based Optmzaton Algorthm u Sun, Yan fu, Lele Kong, Haolang Q,, Helongang Insttute
More informationStrong Efficient Domination in Graphs
P P P IJISET - Internatonal Journal of Innovatve Scence, Engneerng & Technology, Vol Iue 4, June 04 wwwjetcom Strong Effcent Domnaton n Graph ISSN 348-7968 3 NMeenaP P, ASubramananP P, VSwamnathanP PDepartment
More informationBackorder minimization in multiproduct assemble-to-order systems
IIE Tranacton (2005) 37, 763 774 Copyrght C IIE ISSN: 0740-817X prnt / 1545-8830 onlne DOI: 10.1080/07408170590961139 Backorder mnmzaton n multproduct aemble-to-order ytem YINGDONG LU, 1 JING-SHENG SONG
More informationMODELLING OF TRANSIENT HEAT TRANSPORT IN TWO-LAYERED CRYSTALLINE SOLID FILMS USING THE INTERVAL LATTICE BOLTZMANN METHOD
Journal o Appled Mathematc and Computatonal Mechanc 7, 6(4), 57-65 www.amcm.pcz.pl p-issn 99-9965 DOI:.75/jamcm.7.4.6 e-issn 353-588 MODELLING OF TRANSIENT HEAT TRANSPORT IN TWO-LAYERED CRYSTALLINE SOLID
More informationAPPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14
APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor 1 1.1. Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce
More informationResearch Article Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations
Hndaw Publhng Corporaton Mathematcal Problem n Engneerng Volume 205, Artcle ID 893763, page http://dx.do.org/0.55/205/893763 Reearch Artcle Runge-Kutta Type Method for Drectly Solvng Specal Fourth-Order
More informationPredictors Using Partially Conditional 2 Stage Response Error Ed Stanek
Predctor ng Partally Condtonal Stage Repone Error Ed Stane TRODCTO We explore the predctor that wll relt n a mple random ample wth repone error when a dfferent model potlated The model we decrbe here cloely
More informationStrong Markov property: Same assertion holds for stopping times τ.
Brownan moton Let X ={X t : t R + } be a real-valued stochastc process: a famlty of real random varables all defned on the same probablty space. Defne F t = nformaton avalable by observng the process up
More informationScattering of two identical particles in the center-of. of-mass frame. (b)
Lecture # November 5 Scatterng of two dentcal partcle Relatvtc Quantum Mechanc: The Klen-Gordon equaton Interpretaton of the Klen-Gordon equaton The Drac equaton Drac repreentaton for the matrce α and
More informationOn multivariate folded normal distribution
Sankhyā : The Indan Journal o Stattc 03, Volume 75-B, Part, pp. -5 c 03, Indan Stattcal Inttute On multvarate olded normal dtrbuton Ah Kumar Chakraborty and Moutuh Chatterjee Indan Stattcal Inttute, Kolkata,
More informationTAIL BOUNDS FOR SUMS OF GEOMETRIC AND EXPONENTIAL VARIABLES
TAIL BOUNDS FOR SUMS OF GEOMETRIC AND EXPONENTIAL VARIABLES SVANTE JANSON Abstract. We gve explct bounds for the tal probabltes for sums of ndependent geometrc or exponental varables, possbly wth dfferent
More informationWeek 5: Neural Networks
Week 5: Neural Networks Instructor: Sergey Levne Neural Networks Summary In the prevous lecture, we saw how we can construct neural networks by extendng logstc regresson. Neural networks consst of multple
More informationSeismic Reliability Analysis and Topology Optimization of Lifeline Networks
The 4 th World Conference on Earthquake Engneerng October 2-7, 2008, Beng, Chna Semc Relablty Analy and Topology Optmzaton of Lfelne Network ABSTRACT: Je L and We Lu 2 Profeor, Dept. of Buldng Engneerng,
More information