Class Administrivia Motivated Examples

Transcription

1 Class Admnstra Motated Examples

2 Lectures Ths class s offered n Englsh Most of lectures wll be gen usng blacboard. You need to tae note by yourself. Except today s lecture No textboo but wll lst some readng materals. TA: Lst of reference boos s posted onlne. Lst of webstes s also posted. Offce hour: Wednesday 0:00-2:00

3 Gradng 40% homewor (eery two wees) 20% mdterm (tentate date: 4/9) 30% fnal exam/proect (tentate date: 6/2) If your mdterm does not pass a threshold you need to tae the fnal exam. If your mdterm passes a threshold you can choose to do proect or to tae the fnal exam. Proect wll be research orentated ncludng paper readng and mplementaton. 0% attendance

4 Topcs wll be coered. Introducton 2. One-dmensonal optmzaton 3. Unconstraned optmzaton 4. Least square problem 5. Constraned optmzaton 6. Global optmzaton or/and dscrete optmzaton Detals s lsted on the class webste: 32/ndex.html

5 Your goals of ths class Learnng some basc optmzaton concepts and methods as well as ther lmtatons. Ths wll help a lot for you to read papers and to do researches. Knowng the numercal technques behnd the numercal optmzaton methods. Ths can help you to erfy the equatons and to dere new methods.

6 Motated examples Ex: Classfcaton n pattern recognton (SVM) Ex2: Floorplannng n VLSI desgn Ex3: Networ component analyss n system bology Ex4: Localzaton problem n wreless networ Ex5: Topology mappng problem n hgh-performance computng Ex6: Job nterew problem for one arable optmzaton

7 Ex: Classfcaton Gen two sets A={aa2 a} and B={bb2 bm} n R n. Fnd a lne to separate them such that the error s mnmzed. The lne equaton s The errors are a a ) ( 0 f 0 f 0 ) _( a a a a e 0 f 0 f 0 ) ( b b b b e Andrze P. Ruszczyńs Nonlnear optmzaton chap 2006

8 Classfcaton - constrans Constrans If =0 and =0 errors are zero Let = Problem: mn e ( a m ) e ( b ) s.t. Pecewse lnear functon

9 Classfcaton Slac arables Let Slac arables Problem:...m z b... s a z s m z s 0 0 s.t. mn z b e s a e ) ( ) (

10 Classfcaton (SVM) Addng buffers Problem: Ths s called supportng ector machne (SVM)...m z b... s a z s m z s s.t. mn

11 Ex2: Floorplannng An early stage of physcal desgn that determnes module postons shapes and orentatons Module : Wdth and heght: w and h Area: w h Aspect rato: h / w Goal: Fnd wdth w heght A/w and lower left corner coordnates x and y = n such that the area s mnmzed h w Chuan Ln Ha Zhou Chrs Chu 2006 A rest to floorplan optmzaton by Lagrangan relaxaton

12 Constrant graph Horzontal constrant graph Gh of n modules Edge () Gh f module s to the rght of module Vertcal constrant graph Gof n modules Edge () G f module s aboe module

13 Problem formulaton (P area )

14 Ex3:Networ component analyss Networ: a bpartte graph (RGE) R: concentraton of acte form of regulatory protens. G: gene expressons that are obsered through a seres of experments. E: the arrows connectng nodes ndcate whch regulatory protens hae nfluence on whch genes. James C. Lao Rccardo Boscolo Young-Lyeol Yang Lnh My Tran Chara Sabatt and Vwan P. Roychowdhury 2003 Networ component analyss: Reconstructon of regulatory sgnals n bologcal systems

15 Problem formulaton E = AP where EAP are matrces E(NM):each row represents a gene; each column represents an experment A(NL):each column represents the acton of a regulatory proten on all the genes P(L M) matrx wth each row representng the profle of each regulatory proten across experments. E s nown; A and P are what we want to compute

16 Networ constrans Elements of matrx A need to satsfy the zero patterns that specfed by the networ Problem formulaton mn A P s.t. E A( AP ) 0 f edge E

17 Ex4: Localzaton problem For n nodes X={xx2 xn} and m anchor nodes A={aa2 am} gen partal relate dstance nformaton between nodes fnd the postons of {xx2 xn}. Prat Bswas and Ynyu Ye 2004 Semdefnte programmng for ad hoc wreless sensor networ localzaton

18 Eucldean dstance matrx (EDM) If we now the full EDM we can fnd the relate coordnates (up to rotaton). If we now the full EDM and some anchor nodes we can dentfy the absolute coordnates of X. EDM s usually nosy. Only maxmum lelhood postons s aalable.

19 Problem formulaton E E x x d x a d d x x d n ) ( ) ( ) ( ) ( mn Not ery robust. Need more constrans Dstance lower bound Dstance upper bound

20 Problem formulaton Defne:

21 Ex5:Topology mappng problem There are n tass and n processors. The traffc between tass s recorded n a matrx A; The communcaton cost (dstance hops) between processors s recorded n a matrx D. Fnd an assgnment P of tass to processors such that the communcaton cost s mnmzed. The oerall cost The maxmum cost

22 Examples

23 Ex6:Job nterew problem There are N nterewees for one poston. They are nterewed n a random order. The result s ether accepted or reected. The reected one cannot be recalled. If one s accepted he/she cannot be replaced. The decson to accept or reect s based on the relate rans of the nterewee so far. The obect s to select the best applcant.

24 How to mae decson? The obecte functon: f the best one s selected 0 otherwse. If acceptng too early you mght mss better one later; f too late the better one could be reected earler. Basc strategy: passng on the frst canddates and then selectng the next best so far. How to decde the best?

25 Fxed best canddate Suppose the best canddate s at poston The probablty s /N For to be selected the best of (-) must be n the [..]. The probablty s /(-) gen <. So the probablty for to be selected s

26 Fxed If we fxed and sum up all possble > P { N} N N N / N / P {N} Fnd to maxmze P{N}

27 For large enough and N To fnd the best strategy we need to decde what s? P() s called unmodal whch means t has only one maxmum x*; for x<x* f(x) s monotoncally ncreasng and for x>x* f(x) s monotoncally decreasng. N N N N N N N P N N ln ) ln( ln / / } {