Operations Research - Optimization Homework 3: Linear Regression Modeling, Assignments of Operational Research

Download Operations Research - Optimization Homework 3: Linear Regression Modeling and more Assignments Operational Research in PDF only on Docsity! IE 335: Operations Research - Optimization Fall 2008 Homework 3 Due: Wednesday, October 1, 11:30am in class Problem 1 (Bertsimas and Tsitsiklis (1997) - Data fitting, reworded). We are given m data points of the form .ai1; : : : ; a i n; b i / i D 1; : : : ; m: The data .ai1; : : : ; a i n/ for i D 1; : : : ; m represent explanatory factors, and the data b i for i D 1; : : : ; m represent the response. For example, the data might come from m different people, where .ai1; : : : ; a i n/ rep- resents various types of demographic and educational attainment information for person i , and bi represents person i ’s salary. We wish to use these data points to estimate a linear predictive model between the explanatory factors and the response: we want to estimate parameters .x1; : : : ; xn/ such that b  nX j D1 aj xj for explanatory factors .a1; : : : ; an/ and response b. Given a particular parameter vector .x1; : : : ; xn/, the residual, or prediction error, at the i th data point is defined as ˇ̌̌̌ bi nX j D1 aij xj ˇ̌̌̌ : Given a choice between alternative predictive models, one typically chooses a model that “explains” the available data as best as possible, i.e., a model that results in small residuals. (a) One possibility is to minimize the total prediction error mX iD1 ˇ̌̌̌ bi nX j D1 aij xj ˇ̌̌̌ ; with respect to .x1; : : : ; xn/, subject to no constraints. Formulate this optimization problem as a linear programming model. (b) (A little bit harder) In an alternative formulation, one could minimize the largest residual max iD1;:::;m ˇ̌̌̌ bi nX j D1 aij xj ˇ̌̌̌ ; with respect to .x1; : : : ; xn/, subject to no constraints. Formulate this optimization problem as a linear programming model. Problem 2. Rardin, Exercise 4-36 Problem 3. Rardin, Exercise 5-3 (b), (d), (f) 1 Problem 4. Rardin, Exercise 5-5 Problem 5. Rardin, Exercise 5-8 Problem 6. Rardin, Exercise 5-10 Problem 7. Rardin, Exercise 5-15 Problem 8. Rardin, Exercise 5-23 2 's-..- "5 ~ \. J..) I -tY"Oc\"'-c..~ ~ +~ dlA-c.,~ vO-.~ ~ 1-01 t, -t ~'\IV,tv~""t-. - -x ~ -\-x-\f -­s :t . 'X. \ - I -i..z.. - x ~ -+)(r ::::- - \ -:Cl -t X 2,. -t JL") =to -x t -x 1- I -L\ 'Xy :z )\ " J ~~ 0 '0 ¢ "- \ <::l -I \') \ ..J \ () A~ ( ~ ) 10::.. (~: ) \)c... ~ \. - 1-''0 '0 q~ v) \...{) < v. ~~-\-\ ..\,,,,-\: ~""-'6 ~v ~ \ \N ~~~ -1 \ = _~tr ~,( trv , I -x: '-- \IV \"1-' ~ '""l.. 1- ~ -:;( L - X ~, k -tv-v d..-Vvt-----.~ ~ d Co.. vk. VG-"f (-Xu-) \'VI ...vk t~vs-t' l-'l vq \ yO'... • y\.-1"' """"-t\. ,::(.~ \ ~'\ ~ s*~~ ~vv'vtYOV:-\A '\" ~o I ' 1­ \ 5 ~ ) ~, -~~-, p ., v ) ~ ­ (/V'L~) " " '_, I . , , \'vJ ) - ~ \-t Z~2r k '1~ ;; b "a 1. ~'(} ~ ~2 'a ~ ~ '1J ~ 1 ~ ~, '2 1f 7/ 0 A ~ (-I L 'C \ \CoO) ~ ~,~ ~ '> +00 -f..vvv ~ v d~0 ~ ~ ~ . ~ \.1 d lot 1 ~ ~ ~ I d 5 1 ~'r~ l ( ~y l ~I olL~-r. ': IJ ~ ~ \ I '1 t- \ ~- .s­ ~ l ~) ~-y \~ , I\'d ~ \ , ~ t\' ==- ( ,:}, , "a L , \) V) S", l'V~ A'{I...\);=: 0 -1 'Y l \ ) =- \. '2. \ "2.. \J \:» ) p. ) \))~y ~ d-~ I ~ ~ ~ 1 ~( ~,val/ vJ ) , , =i Y(2 :=; (IJ / ~ . e.. , Q) Y\\) = ( 'V \.)~ 1..1hY ~d'L , d Y~ , ,- ,- 'd ~ ) ~~ ) J~ -f,'~+ -tw O (")Mp(lY\J.V\t~ ~t -{O\.(..h... b ~ l, ; C (o[ v\..hc7 v\ ~,:¥R- ~ (foYY-l '>F \? ~l ~~21 f,-,'V\t s -b -t ~ bY't,- \V\.- ..~ (.JY'O \.-)l ~.V\I\. ( '\J l\ ) ~ ( <- "2.. ) , '(' (,,) -= ( ) ,,/ (~ ) ") '\ '----A 1 I VI 2- , ! =( V , ~ / /~ +'Il,. t two f~ ",c,~b\~ b,,-(..~c S o(v.. 'hO\i1> C\y-~ .J(/f-f'f.1lJN\ ~ \ o\VltS) J" k \V't .eltc..; b( -e b~ ~ i c. S'o[\A'1l0 11 "\ ~ "'- ,.-t llo) ~ 2.~ \ + r ~ ?- ) ~ . ~ )\-t ?. 3 't- -t )~ = I ?' ~ \ -t ~ "I ,:;. S­ ? +~~ ~ 5 , (r 2.­ 1 \ I ~ '-I ~ \ / ~ 41 ~~ ~ 0 o o b o ~ ) C. - \ 2 S­ r~(O :;:::;-­ lL.) 1 t 3~ ~ ~ ~ u ~ -r- 'v\J...- "'- -t "2­ r 0 () 0 ) 1) ~ 0 b 0 0 () 0 0 t l. ~ .r J 10 IV B t3 t3 .::;C, ~ IQ)~ t \;))= (') 0 ~ta- .3 0 6~~~,) !? -':> ~l 0 ~ ~ c- a 2J 6 ~ (~t) l\ () f-.J ts - Z- l ~ -""2-­ ~ (') ~ -\ r---­, l+ \ N Lbl ~ (\) -= b ~ \ '­ S 0 \r-~(. b(I) 6 1 ~~ \ ) 0 -1­ -I 0 ~ _' C '6~ 6.1l1r-) 1\ 0 f) - \ ~ G '" [!lJ. 3 I .. N CJ .\- r IS N -d:: / ~ ll) = tj., 3 0 \ (::) 2- S -;::. c· 1'1-) '<:) 0LJ1l1>/ -~ )13 - 1./3 x ~~n' '(J 2/3 -YJ -1 /3 -'I 0 " l-RX 'M \ V\..o,,\-z ~ L '11 7lr-J.~ Ig (? +l M.. "'- ,~ "J I . ~ y- =- t li , 'S) \...e..) ~...e..e... flVV~ (, l1\..) I~ l~) I \ , - v" ..J----t-7 tI'- \ I I I , ­ , .­ c..\?) M-' '" -l 0 lJ \ -t- ~- ~- ~\. t ~ - r- ~ \-t~ ~ L --+- ~ \ 'S"~ ~ , :.1 <a , - ~ d ~ -+- (f '-l ~ , ~ \' ~") J~~' ~4 ~ 0 o Q )\ ~ )