Solutions to exam in SF1811 Optimization, Jan 14, 2015
|
|
- Drusilla Powell
- 5 years ago
- Views:
Transcription
1 Solutons to exam n SF8 Optmzaton, Jan 4, O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O (a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable x j stands for the flow n the lnk from node to node j, and let c = ( c 3, c 4, c 23, c 24 ) T. Then the total cost for the flow s gven by c T x. The flow balance condtons n the nodes can be wrtten Ax = b, where 3 6 A = and b = 4 5 Fnally, the gven drectons of the lnks mply the constrants x. (It s recommended to remove the last row n A and the correspondng last component n b to get a system wthout any redundant equaton.).(b) The four dfferent spannng trees are shown n the followng fgure, together wh the unque lnk flows whch satsfy Ax = b O------O O------O O------O O O \ / \ / \ / 5\ / \ / 3\ / \/ -\ / \/ /\ \ / /\ 6/ \ \ / 4/ \ / \ \ / / \ O O O------O O------O O------O Spannng Spannng Spannng Spannng tree tree 2 tree 3 tree 4 The lnk flows n the frst spannng tree are calculated as follows: The only way to satsfy the supply constrant n node 2 s to let x 23 = 6. Then the only way to satsfy the demand constrant n node 3 s to let x 3 = 2. Then the only way to satsfy the supply constrant n node s to let x 4 = 5. Then the demand constrant n node 4 s also satsfed.
2 The lnk flows for the other spannng trees are calculated n a smlar way. We thus have the followng four basc solutons: x = ( 2, 5, 6, ) T, correspondng to spannng tree number, x = ( 4,,, 6) T, correspondng to spannng tree number 2, x = ( 3,,, 5) T, correspondng to spannng tree number 3, and x = (, 3, 4, 2) T, correspondng to spannng tree number 4. All these four solutons satsfy Ax = b, but only the last two satsfy x. Thus, the basc solutons correspondng to spannng trees 3 and 4 are feasble basc solutons, whle the basc solutons correspondng to spannng trees and 2 are nfeasble basc solutons..(c) For a gven feasble basc soluton, the smplex multplers y for the dfferent nodes are calculated from y 4 = and y y j = c j for all lnks (, j) n the correspondng spannng tree. For the feasble basc soluton correspondng to spannng tree 3, we get y 4 =, y 2 = y 4 + c 24 = c 24, y 3 = y 2 c 23 = c 24 c 23, y = y 3 + c 3 = c 24 c 23 + c 3. The reduced cost for the only non-basc varable s then gven by r 4 = c 4 y + y 4 = c 4 c 24 + c 23 c 3. Thus, f c 3 c 4 c 23 + c 24 < then r 4 >, and then x = (3,,, 5) T s the unque optmal soluton to the consdered problem. For the feasble basc soluton correspondng to spannng tree 4, we get y 4 =, y 2 = y 4 + c 24 = c 24, y = y 4 + c 4 = c 4, y 3 = y 2 c 23 = c 24 c 23, The reduced cost for the only non-basc varable s then gven by r 3 = c 3 y + y 3 = c 3 c 4 + c 24 c 23. Thus, f c 3 c 4 c 23 + c 24 > then r 3 >, and then x = (, 3, 4, 2) T s the unque optmal soluton to the consdered problem. If c 3 c 4 c 23 + c 24 = then both the above feasble basc solutons are optmal solutons, and then every convex combnaton of these two solutons,.e. x = t (3,,, 5) T + ( t) (, 3, 4, 2) T, where t [, ], s also an optmal soluton snce the constrants and the objectve functon are lnear. As an example, the followng three solutons (correspondng to t = /3, /2 and 2/3) are optmal solutons for the case that c 3 c 4 c 23 + c 24 = : x = (, 2, 3, 3) T, x = (5, 5, 25, 35) T and x = (2,, 2, 4) T. 2
3 2.(a) We have an LP problem on the standard form where A = 2 3 4, b = mnmera c T x då Ax = b, x, ( ) 8 and c 4 T = (4, 4, 2, 4, 4). The startng soluton should have the basc varables x and x 5, whch means that β = (, 5) and ν = (2, 3, 4) The correspondng basc matrx s A β =, whle A 4 ν =. 3 2 The values of the current basc varables are gven by x β = b, where the vector b s calculated from the system A β b = b,.e. ) ( ) ) ( ) 4 ( b 8 ( b =, wth the soluton b = =. 4 b2 4 b2 2 The vector y wth smplex multplers s obtaned by the system A T β y = c β,.e. ( ) ( ) ( ) ( ) 4 y 4 y =, wth the soluton y = =. 4 4 y 2 Then the reduced costs for the non-basc varables are obtaned from 2 3 r T ν = c T ν y T A ν = (4, 2, 4) (, ) = (, 2, ). 3 2 Snce r ν2 = r 3 = 2 s smallest, and <, we let x 3 become the new basc varable. Then we should calculate the vector ā 3 from the system A β ā 3 = a 3,.e. ) ( ) ) ( ) 4 (ā3 2 (ā3.5 =, wth the soluton ā = =..5 ā 23 The largest permtted value of the new basc varable x 3 s then gven by { } { } b t max = mn ā 3 > = mn ā 3.5, 2 =.5.5 = b. ā 3 Mnmzng ndex s =, whch mples that x β = x should no longer be a basc varable. Its place as basc varable s taken by x 3, so that β = (3, 5) and ν = (2,, 4) The correspondng basc matrx s A β =, whle A 2 ν =. 3 4 The values of the current basc varables are x β = b, where the vector b s calculated from the system A β b = b,.e. ) ( ) ) ( ) 2 4 ( b 8 ( b 2 =, wth the soluton b = =. 2 b2 4 b2 y 2 ā 23 3
4 The vector y wth smplex multplers s obtaned from the system A T β y = c β,.e. ( ) ( ) ( ) ( ) 2 2 y 2 y =, wth the soluton y = =. 4 4 y 2 Then the reduced costs for the non-basc varables are obtaned from 3 r T ν = c T ν y T A ν = (4, 4, 4) (, ) = (3, 4, ). 3 4 Snce r ν the current feasble basc soluton s optmal. Thus, x = (,, 2,, ) T s an optmal soluton, wth optmal value c T x = 8. y 2 2.(b) If the prmal problem s on the standard form mnmze c T x subject to Ax = b, x, the correspondng dual problem s: maxmze b T y subject to A T y c, whch becomes maxmze 8y + 4y 2 subject to 4y 2 4, y + 3y 2 4, 2y + 2y 2 2, 3y + y 2 4, 4y 4. Ths dual problem can be llustrated by drawng the constrants and some level lnes for the objectve functon n a coordnate system wth y and y 2 on the axes. (The fgure s omtted here.) It s well known that an optmal soluton to ths problem s gven by the vector y of smplex multplers for the optmal basc soluton n (a) above,.e. y = (, ) T. Alternatvely, ths can be seen from the fgure (whch s omtted here). Check: It s easy to verfy that y = (, ) T satsfes the dual constrants, wth dual objectve value 8y + 4y 2 = 8 = the optmal value of the prmal problem. Thus, y = (, ) T s an optmal soluton to the dual problem. 4
5 2.(c) If the second constrant n the prmal problem s removed, the correspondng dual problem becomes maxmze 8y subject to y 4, y 4, 2y 2, 3y 4, 4y 4. The optmal soluton to ths problem s clearly y =, wth the optmal value 8y = 8. But then the optmal value of the reduced prmal problem must also be = 8. Snce the optmal soluton x = (,, 2,, ) T from (a) above s feasble also to the reduced prmal problem, and stll has the objectve value c T x = 8, t follows that x = (,, 2,, ) T s an optmal soluton also to the reduced prmal problem! (But not a basc soluton. Two optmal basc solutons are now x = (,, 4,, ) T and x = (,,,, 2) T, wth objectve values = 8.) 2.(d) If the frst constrant n the prmal problem s removed, the correspondng dual problem becomes maxmze 4y subject to 4y 4, 3y 4, 2y 2, y 4, y 4. The optmal soluton to ths problem s clearly y =, wth the optmal value 4y = 4, and then the optmal value of the reduced prmal problem must also be = 4. But the optmal soluton x = (,, 2,, ) T from (a) above stll has the objectve value c T x = 8 > 4, so t can not be an optmal soluton to the reduced prmal problem! (Two optmal basc solutons are now x = (,, 2,, ) T and x = (,,,, ) T, wth objectve values = 4.) 5
6 3.(a) The objectve functon s f(x) = 2 xt Hx + c T x, wth H = 2, c =. LDL T -factorzaton of H gves H = LDL T = Snce there s a negatve dagonal element n D, the matrx H s not postve semdefnte, whch n turn mples that there s no optmal soluton to the problemen of mnmzng f(x) wthout constrants. (Wth e.g. d = (,, ) T, f(t d) = t 2 when t.) 3.(b) We now have a QP problem wth equalty constrants,.e. a problem of the form mnmze 2 xt Hx + c T x subject to Ax = b, wth H and c as above, A = [ ] and b =. The general soluton to Ax = b,.e. to x x 2 + x 3 =, s gven by x x 2 = + v + v 2, for arbtrary values on v and v 2, x 3 whch means that x = s a feasble soluton, and Z = s a matrx whos columns form a bass for the null space of A. After the varable change x = x+zv we should solve the system (Z T HZ)v = Z T (H x+c), provded that Z T HZ s at least postve semdefnte. [ We have that Z T 2 HZ = 2 4 The system (Z T HZ)v = Z T (H x + c) becomes ], whch s postve semdefnte (but not postve defnte). [ ]( ) ( ) 2 v =, 2 4 v 2 ( ) 2t wth the solutons ˆv(t) =, for arbtrary values on the real number t, whch mples t t that ˆx(t) = x + Zˆv(t) = 2t, for t IR, are the (nfnte number of) optmal solutons. t Note that f(ˆx(t)) = for all t IR. 6
7 3.(c) Agan, we have a problem on the form: mnmze 2 xt Hx + c T x subject to Ax = b, ( ) wth H and c as above, A =, and b =. x The general soluton to Ax = b s now x 2 = v, for v IR, whch mples that x 3 x = s a feasble soluton, and z = form a bass for the null space of A. After the varable change x = x + zv, we should solve the system (z T Hz)v = z T (H x + c), provded that z T Hz s at least. We have that z T Hz = 6 >, so the system (z T Hz)v = z T (H x + c) becomes 6v =, wth the unque soluton ˆv =, so that ˆx = x + z ˆv = s the unque optmal soluton. 7
8 4.(a) The Lagrange functon for the consdered problem s gven by L(x, y) = 2 (x q)t (x q) + y T (b Ax), wth x IR n and y IR m. The Lagrange relaxed problem PR y s defned, for a gven y, as the problem of mnmzng L(x, y) wth respect to x IR n. Snce L(x, y) = 2 xt I x (A T y + q) T x + b T y + 2 qt q, and the unt matrx I s postve defnte, the unque optmal soluton to PR y s gven by x(y) = A T y + q. Then the dual objectve functon becomes ϕ(y) = L( x(y), y) = 2 (AT y + q) T (A T y + q) + b T y + 2 qt q = = 2 yt AA T y + (b Aq) T y. [ 4.(b) From now on, A = ( 4 6 Then AA T = and b Aq = 4 3 ] ( 6, b = 3 ) ( ) 2 = 4 ) ( 4 and q = (, 2, 2, ) T. so that the dual objectve functon becomes ϕ(y) = 2y 2 2y y y 2. Alternatve calculaton of the dual functon, wthout usng the results from (a): The consdered problem s: mnmze f(x) subject to g (x) and g 2 (x), where f(x) = 2 (x ) (x 2 2) (x 3 2) (x 4 ) 2, g (x) = 6 x x 2 + x 3 x 4 and g 2 (x) = 3 x x 2 x 3 + x 4. The Lagrange functon then becomes: L(x, y) = 2 (x ) (x 2 2) (x 3 2) (x 4 ) 2 + y (6 x x 2 + x 3 x 4 ) + y 2 (3 x x 2 x 3 + x 4 ) = = 2 (x ) 2 (y +y 2 ) x + 2 (x 2 2) 2 (y +y 2 ) x (x 3 2) 2 (y 2 y ) x (x 4 ) 2 (y y 2 ) x 4 + 6y + 3y 2. Mnmzng ths wth respect to x gves: x (y) = +y +y 2, x 2 (y) = 2+y +y 2, x 3 (y) = 2+y 2 y, x 4 (y) = +y y 2, so that x(y) = (+y +y 2, 2+y +y 2, 2+y 2 y, +y y 2 ) T, and then the dual functon becomes ϕ(y) = L( x(y), y) = 2 ( x(y) ) 2 (y +y 2 ) x(y) + 2 ( x(y) 2 2) 2 (y +y 2 ) x(y) ( x(y) 3 2) 2 (y 2 y ) x(y) ( x(y) 4 ) 2 (y y 2 ) x(y) 4 + 6y + 3y 2 = 2 (y +y 2 ) 2 (y +y 2 ) (y +y 2 ) (y +y 2 ) 2 2(y +y 2 ) (y +y 2 ) (y 2 y 2 ) 2 2(y 2 y ) (y 2 y ) (y y 2 ) 2 (y y 2 ) (y y 2 ) 2 + 6y + 3y 2 = = 2y 2 2y y y 2, as above. ), 8
9 The dual problem then becomes: D: maxmze ϕ(y) = 2y 2 2y y y 2 subject to y and y 2, whch decomposes nto the two separate problems D : maxmze 2y 2 + 4y subject to y, and D 2 : maxmze 2y 2 2 y 2 subject to y 2. Clearly, the optmal soluton to the frst problem s ŷ =, whle the optmal soluton to the second problem s ŷ 2 =. Thus, ŷ = (, ) T s the unque optmal soluton to D, wth ϕ(ŷ) = 2. 4.(c) Let ˆx = x(ŷ) = (+ŷ +ŷ 2, 2+ŷ +ŷ 2, 2+ŷ 2 ŷ, +ŷ ŷ 2 ) T = (2, 3,, 2) T. Then Aˆx b = (6, 4) T (6, 3) T, so ˆx s a feasble soluton to the prmal problem. Further, the prmal objectve value s f(ˆx) = 2 (ˆx q)t (ˆx q) = 2 (,,, )T 2 = 2. Snce ˆx s feasble to P and f(ˆx) = ϕ(ŷ), we conclude that ˆx s an optmal soluton to P. x 4.(d) Snce f(x) T = x 2 2 x 3 2, g (x) T =, g 2(x) T =, x 4 the KKT condtons become: x (KKT-): x 2 2 x y + y 2 = x 4 (KKT-2): 6 x x 2 + x 3 x 4 and 3 x x 2 x 3 + x 4. (KKT-3): y and y 2. (KKT-4): y (6 x x 2 + x 3 x 4 ) = and y 2 (3 x x 2 x 3 + x 4 ) =.. Wth x = ˆx = (2, 3,, 2) T and y = ŷ = (, ) T, we get that 2 (KKT-): =. OK! 2 (KKT-2): (KKT-3): (KKT-4): = and =. OK! and. OK! = and ( ) =. OK! Thus, ˆx and ŷ satsfy the KKT condtons, and snce the consdered problem s a convex problem, we can conclude (agan) that ˆx s a global optmal soluton. 9
10 5.(a) Change notaton and let the varable vector be called x,.e. x = (x, x 2, x 3 ) T = (x, y, r) T. m Then f(x) = 2 h (x) 2 = 2 h(x)t h(x), where = h (x) = (x a ) 2 + (x 2 b ) 2 x 3 and h(x) = (h (x),..., h m (x)) T. The gradent of h s gven by ( h (x) = x a (x a ) 2 + (x 2 b ) 2, x 2 b (x a ) 2 + (x 2 b ) 2, ), and h(x) s the m 3 matrx wth these gradents as rows. Wth the gven data, we get that f(x) = 2 (h (x) 2 + h 2 (x) 2 + h 3 (x) 2 + h 4 (x) 2 ), where h (x) = (x 5) 2 + x 2 2 x 3, h 2 (x) = x 2 + (x 2 6) 2 x 3, h 3 (x) = (x + 4) 2 + x 2 2 x 3, h 4 (x) = x 2 + (x 2 + 5) 2 x 3. The startng pont should be x () = (,, 5) T. and then h(x () ) =, f(x() ) = and h(x () ) = In Gauss-Newtons method, we should solve h(x () ) T h(x () )d = h(x () ) T h(x () ), 2 d.5 whch becomes 2 d 2 =, wth the soluton d () =.5. 4 d 3 We try t =, so that x (2) = x () + t d () = x () + d () = (.5,.5, 5) T. Then h(x (2) ) = ( , , , ) T = 2 ( 82, 22, 82, 22 ) T 2 (,,, )T, so that f(x (2) ) = 2 h(x(2) ) T h(x (2) ) 8 ( ) = 2 < = f(x() ), Thus, the choce t = s accepted, and x (2) = (.5,.5, 5) T s the next teraton pont..
11 5.(b) Let (x, x 2 ) = the locaton of the (common) center of C and C 2, z = the square of the radus of the small crcle C, and z 2 = the square of the radus of the large crcle C 2. Then the problem can be formulated as follows n the varables are x, x 2, z and z 2 : mnmze πz 2 πz subject to (x a ) 2 + (x 2 b ) 2 z, =,..., m, (x a ) 2 + (x 2 b ) 2 z 2, =,..., m. 5.(c) For each gven pont (x, x 2 ) T IR 2, the correspondng optmal values of z and z 2 n the above problem are clearly ẑ (x, x 2 ) = mn{(x a ) 2 + (x 2 b ) 2 } and ẑ (x, x 2 ) = max{(x a ) 2 + (x 2 b ) 2 }. Thus, the above problem can be wrtten mnmze π max{(x a ) 2 + (x 2 b ) 2 } π mn{(x a ) 2 + (x 2 b ) 2 }, whch s a problem n just the two varables x and x 2. (However, the objectve functon s not dfferentable, so ths s not a correct answer to 5.(b).) But mn{(x a ) 2 + (x 2 b ) 2 } = mn { 2a x + a 2 2b x 2 + b 2 }, and x 2 + x mn max {x 2 2a x + a 2 + x 2 2 2b x 2 + b 2 } = {(x a ) 2 + (x 2 b ) 2 } = max{x 2 2a x + a 2 + x 2 2 2b x 2 + b 2 } = x 2 + x max { 2a x + a 2 2b x 2 + b 2 }. Therefore, max{(x a ) 2 + (x 2 b ) 2 } mn{(x a ) 2 + (x 2 b ) 2 } = max{ 2a x + a 2 2b x 2 + b 2 } mn{ 2a x + a 2 2b x 2 + b 2 }, so the above problem can be wrtten mnmze π max{ 2a x + a 2 2b x 2 + b 2 } π mn{ 2a x + a 2 2b x 2 + b 2 }, whch may equvalently be wrtten mnmze πw 2 πw subject to w + 2a x + 2b x 2 a 2 + b 2, =,..., m, w 2 + 2a x + 2b x 2 a 2 + b 2, =,..., m, whch s an LP problem n the varables x, x 2, w and w 2.
MMA 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 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 informationSolutions HW #2. minimize. Ax = b. Give the dual problem, and make the implicit equality constraints explicit. Solution.
Solutons HW #2 Dual of general LP. Fnd the dual functon of the LP mnmze subject to c T x Gx h Ax = b. Gve the dual problem, and make the mplct equalty constrants explct. Soluton. 1. The Lagrangan s L(x,
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 informationSELECTED SOLUTIONS, SECTION (Weak duality) Prove that the primal and dual values p and d defined by equations (4.3.2) and (4.3.3) satisfy p d.
SELECTED SOLUTIONS, SECTION 4.3 1. Weak dualty Prove that the prmal and dual values p and d defned by equatons 4.3. and 4.3.3 satsfy p d. We consder an optmzaton problem of the form The Lagrangan for ths
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationCollege of Computer & Information Science Fall 2009 Northeastern University 20 October 2009
College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:
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 informationLagrange Multipliers Kernel Trick
Lagrange Multplers Kernel Trck Ncholas Ruozz Unversty of Texas at Dallas Based roughly on the sldes of Davd Sontag General Optmzaton A mathematcal detour, we ll come back to SVMs soon! subject to: f x
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 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 informationCOS 521: Advanced Algorithms Game Theory and Linear Programming
COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton
More informationSingular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
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 informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpenCourseWare http://ocw.mt.edu 6.854J / 18.415J Advanced Algorthms Fall 2008 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 18.415/6.854 Advanced Algorthms
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 informationConvex Optimization. Optimality conditions. (EE227BT: UC Berkeley) Lecture 9 (Optimality; Conic duality) 9/25/14. Laurent El Ghaoui.
Convex Optmzaton (EE227BT: UC Berkeley) Lecture 9 (Optmalty; Conc dualty) 9/25/14 Laurent El Ghaou Organsatonal Mdterm: 10/7/14 (1.5 hours, n class, double-sded cheat sheet allowed) Project: Intal proposal
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationAppendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems
More informationWhich Separator? Spring 1
Whch Separator? 6.034 - Sprng 1 Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng 3 Margn of a pont " # y (w $ + b) proportonal
More informationNP-Completeness : Proofs
NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem
More information10-801: Advanced Optimization and Randomized Methods Lecture 2: Convex functions (Jan 15, 2014)
0-80: Advanced Optmzaton and Randomzed Methods Lecture : Convex functons (Jan 5, 04) Lecturer: Suvrt Sra Addr: Carnege Mellon Unversty, Sprng 04 Scrbes: Avnava Dubey, Ahmed Hefny Dsclamer: These notes
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More information14 Lagrange Multipliers
Lagrange Multplers 14 Lagrange Multplers The Method of Lagrange Multplers s a powerful technque for constraned optmzaton. Whle t has applcatons far beyond machne learnng t was orgnally developed to solve
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 informationLecture 20: November 7
0-725/36-725: Convex Optmzaton Fall 205 Lecturer: Ryan Tbshran Lecture 20: November 7 Scrbes: Varsha Chnnaobreddy, Joon Sk Km, Lngyao Zhang Note: LaTeX template courtesy of UC Berkeley EECS dept. Dsclamer:
More informationProseminar Optimierung II. Victor A. Kovtunenko SS 2012/2013: LV
Prosemnar Optmerung II Vctor A. Kovtunenko Insttute for Mathematcs and Scentfc Computng, Karl-Franzens Unversty of Graz, Henrchstr. 36, 8010 Graz, Austra; Lavrent ev Insttute of Hydrodynamcs, Sberan Dvson
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
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 informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
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 informationOPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming
OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationLecture 10 Support Vector Machines. Oct
Lecture 10 Support Vector Machnes Oct - 20-2008 Lnear Separators Whch of the lnear separators s optmal? Concept of Margn Recall that n Perceptron, we learned that the convergence rate of the Perceptron
More informationFeature Selection: Part 1
CSE 546: Machne Learnng Lecture 5 Feature Selecton: Part 1 Instructor: Sham Kakade 1 Regresson n the hgh dmensonal settng How do we learn when the number of features d s greater than the sample sze n?
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationThe Second Anti-Mathima on Game Theory
The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player
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 informationLecture 17: Lee-Sidford Barrier
CSE 599: Interplay between Convex Optmzaton and Geometry Wnter 2018 Lecturer: Yn Tat Lee Lecture 17: Lee-Sdford Barrer Dsclamer: Please tell me any mstake you notced. In ths lecture, we talk about the
More informationMaximal Margin Classifier
CS81B/Stat41B: Advanced Topcs n Learnng & Decson Makng Mamal Margn Classfer Lecturer: Mchael Jordan Scrbes: Jana van Greunen Corrected verson - /1/004 1 References/Recommended Readng 1.1 Webstes www.kernel-machnes.org
More informatione - c o m p a n i o n
OPERATIONS RESEARCH http://dxdoorg/0287/opre007ec e - c o m p a n o n ONLY AVAILABLE IN ELECTRONIC FORM 202 INFORMS Electronc Companon Generalzed Quantty Competton for Multple Products and Loss of Effcency
More informationFIRST AND SECOND ORDER NECESSARY OPTIMALITY CONDITIONS FOR DISCRETE OPTIMAL CONTROL PROBLEMS
Yugoslav Journal of Operatons Research 6 (6), umber, 53-6 FIRST D SECOD ORDER ECESSRY OPTIMLITY CODITIOS FOR DISCRETE OPTIML COTROL PROBLEMS Boban MRIKOVIĆ Faculty of Mnng and Geology, Unversty of Belgrade
More informationSupport Vector Machines CS434
Support Vector Machnes CS434 Lnear Separators Many lnear separators exst that perfectly classfy all tranng examples Whch of the lnear separators s the best? + + + + + + + + + Intuton of Margn Consder ponts
More informationSupplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso
Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed
More informationLECTURE 9 CANONICAL CORRELATION ANALYSIS
LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of
More informationEconomics 101. Lecture 4 - Equilibrium and Efficiency
Economcs 0 Lecture 4 - Equlbrum and Effcency Intro As dscussed n the prevous lecture, we wll now move from an envronment where we looed at consumers mang decsons n solaton to analyzng economes full of
More informationA 2D Bounded Linear Program (H,c) 2D Linear Programming
A 2D Bounded Lnear Program (H,c) h 3 v h 8 h 5 c h 4 h h 6 h 7 h 2 2D Lnear Programmng C s a polygonal regon, the ntersecton of n halfplanes. (H, c) s nfeasble, as C s empty. Feasble regon C s unbounded
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 informationArmy Ants Tunneling for Classical Simulations
Electronc Supplementary Materal (ESI) for Chemcal Scence. Ths journal s The Royal Socety of Chemstry 2014 electronc supplementary nformaton (ESI) for Chemcal Scence Army Ants Tunnelng for Classcal Smulatons
More informationMath1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
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 information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More information15 Lagrange Multipliers
15 The Method of s a powerful technque for constraned optmzaton. Whle t has applcatons far beyond machne learnng t was orgnally developed to solve physcs equatons), t s used for several ey dervatons n
More informationTopic 5: Non-Linear Regression
Topc 5: Non-Lnear Regresson The models we ve worked wth so far have been lnear n the parameters. They ve been of the form: y = Xβ + ε Many models based on economc theory are actually non-lnear n the parameters.
More informationThe General Nonlinear Constrained Optimization Problem
St back, relax, and enjoy the rde of your lfe as we explore the condtons that enable us to clmb to the top of a concave functon or descend to the bottom of a convex functon whle constraned wthn a closed
More informationAnd the full matrix is the concatenation of the two:
Appendx A--A Step n the Dual Newton Method Alesandar Donev, 1/17/00 Ths Maple worsheet llustrates the basc step n the Dual Newton Method for convex networ optmzaton when the arcs have a superconductor-le
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More informationThe lower and upper bounds on Perron root of nonnegative irreducible matrices
Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College
More informationLecture 3: Dual problems and Kernels
Lecture 3: Dual problems and Kernels C4B Machne Learnng Hlary 211 A. Zsserman Prmal and dual forms Lnear separablty revsted Feature mappng Kernels for SVMs Kernel trck requrements radal bass functons SVM
More informationThe exam is closed book, closed notes except your one-page cheat sheet.
CS 89 Fall 206 Introducton to Machne Learnng Fnal Do not open the exam before you are nstructed to do so The exam s closed book, closed notes except your one-page cheat sheet Usage of electronc devces
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationProblem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?
Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2
More informationSTAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16
STAT 39: MATHEMATICAL COMPUTATIONS I FALL 218 LECTURE 16 1 why teratve methods f we have a lnear system Ax = b where A s very, very large but s ether sparse or structured (eg, banded, Toepltz, banded plus
More informationCalculation of time complexity (3%)
Problem 1. (30%) Calculaton of tme complexty (3%) Gven n ctes, usng exhaust search to see every result takes O(n!). Calculaton of tme needed to solve the problem (2%) 40 ctes:40! dfferent tours 40 add
More informationECE559VV Project Report
ECE559VV Project Report (Supplementary Notes Loc Xuan Bu I. MAX SUM-RATE SCHEDULING: THE UPLINK CASE We have seen (n the presentaton that, for downlnk (broadcast channels, the strategy maxmzng the sum-rate
More informationPerfect Competition and the Nash Bargaining Solution
Perfect Competton and the Nash Barganng Soluton Renhard John Department of Economcs Unversty of Bonn Adenauerallee 24-42 53113 Bonn, Germany emal: rohn@un-bonn.de May 2005 Abstract For a lnear exchange
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
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 informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
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 informationfind (x): given element x, return the canonical element of the set containing x;
COS 43 Sprng, 009 Dsjont Set Unon Problem: Mantan a collecton of dsjont sets. Two operatons: fnd the set contanng a gven element; unte two sets nto one (destructvely). Approach: Canoncal element method:
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationVARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES
VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES BÂRZĂ, Slvu Faculty of Mathematcs-Informatcs Spru Haret Unversty barza_slvu@yahoo.com Abstract Ths paper wants to contnue
More informationChapter Newton s Method
Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve
More informationError Bars in both X and Y
Error Bars n both X and Y Wrong ways to ft a lne : 1. y(x) a x +b (σ x 0). x(y) c y + d (σ y 0) 3. splt dfference between 1 and. Example: Prmordal He abundance: Extrapolate ft lne to [ O / H ] 0. [ He
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 informationMixed Taxation and Production Efficiency
Floran Scheuer 2/23/2016 Mxed Taxaton and Producton Effcency 1 Overvew 1. Unform commodty taxaton under non-lnear ncome taxaton Atknson-Stgltz (JPubE 1976) Theorem Applcaton to captal taxaton 2. Unform
More informationCalculus of Variations Basics
Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationImportant Instructions to the Examiners:
Summer 0 Examnaton Subject & Code: asc Maths (70) Model Answer Page No: / Important Instructons to the Examners: ) The Answers should be examned by key words and not as word-to-word as gven n the model
More informationDynamic Systems on Graphs
Prepared by F.L. Lews Updated: Saturday, February 06, 200 Dynamc Systems on Graphs Control Graphs and Consensus A network s a set of nodes that collaborates to acheve what each cannot acheve alone. A network,
More informationCHAPTER 7 CONSTRAINED OPTIMIZATION 2: SQP AND GRG
Chapter 7: Constraned Optmzaton CHAPER 7 CONSRAINED OPIMIZAION : SQP AND GRG Introducton In the prevous chapter we eamned the necessary and suffcent condtons for a constraned optmum. We dd not, however,
More information8/25/17. Data Modeling. Data Modeling. Data Modeling. Patrice Koehl Department of Biological Sciences National University of Singapore
8/5/17 Data Modelng Patrce Koehl Department of Bologcal Scences atonal Unversty of Sngapore http://www.cs.ucdavs.edu/~koehl/teachng/bl59 koehl@cs.ucdavs.edu Data Modelng Ø Data Modelng: least squares Ø
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 informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationMACHINE APPLIED MACHINE LEARNING LEARNING. Gaussian Mixture Regression
11 MACHINE APPLIED MACHINE LEARNING LEARNING MACHINE LEARNING Gaussan Mture Regresson 22 MACHINE APPLIED MACHINE LEARNING LEARNING Bref summary of last week s lecture 33 MACHINE APPLIED MACHINE LEARNING
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 information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More informatione i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
More information