Optimization Methods MIT 2.098/6.255/ Final exam
|
|
- Nathaniel Alexander
- 5 years ago
- Views:
Transcription
1 Optimizatio Methods MIT 2.098/6.255/ Fial exam Date Give: December 19th, 2006 P1. [30 pts] Classify the followig statemets as true or false. All aswers must be well-justified, either through a short explaatio, or a couterexample. Uless stated otherwise, all LP problems are i stadard form. (a) If there is a uique primal optimal solutio to a liear programmig problem, the the reduced costs of all the obasic variables are strictly positive. (b) For a etwork flow problem with capacity costraits, there always exists a optimal solutio that is tree-structured. (c) Whe miimizig a covex fuctio over a covex set, the optimal solutio is always o the boudary of the set. (d) For a covex optimizatio problem with costraits, if a feasible poit satisfies the KKT coditios the it is a global optimum. (e) For a oliear optimizatio problem, if Newto s method coverges, the it coverges to a local miimum. (f) (g) For a LP i stadard form, if c 0 the the primal is either bouded or ifeasible. O very degeerate LP problems, the simplex method performs better tha iterior poit methods. (h) The primal iterates x k geerated by the affie scalig algorithm are always i the iterior of the primal feasible set. (i) The feasible set of a semidefiite programmig problem is always covex. (j) For a quadratic fuctio f(x) = x T Ax + b T x + c, the covergece rate of Newto s method depeds o the coditio umber of the matrix A. Solutio: (a) FALSE. This is true oly if the uique optimal solutio is odegeerate. (b) FALSE. I the capacitated case, optimal solutios do ot ecessarily have to be trees, a simple couterexample is a etwork with odes {A, B, C, D} ad edges A B, A C, B D, C D of uit capacity. (c) FALSE. Solutios ca be i the iterior. As a example, cosider miimizig x 2 o [ 1, 1]. (d) TRUE. This is prove i the book ad i the lecture otes. (e) FALSE. Newto s method ca coverge to global maxima. (f) TRUE. If c is oegative, the if the problem is feasible we have c T x 0, ad thus is it bouded. (g) FALSE. For degeerate problems, iterior poit methods are a much better choice. (h) TRUE. By costructio, the primal iterates i the affie scalig method (with β < 1) are always strictly feasible. (i) TRUE. The feasible set of a SDP problem is covex, sice it is the itersectio of two covex sets (a affie subspace ad the coe of PSD matrices). (j) FALSE. For quadratic fuctios, Newto s method coverges exactly i oe iteratio. 1
2 P2. [25 pts] You are plaig o havig access to a car for the ext N years, where N is a fixed umber. The price of a ew car is P dollars. For reliability reasos, you will oly ow relatively ew cars, at most m years old. The yearly cost of repairig ad maitaiig a car durig its kth year is r k, ad it satisfies r 1 < r 2 < < r m (i.e., it icreases over time). At the ed of ay give year, you have the optio of exchagig your k-year old car for a ew oe, with the correspodig trade-i value t k of your old car satisfyig t 1 > t 2 > > t m (i.e., depreciatig over time). You wat to fid the most ecoomical sequece of buys ad trade-is, i.e., wat to miimize the total cost over the N-year period. This icludes all the moey spet, either i buyig or repairig (otice that you ca sell your car at the ed of the N years). (a) Propose a shortest path (or etwork flow) formulatio for this problem. (b) Propose a dyamic programmig formulatio for this problem. Express clearly what are the state ad decisio variables, ad the correspodig iteratio. (c) Use your DP formulatio to solve the problem for the followig data: N = 5, m = 3, P = 20000, the repairig costs ad the trade-i values r 1 = 1200, r 2 = 1600, r 3 = 2400, t 1 = 16000, t 2 = 12000, t 3 = What is the optimal sequece of actios? Is it uique? Solutio: (a) Defie a etwork whose odes are o a N m grid with two additioal odes: a source ode s ad a sik ode t. A ode o the gride is idexed by (time idex) t = 1,..., N ad (car age) a = 1,..., m. From ode (t, a), there is a directed edge to ode (t + 1, a + 1) with cost r a if t < N ad a < m (car is maitaied for oe year) to ode (t + 1, 1) with cost P t k + r 1 if t < N (car is traded i) to a sik ode t if t = N with cost t a. Node s is coected to the grid ode (1, 1) with cost P + r 1. Node s a supply of 1 ad ode t a demad of 1. A mi-cost flow solutio will correspod to a optimal purchase/maiteace pla for the car over the time horizo N (b) Let the time idex t ru from 1 to N. Defie the state a at time t as the age of the car at the ed of the curret period. Let V t (a) be the expected cost give that the car is a-year old at the ed of the time period t. At time t < N, if a = m, the car has to be exchaged ad maitaied for oe year with cost P t m + r 1, whereas there are two available optios i state a {1,..., m 1}: maitai the car, with a cost of r a, leadig to (a + 1)-year old car at the ext period trade-i the car for a ew oe ad maitai it for a year, with a cost P t a + r 1, leadig to a oe-year old car i the ext period. At time t = N, the car is sold for its value t a. Bellma s equatios for this problem are V N (a) = t a, V t (m) = P t m + r 1 + V t+1 (1), V t (a) = mi (r a + V t+1 (a + 1), P t a + r 1 + V t+1 (1)), a = 1,..., m, t < N a = 1,..., m 1, t < N. 2
3 (c) Solvig Bellma s recursio yields the optimal cost of $26,
4 P3. [20 pts] Cosider the followig trasshipmet problem: The supply odes are A, B, the demad odes are D, E, ad the trasshipmet ode is ode C. There are four ukows a, b, c ad d. The supply/demad amouts i the differet odes are: A : a, B : 400, D : b, E : 200, where as usual a positive amout idicates supply ad a egative amout idicates demad. We are iterested i fidig a optimal (miimum cost) trasshipmet pla. (a) State coditios o a, b, c, d such that the above problem is feasible. (b) Cosider the spaig tree give by the edges {(A, D), (B, C), (C, D), (C, E)}. State coditios o a, b, c, d for which the spaig tree solutio will be feasible. (c) State coditios o a, b, c, d for which the spaig tree solutio will be optimal. (d) State coditios o a, b, c, d for which there will be multiple solutios, icludig the spaig tree solutio idicated above. Solutio: (a) For the problem to be solvable, supply ad demad must balace, givig the ecessary coditio a = b, or equivaletly, b a = 200. (b) If the solutio has the structure of the idicated spaig tree, the the followig relatios must hold: a = b. Here we ca calculate all flows o this spaig tree ad the coditio agai is the flow balace coditio. (c) The optimality coditio for the spaig tree solutio is that all reduced costs of o-basic arcs are o-egative. I order to calculate the reduced costs, we eed to calculate ode potetials. Without loss of geerality, set p C = 0. Kowig the fact that c ij = c ij (p i p j ) = 0 for all basic arcs (i, j), we obtai p B = 5, p D = 3, p A = 1, ad p E = 2. The reduced costs for o-basic arcs the ca be calculated as follows: c BA = c 6, c CA = d 1, c BE = 2, c ED = 10 Thus i additio to the feasibility coditio b = a + 200, we obtai the optimality coditio for the idicated spaig tree solutio: { c 6 0. d 1 0 (d) We eed to have reduced costs of some o-basic arcs to be zero i order to have multiple optimal solutios, icludig this spaig tree solutio. It meas either c = 6 or d = 1 (i additio to the feasibility ad optimality coditios metioed previously i (c)). We ca see that there is a cost-equivalet path from B to D via A as compared to the path B C D i both cases. Thus, other optimal solutios ca be costructed by reroutig flows from B to D via A if { either c = 6 or d { = 1. The fial coditios for multiple optimal solutios are: a = b, ad c 6 = 0 c 6 0 or d 1 0 d 1 = 0. 4
5 P4. [25 pts] Cosider a set of poits {(x 1, y 1 ),..., (x, y )} i the plae. We wat to fid a poit (x, y) such that the sum of the Euclidea distaces from this poit to all the other poits is miimized. (a) Give a oliear optimizatio formulatio of this problem. (b) Is the objective fuctio differetiable? Is this a covex optimizatio problem? (c) Write the correspodig optimality coditios. Give a geometric iterpretatio of this coditio. (d) Are the optimality coditios ecessary? Sufficiet? State clearly your assumptios. (e) Provide a semidefiite programmig formulatio of this problem. All aswers ad explaatios must be fully justified. Solutio: (a) A simple formulatio as a ucostraied oliear optimizatio problem is the followig: mi (x x i ) 2 + (y y i ) 2. x,y (b) The objective fuctio is differetiable everywhere, except at the poits where (x, y) is equal to oe of the (x i, y i ). The objective fuctio is covex, sice it is a sum of covex fuctios. (c) The optimality coditios are obtaied by settig the gradiet equal to zero (we assume that the miimum occurs at a differetiable poit). f x x i = = 0 x (x, y) (x i, y i ) f y y i = = 0 y (x, y) (x i, y i ) This coditio ca be iterpreted as requirig the sum of the ormalized vectors from the poit (x, y) to the (x i, y i ) to be equal to zero. For istace, i the case = 2, the ay poit i the lie segmet betwee the two give poits will be optimal. Similarly, for = 3, the optimality coditio implies that the agles betwee the vectors from (x, y) to the other poits are all equal. (d) The optimality coditios are sufficiet, because the problem is covex. They are ecessary if the miimum occurs at a differetiable poit. (e) A semidefiite formulatio of the problem ca be easily obtaied if we itroduce slack variables d i ad formulate the problem as: mi d i s.t. (x x i ) 2 + (y y i ) 2 d 2 i i = 1,...,. The costraits are equivalet to: [ ] T [ ] [ ] d 2 x x i 1 0 x x i i 0 y y i 0 1 y y i Dividig by d i ad usig Schur complemets, we ca rewrite this as: d i x x i y y i mi d i s.t. x x i d i 0 0, i = 1,...,. which is a SDP formulatio. y y i 0 d i 5
6 1 MIT OpeCourseWare J / 6.255J Optimizatio Methods Fall 2009 For iformatio about citig these materials or our Terms of Use, visit: -
Differentiable Convex Functions
Differetiable Covex Fuctios The followig picture motivates Theorem 11. f ( x) f ( x) f '( x)( x x) ˆx x 1 Theorem 11 : Let f : R R be differetiable. The, f is covex o the covex set C R if, ad oly if for
More informationDefinitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.
Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,
More informationIntroduction to Optimization Techniques. How to Solve Equations
Itroductio to Optimizatio Techiques How to Solve Equatios Iterative Methods of Optimizatio Iterative methods of optimizatio Solutio of the oliear equatios resultig form a optimizatio problem is usually
More informationLinear Programming and the Simplex Method
Liear Programmig ad the Simplex ethod Abstract This article is a itroductio to Liear Programmig ad usig Simplex method for solvig LP problems i primal form. What is Liear Programmig? Liear Programmig is
More information15.081J/6.251J Introduction to Mathematical Programming. Lecture 21: Primal Barrier Interior Point Algorithm
508J/65J Itroductio to Mathematical Programmig Lecture : Primal Barrier Iterior Poit Algorithm Outlie Barrier Methods Slide The Cetral Path 3 Approximatig the Cetral Path 4 The Primal Barrier Algorithm
More informationInteger Linear Programming
Iteger Liear Programmig Itroductio Iteger L P problem (P) Mi = s. t. a = b i =,, m = i i 0, iteger =,, c Eemple Mi z = 5 s. t. + 0 0, 0, iteger F(P) = feasible domai of P Itroductio Iteger L P problem
More informationSolutions for the Exam 9 January 2012
Mastermath ad LNMB Course: Discrete Optimizatio Solutios for the Exam 9 Jauary 2012 Utrecht Uiversity, Educatorium, 15:15 18:15 The examiatio lasts 3 hours. Gradig will be doe before Jauary 23, 2012. Studets
More informationIP Reference guide for integer programming formulations.
IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more
More informationLINEAR PROGRAMMING II
LINEAR PROGRAMMING II Patrik Forssé Office: 2404 Phoe: 08/47 29 66 E-mail: patrik@tdb.uu.se HE LINEAR PROGRAMMING PROBLEM (LP) A LP-problem ca be formulated as: mi c subject a + am + g + g p + + + c to
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More information10-701/ Machine Learning Mid-term Exam Solution
0-70/5-78 Machie Learig Mid-term Exam Solutio Your Name: Your Adrew ID: True or False (Give oe setece explaatio) (20%). (F) For a cotiuous radom variable x ad its probability distributio fuctio p(x), it
More information1 Duality revisited. AM 221: Advanced Optimization Spring 2016
AM 22: Advaced Optimizatio Sprig 206 Prof. Yaro Siger Sectio 7 Wedesday, Mar. 9th Duality revisited I this sectio, we will give a slightly differet perspective o duality. optimizatio program: f(x) x R
More informationIntroduction to Machine Learning DIS10
CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig
More informationThe Method of Least Squares. To understand least squares fitting of data.
The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More informationMathematical Foundations -1- Sets and Sequences. Sets and Sequences
Mathematical Foudatios -1- Sets ad Sequeces Sets ad Sequeces Methods of proof 2 Sets ad vectors 13 Plaes ad hyperplaes 18 Liearly idepedet vectors, vector spaces 2 Covex combiatios of vectors 21 eighborhoods,
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationBoosting. Professor Ameet Talwalkar. Professor Ameet Talwalkar CS260 Machine Learning Algorithms March 1, / 32
Boostig Professor Ameet Talwalkar Professor Ameet Talwalkar CS260 Machie Learig Algorithms March 1, 2017 1 / 32 Outlie 1 Admiistratio 2 Review of last lecture 3 Boostig Professor Ameet Talwalkar CS260
More informationsubject to A 1 x + A 2 y b x j 0, j = 1,,n 1 y j = 0 or 1, j = 1,,n 2
Additioal Brach ad Boud Algorithms 0-1 Mixed-Iteger Liear Programmig The brach ad boud algorithm described i the previous sectios ca be used to solve virtually all optimizatio problems cotaiig iteger variables,
More informationMath 116 Practice for Exam 3
Math 6 Practice for Exam Geerated October 0, 207 Name: SOLUTIONS Istructor: Sectio Number:. This exam has 7 questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur
More informationOptimization Methods: Linear Programming Applications Assignment Problem 1. Module 4 Lecture Notes 3. Assignment Problem
Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem Itroductio Module 4 Lecture Notes 3 Assigmet Problem I the previous lecture, we discussed about oe of the bech mark problems called trasportatio
More informationThe Simplex algorithm: Introductory example. The Simplex algorithm: Introductory example (2)
Discrete Mathematics for Bioiformatics WS 07/08, G. W. Klau, 23. Oktober 2007, 12:21 1 The Simplex algorithm: Itroductory example The followig itroductio to the Simplex algorithm is from the book Liear
More informationHomework Set #3 - Solutions
EE 15 - Applicatios of Covex Optimizatio i Sigal Processig ad Commuicatios Dr. Adre Tkaceko JPL Third Term 11-1 Homework Set #3 - Solutios 1. a) Note that x is closer to x tha to x l i the Euclidea orm
More informationBrief Review of Functions of Several Variables
Brief Review of Fuctios of Several Variables Differetiatio Differetiatio Recall, a fuctio f : R R is differetiable at x R if ( ) ( ) lim f x f x 0 exists df ( x) Whe this limit exists we call it or f(
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationA New Solution Method for the Finite-Horizon Discrete-Time EOQ Problem
This is the Pre-Published Versio. A New Solutio Method for the Fiite-Horizo Discrete-Time EOQ Problem Chug-Lu Li Departmet of Logistics The Hog Kog Polytechic Uiversity Hug Hom, Kowloo, Hog Kog Phoe: +852-2766-7410
More informationTHE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.
THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of
More informationLinear Programming! References! Introduction to Algorithms.! Dasgupta, Papadimitriou, Vazirani. Algorithms.! Cormen, Leiserson, Rivest, and Stein.
Liear Programmig! Refereces! Dasgupta, Papadimitriou, Vazirai. Algorithms.! Corme, Leiserso, Rivest, ad Stei. Itroductio to Algorithms.! Slack form! For each costrait i, defie a oegative slack variable
More informationEfficient GMM LECTURE 12 GMM II
DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationMath 508 Exam 2 Jerry L. Kazdan December 9, :00 10:20
Math 58 Eam 2 Jerry L. Kazda December 9, 24 9: :2 Directios This eam has three parts. Part A has 8 True/False questio (2 poits each so total 6 poits), Part B has 5 shorter problems (6 poits each, so 3
More informationSupport vector machine revisited
6.867 Machie learig, lecture 8 (Jaakkola) 1 Lecture topics: Support vector machie ad kerels Kerel optimizatio, selectio Support vector machie revisited Our task here is to first tur the support vector
More informationMathematics: Paper 1
GRADE 1 EXAMINATION JULY 013 Mathematics: Paper 1 EXAMINER: Combied Paper MODERATORS: JE; RN; SS; AVDB TIME: 3 Hours TOTAL: 150 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This questio paper cosists
More informationSupplemental Material: Proofs
Proof to Theorem Supplemetal Material: Proofs Proof. Let be the miimal umber of traiig items to esure a uique solutio θ. First cosider the case. It happes if ad oly if θ ad Rak(A) d, which is a special
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationLinear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d
Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More informationAP Calculus BC Review Applications of Derivatives (Chapter 4) and f,
AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)
More informationON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS * M. JA]IMOVI], I. KRNI] 1.
Yugoslav Joural of Operatios Research 1 (00), Number 1, 49-60 ON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS M. JA]IMOVI], I. KRNI] Departmet of Mathematics
More informationMachine Learning Brett Bernstein
Machie Learig Brett Berstei Week 2 Lecture: Cocept Check Exercises Starred problems are optioal. Excess Risk Decompositio 1. Let X = Y = {1, 2,..., 10}, A = {1,..., 10, 11} ad suppose the data distributio
More informationLinear Support Vector Machines
Liear Support Vector Machies David S. Roseberg The Support Vector Machie For a liear support vector machie (SVM), we use the hypothesis space of affie fuctios F = { f(x) = w T x + b w R d, b R } ad evaluate
More information(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is
Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationSimple Polygons of Maximum Perimeter Contained in a Unit Disk
Discrete Comput Geom (009) 1: 08 15 DOI 10.1007/s005-008-9093-7 Simple Polygos of Maximum Perimeter Cotaied i a Uit Disk Charles Audet Pierre Hase Frédéric Messie Received: 18 September 007 / Revised:
More informationPUTNAM TRAINING INEQUALITIES
PUTNAM TRAINING INEQUALITIES (Last updated: December, 207) Remark This is a list of exercises o iequalities Miguel A Lerma Exercises If a, b, c > 0, prove that (a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca
More informationMath 312 Lecture Notes One Dimensional Maps
Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions
Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x
More informationECE 8527: Introduction to Machine Learning and Pattern Recognition Midterm # 1. Vaishali Amin Fall, 2015
ECE 8527: Itroductio to Machie Learig ad Patter Recogitio Midterm # 1 Vaishali Ami Fall, 2015 tue39624@temple.edu Problem No. 1: Cosider a two-class discrete distributio problem: ω 1 :{[0,0], [2,0], [2,2],
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More information4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3
Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationNUMERICAL METHODS FOR SOLVING EQUATIONS
Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationLecture 4: Grassmannians, Finite and Affine Morphisms
18.725 Algebraic Geometry I Lecture 4 Lecture 4: Grassmaias, Fiite ad Affie Morphisms Remarks o last time 1. Last time, we proved the Noether ormalizatio lemma: If A is a fiitely geerated k-algebra, the,
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationVector Quantization: a Limiting Case of EM
. Itroductio & defiitios Assume that you are give a data set X = { x j }, j { 2,,, }, of d -dimesioal vectors. The vector quatizatio (VQ) problem requires that we fid a set of prototype vectors Z = { z
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More informationLecture 5. Power properties of EL and EL for vectors
Stats 34 Empirical Likelihood Oct.8 Lecture 5. Power properties of EL ad EL for vectors Istructor: Art B. Owe, Staford Uiversity. Scribe: Jigshu Wag Power properties of empirical likelihood Power of the
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory
1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.
More informationECONOMIC OPERATION OF POWER SYSTEMS
ECOOMC OEATO OF OWE SYSTEMS TOUCTO Oe of the earliest applicatios of o-lie cetralized cotrol was to provide a cetral facility, to operate ecoomically, several geeratig plats supplyig the loads of the system.
More information15.083J/6.859J Integer Optimization. Lecture 3: Methods to enhance formulations
15.083J/6.859J Iteger Optimizatio Lecture 3: Methods to ehace formulatios 1 Outlie Polyhedral review Slide 1 Methods to geerate valid iequalities Methods to geerate facet defiig iequalities Polyhedral
More information5.1. The Rayleigh s quotient. Definition 49. Let A = A be a self-adjoint matrix. quotient is the function. R(x) = x,ax, for x = 0.
40 RODICA D. COSTIN 5. The Rayleigh s priciple ad the i priciple for the eigevalues of a self-adjoit matrix Eigevalues of self-adjoit matrices are easy to calculate. This sectio shows how this is doe usig
More information6.867 Machine learning
6.867 Machie learig Mid-term exam October, ( poits) Your ame ad MIT ID: Problem We are iterested here i a particular -dimesioal liear regressio problem. The dataset correspodig to this problem has examples
More informationMarkov Decision Processes
Markov Decisio Processes Defiitios; Statioary policies; Value improvemet algorithm, Policy improvemet algorithm, ad liear programmig for discouted cost ad average cost criteria. Markov Decisio Processes
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More informationEconomics 241B Relation to Method of Moments and Maximum Likelihood OLSE as a Maximum Likelihood Estimator
Ecoomics 24B Relatio to Method of Momets ad Maximum Likelihood OLSE as a Maximum Likelihood Estimator Uder Assumptio 5 we have speci ed the distributio of the error, so we ca estimate the model parameters
More informationAlternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.
0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationAssignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1
Assigmet : Real Numbers, Sequeces. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a upper boud of A for every N. 2. Let y (, ) ad x (, ). Evaluate
More informationRank Modulation with Multiplicity
Rak Modulatio with Multiplicity Axiao (Adrew) Jiag Computer Sciece ad Eg. Dept. Texas A&M Uiversity College Statio, TX 778 ajiag@cse.tamu.edu Abstract Rak modulatio is a scheme that uses the relative order
More informationCS537. Numerical Analysis and Computing
CS57 Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 Jauary 9 9 What is the Root May physical system ca be
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 3 9/11/2013. Large deviations Theory. Cramér s Theorem
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 3 9//203 Large deviatios Theory. Cramér s Theorem Cotet.. Cramér s Theorem. 2. Rate fuctio ad properties. 3. Chage of measure techique.
More informationLinear Elliptic PDE s Elliptic partial differential equations frequently arise out of conservation statements of the form
Liear Elliptic PDE s Elliptic partial differetial equatios frequetly arise out of coservatio statemets of the form B F d B Sdx B cotaied i bouded ope set U R. Here F, S deote respectively, the flux desity
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationis also known as the general term of the sequence
Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie
More informationINEQUALITIES BJORN POONEN
INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad
More informationSupport Vector Machines and Kernel Methods
Support Vector Machies ad Kerel Methods Daiel Khashabi Fall 202 Last Update: September 26, 206 Itroductio I Support Vector Machies the goal is to fid a separator betwee data which has the largest margi,
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationCSCI567 Machine Learning (Fall 2014)
CSCI567 Machie Learig (Fall 2014) Drs. Sha & Liu {feisha,yaliu.cs}@usc.edu October 14, 2014 Drs. Sha & Liu ({feisha,yaliu.cs}@usc.edu) CSCI567 Machie Learig (Fall 2014) October 14, 2014 1 / 49 Outlie Admiistratio
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationMath 5311 Problem Set #5 Solutions
Math 5311 Problem Set #5 Solutios March 9, 009 Problem 1 O&S 11.1.3 Part (a) Solve with boudary coditios u = 1 0 x < L/ 1 L/ < x L u (0) = u (L) = 0. Let s refer to [0, L/) as regio 1 ad (L/, L] as regio.
More informationProblem Set 4 Due Oct, 12
EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More information18.657: Mathematics of Machine Learning
8.657: Mathematics of Machie Learig Lecturer: Philippe Rigollet Lecture 0 Scribe: Ade Forrow Oct. 3, 05 Recall the followig defiitios from last time: Defiitio: A fuctio K : X X R is called a positive symmetric
More informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
More information