DEPARTMENT OF STATISTICS AND OPERATIONS RESEARCH OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION. Part I: Short Questions
|
|
- Dominic Simon
- 5 years ago
- Views:
Transcription
1 DEPARTMENT OF STATISTICS AND OPERATIONS RESEARCH OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION Part I: Short Questions August 12, :00 am - 12 pm General Instructions This examination is closed-book and consists of five equally-weighted questions. Do all five problems. Present your answers as clearly and concisely as you are able.
2 1. Consider the following LP: The points maximize 20x x x 3 subject to 3x 1 + 4x 2 + 6x x 1 + 6x 2 + 5x x 1 + 6x 2 + 5x x 1, x 2, x x = 35 y = are an optimal primal-dual solution of the above LP. Without reconstructing the entire tableau: (a) Show that x and y are in fact optimal to their respective LPs. (b) Determine the range of values for the coefficient of x 1 in the objective (currently set at 20) such that x remains optimal. (c) Determine the range of values for the coefficient of x 1 in the second constraint (currently set at 5) such that x remains optimal. (d) Determine the range of values for the rhs in the third inequality (currently set at 400) such that y remains optimal. 2. A subvector of a vector a R n is a vector of the form (a k, a k+1,..., a l ), where k l. The subvector (a k, a k+1,..., a l ) is said to be increasing, if a i < a i+1 for all i with k i < l. You are given a R n with all components distinct. (a) Describe an O(n) dynamic programming algorithm to find the longest increasing subvector of a. Note that an O(n 2 ) algorithm is trivial to come up with, so such a solution will receive no credit. (b) Describe a directed graph, with O(n) nodes, in which finding longest paths will accomplish the same as running the algorithm in part (a).
3 3. Let n and k be positive integers with k n, and a R n with a 1 > a 2 > > a n. Denote by e the vector of all ones. Consider the LP min st. kz + e T u ze + u a u 0, z free. (P) (a) Write out the dual of (P). (b) Prove that every optimal solution of the dual has only integer components. (c) Determine explicitly the optimal value of (P). 4. Let S = {v 1,..., v m } be a set of vectors in R n. Show how one application of the Gauss-Jordan Reduction Method can be used to find a basis for both span(s) and null(s), and use this to show that dim(span(s))+dim(null(s)) = n. 5. Suppose that x 1,..., x 12 are 0 1 variables. (a) Using extra 0-1 variables, write constraints that will force the following: At least one out of and is true. x x 5 2 (0.1) x x 12 4 (0.2) (b) Using extra 0-1 variables, write constraints that will force the following: Exactly one out of (0.1) and (0.2) is true. For full credit, use the smallest possible big-m constants.
4 DEPARTMENT OF OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION Part II Long Questions August 12, :00 pm - 4:30 pm General Instructions This examination is open-book and consists of two equally-weighted questions. You are to hand in the solutions to both problems. It is expected that your answers will be presented in a clear and concise form. Use of the internet is not permitted.
5 1. Let f be a concave differentiable function from R n to R, let g i, i = 1,..., m, be a set of convex differentiable functions from R n to R, and let b R m be an m-vector. Consider the nonlinear program (NLP b ) z b = max f(x) s.t. g i (x) b i, i = 1,..., m (a) Give the KKT conditions for a point x to be optimal to (NLP b ), using multipliers λ 1,..., λ m. (b) Now consider the nonlinear program (NLP c ) z c = max f(x) s.t. g i (x) c i, i = 1,..., m where c R m is unrelated to b. Give an upper bound for z c in terms of b, c, z b, and the associated multipliers λ i. Use the fact that a differentiable concave function f : R n R satisfies f(x) f( x) + (x x) f( x) and a differentiable convex function g : R n R satisfies g(x) g( x) + (x x) g( x) for each x R n, where denotes the gradient.
6 2. Consider a directed network G = (N, A) with m arcs, nonnegative capacities u e on each arc e A, and specified source and sink nodes s and t, respectively. We are interested in finding the maximum flow from s to t. To do this, let Γ be the set of p (s, t)-paths in G, and consider solving this problem by assigning flow f P to each of the p paths P Γ in such a way that capacities are respected and the maximum total flow is obtained. This results in LP max P Γ f P (F P ) P Γ, e P f P u e, e A f P 0, P Γ Since there may be prohibitively many (s, t)-paths in G for (F P ) to be solved explicitly, we would like to solve this problem using delayed column generation. (a) Write (F P ) in equality form by adding slack variables s e, e A. Give a feasible starting basis for this system, and explain why it is feasible. (b) After performing some Revised Simplex pivots, we get feasible basis ˆB with m elements made up of r path variables corresponding to paths P 1,..., P r and m r slack variables corresponding to arcs e 1,..., e m r. Write an explicit set of equations that will determine the shadow prices ŷ = (ŷ e : e A) corresponding to ˆB. (c) Next write the associated reduced costs associated with the shadow prices ŷ found in (b). Denote these reduced costs by γ P, P Γ and γ e, e A, respectively. Tell when these reduced costs indicate optimality of the tableau, in terms of the network G itself. (d) Given the reduced costs, describe an algorithm that in O( A 2 ) steps will either verify that the current solution is optimal; or, find an entering, and leaving variable of the next Revised Simplex pivot.
21. Solve the LP given in Exercise 19 using the big-m method discussed in Exercise 20.
Extra Problems for Chapter 3. Linear Programming Methods 20. (Big-M Method) An alternative to the two-phase method of finding an initial basic feasible solution by minimizing the sum of the artificial
More informationReview Solutions, Exam 2, Operations Research
Review Solutions, Exam 2, Operations Research 1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the dual, then... HINT: Consider the quantity y T Ax. SOLUTION: To
More informationSlack Variable. Max Z= 3x 1 + 4x 2 + 5X 3. Subject to: X 1 + X 2 + X x 1 + 4x 2 + X X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0
Simplex Method Slack Variable Max Z= 3x 1 + 4x 2 + 5X 3 Subject to: X 1 + X 2 + X 3 20 3x 1 + 4x 2 + X 3 15 2X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0 Standard Form Max Z= 3x 1 +4x 2 +5X 3 + 0S 1 + 0S 2
More informationLinear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004
Linear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004 1 In this section we lean about duality, which is another way to approach linear programming. In particular, we will see: How to define
More informationIntroduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.1-4.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following
More informationNetwork Flows. 7. Multicommodity Flows Problems. Fall 2010 Instructor: Dr. Masoud Yaghini
In the name of God Network Flows 7. Multicommodity Flows Problems 7.3 Column Generation Approach Fall 2010 Instructor: Dr. Masoud Yaghini Path Flow Formulation Path Flow Formulation Let first reformulate
More informationWorked Examples for Chapter 5
Worked Examples for Chapter 5 Example for Section 5.2 Construct the primal-dual table and the dual problem for the following linear programming model fitting our standard form. Maximize Z = 5 x 1 + 4 x
More informationMath Models of OR: Sensitivity Analysis
Math Models of OR: Sensitivity Analysis John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 8 USA October 8 Mitchell Sensitivity Analysis / 9 Optimal tableau and pivot matrix Outline Optimal
More informationOPTIMISATION /09 EXAM PREPARATION GUIDELINES
General: OPTIMISATION 2 2008/09 EXAM PREPARATION GUIDELINES This points out some important directions for your revision. The exam is fully based on what was taught in class: lecture notes, handouts and
More information(includes both Phases I & II)
Minimize z=3x 5x 4x 7x 5x 4x subject to 2x x2 x4 3x6 0 x 3x3 x4 3x5 2x6 2 4x2 2x3 3x4 x5 5 and x 0 j, 6 2 3 4 5 6 j ecause of the lack of a slack variable in each constraint, we must use Phase I to find
More informationMVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis
MVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis Ann-Brith Strömberg 2017 03 29 Lecture 4 Linear and integer optimization with
More informationOPTIMISATION 2007/8 EXAM PREPARATION GUIDELINES
General: OPTIMISATION 2007/8 EXAM PREPARATION GUIDELINES This points out some important directions for your revision. The exam is fully based on what was taught in class: lecture notes, handouts and homework.
More informationUnderstanding the Simplex algorithm. Standard Optimization Problems.
Understanding the Simplex algorithm. Ma 162 Spring 2011 Ma 162 Spring 2011 February 28, 2011 Standard Optimization Problems. A standard maximization problem can be conveniently described in matrix form
More informationChap6 Duality Theory and Sensitivity Analysis
Chap6 Duality Theory and Sensitivity Analysis The rationale of duality theory Max 4x 1 + x 2 + 5x 3 + 3x 4 S.T. x 1 x 2 x 3 + 3x 4 1 5x 1 + x 2 + 3x 3 + 8x 4 55 x 1 + 2x 2 + 3x 3 5x 4 3 x 1 ~x 4 0 If we
More informationCO 250 Final Exam Guide
Spring 2017 CO 250 Final Exam Guide TABLE OF CONTENTS richardwu.ca CO 250 Final Exam Guide Introduction to Optimization Kanstantsin Pashkovich Spring 2017 University of Waterloo Last Revision: March 4,
More informationRelation of Pure Minimum Cost Flow Model to Linear Programming
Appendix A Page 1 Relation of Pure Minimum Cost Flow Model to Linear Programming The Network Model The network pure minimum cost flow model has m nodes. The external flows given by the vector b with m
More information2.098/6.255/ Optimization Methods Practice True/False Questions
2.098/6.255/15.093 Optimization Methods Practice True/False Questions December 11, 2009 Part I For each one of the statements below, state whether it is true or false. Include a 1-3 line supporting sentence
More informationPart 1. The Review of Linear Programming
In the name of God Part 1. The Review of Linear Programming 1.5. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Formulation of the Dual Problem Primal-Dual Relationship Economic Interpretation
More informationThe Dual Simplex Algorithm
p. 1 The Dual Simplex Algorithm Primal optimal (dual feasible) and primal feasible (dual optimal) bases The dual simplex tableau, dual optimality and the dual pivot rules Classical applications of linear
More informationYinyu Ye, MS&E, Stanford MS&E310 Lecture Note #06. The Simplex Method
The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapters 2.3-2.5, 3.1-3.4) 1 Geometry of Linear
More informationSummary of the simplex method
MVE165/MMG631,Linear and integer optimization with applications The simplex method: degeneracy; unbounded solutions; starting solutions; infeasibility; alternative optimal solutions Ann-Brith Strömberg
More informationLecture 5 Simplex Method. September 2, 2009
Simplex Method September 2, 2009 Outline: Lecture 5 Re-cap blind search Simplex method in steps Simplex tableau Operations Research Methods 1 Determining an optimal solution by exhaustive search Lecture
More informationMath Models of OR: Some Definitions
Math Models of OR: Some Definitions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Some Definitions 1 / 20 Active constraints Outline 1 Active constraints
More informationPrelude to the Simplex Algorithm. The Algebraic Approach The search for extreme point solutions.
Prelude to the Simplex Algorithm The Algebraic Approach The search for extreme point solutions. 1 Linear Programming-1 x 2 12 8 (4,8) Max z = 6x 1 + 4x 2 Subj. to: x 1 + x 2
More informationLINEAR PROGRAMMING 2. In many business and policy making situations the following type of problem is encountered:
LINEAR PROGRAMMING 2 In many business and policy making situations the following type of problem is encountered: Maximise an objective subject to (in)equality constraints. Mathematical programming provides
More informationLesson 27 Linear Programming; The Simplex Method
Lesson Linear Programming; The Simplex Method Math 0 April 9, 006 Setup A standard linear programming problem is to maximize the quantity c x + c x +... c n x n = c T x subject to constraints a x + a x
More informationNonlinear Programming (Hillier, Lieberman Chapter 13) CHEM-E7155 Production Planning and Control
Nonlinear Programming (Hillier, Lieberman Chapter 13) CHEM-E7155 Production Planning and Control 19/4/2012 Lecture content Problem formulation and sample examples (ch 13.1) Theoretical background Graphical
More informationSummary of the simplex method
MVE165/MMG630, The simplex method; degeneracy; unbounded solutions; infeasibility; starting solutions; duality; interpretation Ann-Brith Strömberg 2012 03 16 Summary of the simplex method Optimality condition:
More informationFarkas Lemma, Dual Simplex and Sensitivity Analysis
Summer 2011 Optimization I Lecture 10 Farkas Lemma, Dual Simplex and Sensitivity Analysis 1 Farkas Lemma Theorem 1. Let A R m n, b R m. Then exactly one of the following two alternatives is true: (i) x
More informationIE 400: Principles of Engineering Management. Simplex Method Continued
IE 400: Principles of Engineering Management Simplex Method Continued 1 Agenda Simplex for min problems Alternative optimal solutions Unboundedness Degeneracy Big M method Two phase method 2 Simplex for
More informationThe Simplex Method. Lecture 5 Standard and Canonical Forms and Setting up the Tableau. Lecture 5 Slide 1. FOMGT 353 Introduction to Management Science
The Simplex Method Lecture 5 Standard and Canonical Forms and Setting up the Tableau Lecture 5 Slide 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard
More information1 Review Session. 1.1 Lecture 2
1 Review Session Note: The following lists give an overview of the material that was covered in the lectures and sections. Your TF will go through these lists. If anything is unclear or you have questions
More informationThe use of shadow price is an example of sensitivity analysis. Duality theory can be applied to do other kind of sensitivity analysis:
Sensitivity analysis The use of shadow price is an example of sensitivity analysis. Duality theory can be applied to do other kind of sensitivity analysis: Changing the coefficient of a nonbasic variable
More informationIE 5531 Midterm #2 Solutions
IE 5531 Midterm #2 s Prof. John Gunnar Carlsson November 9, 2011 Before you begin: This exam has 9 pages and a total of 5 problems. Make sure that all pages are present. To obtain credit for a problem,
More informationWeek_4: simplex method II
Week_4: simplex method II 1 1.introduction LPs in which all the constraints are ( ) with nonnegative right-hand sides offer a convenient all-slack starting basic feasible solution. Models involving (=)
More informationMarch 13, Duality 3
15.53 March 13, 27 Duality 3 There are concepts much more difficult to grasp than duality in linear programming. -- Jim Orlin The concept [of nonduality], often described in English as "nondualism," is
More information(P ) Minimize 4x 1 + 6x 2 + 5x 3 s.t. 2x 1 3x 3 3 3x 2 2x 3 6
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. Problem 1 Consider
More informationminimize x subject to (x 2)(x 4) u,
Math 6366/6367: Optimization and Variational Methods Sample Preliminary Exam Questions 1. Suppose that f : [, L] R is a C 2 -function with f () on (, L) and that you have explicit formulae for
More informationIP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, You wish to solve the IP below with a cutting plane technique.
IP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, 31 14. You wish to solve the IP below with a cutting plane technique. Maximize 4x 1 + 2x 2 + x 3 subject to 14x 1 + 10x 2 + 11x 3 32 10x 1 +
More informationPart IB Optimisation
Part IB Optimisation Theorems Based on lectures by F. A. Fischer Notes taken by Dexter Chua Easter 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationF 1 F 2 Daily Requirement Cost N N N
Chapter 5 DUALITY 5. The Dual Problems Every linear programming problem has associated with it another linear programming problem and that the two problems have such a close relationship that whenever
More informationmin 4x 1 5x 2 + 3x 3 s.t. x 1 + 2x 2 + x 3 = 10 x 1 x 2 6 x 1 + 3x 2 + x 3 14
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. If necessary,
More informationSensitivity Analysis and Duality in LP
Sensitivity Analysis and Duality in LP Xiaoxi Li EMS & IAS, Wuhan University Oct. 13th, 2016 (week vi) Operations Research (Li, X.) Sensitivity Analysis and Duality in LP Oct. 13th, 2016 (week vi) 1 /
More informationMATH 445/545 Homework 2: Due March 3rd, 2016
MATH 445/545 Homework 2: Due March 3rd, 216 Answer the following questions. Please include the question with the solution (write or type them out doing this will help you digest the problem). I do not
More informationOptimization for Communications and Networks. Poompat Saengudomlert. Session 4 Duality and Lagrange Multipliers
Optimization for Communications and Networks Poompat Saengudomlert Session 4 Duality and Lagrange Multipliers P Saengudomlert (2015) Optimization Session 4 1 / 14 24 Dual Problems Consider a primal convex
More informationSimplex Method for LP (II)
Simplex Method for LP (II) Xiaoxi Li Wuhan University Sept. 27, 2017 (week 4) Operations Research (Li, X.) Simplex Method for LP (II) Sept. 27, 2017 (week 4) 1 / 31 Organization of this lecture Contents:
More informationNote 3: LP Duality. If the primal problem (P) in the canonical form is min Z = n (1) then the dual problem (D) in the canonical form is max W = m (2)
Note 3: LP Duality If the primal problem (P) in the canonical form is min Z = n j=1 c j x j s.t. nj=1 a ij x j b i i = 1, 2,..., m (1) x j 0 j = 1, 2,..., n, then the dual problem (D) in the canonical
More informationApril 2003 Mathematics 340 Name Page 2 of 12 pages
April 2003 Mathematics 340 Name Page 2 of 12 pages Marks [8] 1. Consider the following tableau for a standard primal linear programming problem. z x 1 x 2 x 3 s 1 s 2 rhs 1 0 p 0 5 3 14 = z 0 1 q 0 1 0
More informationSystems Analysis in Construction
Systems Analysis in Construction CB312 Construction & Building Engineering Department- AASTMT by A h m e d E l h a k e e m & M o h a m e d S a i e d 3. Linear Programming Optimization Simplex Method 135
More informationMidterm Review. Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.
Midterm Review Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapter 1-4, Appendices) 1 Separating hyperplane
More informationWritten Examination
Division of Scientific Computing Department of Information Technology Uppsala University Optimization Written Examination 202-2-20 Time: 4:00-9:00 Allowed Tools: Pocket Calculator, one A4 paper with notes
More informationDr. S. Bourazza Math-473 Jazan University Department of Mathematics
Dr. Said Bourazza Department of Mathematics Jazan University 1 P a g e Contents: Chapter 0: Modelization 3 Chapter1: Graphical Methods 7 Chapter2: Simplex method 13 Chapter3: Duality 36 Chapter4: Transportation
More information(includes both Phases I & II)
(includes both Phases I & II) Dennis ricker Dept of Mechanical & Industrial Engineering The University of Iowa Revised Simplex Method 09/23/04 page 1 of 22 Minimize z=3x + 5x + 4x + 7x + 5x + 4x subject
More informationSAMPLE QUESTIONS. b = (30, 20, 40, 10, 50) T, c = (650, 1000, 1350, 1600, 1900) T.
SAMPLE QUESTIONS. (a) We first set up some constant vectors for our constraints. Let b = (30, 0, 40, 0, 0) T, c = (60, 000, 30, 600, 900) T. Then we set up variables x ij, where i, j and i + j 6. By using
More informationChapter 1: Linear Programming
Chapter 1: Linear Programming Math 368 c Copyright 2013 R Clark Robinson May 22, 2013 Chapter 1: Linear Programming 1 Max and Min For f : D R n R, f (D) = {f (x) : x D } is set of attainable values of
More informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 5: The Simplex method, continued Prof. John Gunnar Carlsson September 22, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 22, 2010
More informationLecture 11: Post-Optimal Analysis. September 23, 2009
Lecture : Post-Optimal Analysis September 23, 2009 Today Lecture Dual-Simplex Algorithm Post-Optimal Analysis Chapters 4.4 and 4.5. IE 30/GE 330 Lecture Dual Simplex Method The dual simplex method will
More informationCOT 6936: Topics in Algorithms! Giri Narasimhan. ECS 254A / EC 2443; Phone: x3748
COT 6936: Topics in Algorithms! Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 giri@cs.fiu.edu https://moodle.cis.fiu.edu/v2.1/course/view.php?id=612 Gaussian Elimination! Solving a system of simultaneous
More informationStandard Form An LP is in standard form when: All variables are non-negativenegative All constraints are equalities Putting an LP formulation into sta
Chapter 4 Linear Programming: The Simplex Method An Overview of the Simplex Method Standard Form Tableau Form Setting Up the Initial Simplex Tableau Improving the Solution Calculating the Next Tableau
More informationReview Questions, Final Exam
Review Questions, Final Exam A few general questions 1. What does the Representation Theorem say (in linear programming)? 2. What is the Fundamental Theorem of Linear Programming? 3. What is the main idea
More informationSection Notes 9. Midterm 2 Review. Applied Math / Engineering Sciences 121. Week of December 3, 2018
Section Notes 9 Midterm 2 Review Applied Math / Engineering Sciences 121 Week of December 3, 2018 The following list of topics is an overview of the material that was covered in the lectures and sections
More informationLinear Programming: Simplex
Linear Programming: Simplex Stephen J. Wright 1 2 Computer Sciences Department, University of Wisconsin-Madison. IMA, August 2016 Stephen Wright (UW-Madison) Linear Programming: Simplex IMA, August 2016
More informationc) Place the Coefficients from all Equations into a Simplex Tableau, labeled above with variables indicating their respective columns
BUILDING A SIMPLEX TABLEAU AND PROPER PIVOT SELECTION Maximize : 15x + 25y + 18 z s. t. 2x+ 3y+ 4z 60 4x+ 4y+ 2z 100 8x+ 5y 80 x 0, y 0, z 0 a) Build Equations out of each of the constraints above by introducing
More information- Well-characterized problems, min-max relations, approximate certificates. - LP problems in the standard form, primal and dual linear programs
LP-Duality ( Approximation Algorithms by V. Vazirani, Chapter 12) - Well-characterized problems, min-max relations, approximate certificates - LP problems in the standard form, primal and dual linear programs
More informationDual Basic Solutions. Observation 5.7. Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP:
Dual Basic Solutions Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP: Observation 5.7. AbasisB yields min c T x max p T b s.t. A x = b s.t. p T A apple c T x 0 aprimalbasicsolutiongivenbyx
More informationLecture #21. c T x Ax b. maximize subject to
COMPSCI 330: Design and Analysis of Algorithms 11/11/2014 Lecture #21 Lecturer: Debmalya Panigrahi Scribe: Samuel Haney 1 Overview In this lecture, we discuss linear programming. We first show that the
More informationThe Simplex Method. Formulate Constrained Maximization or Minimization Problem. Convert to Standard Form. Convert to Canonical Form
The Simplex Method 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard Form Convert to Canonical Form Set Up the Tableau and the Initial Basic Feasible Solution
More informationSolving Dual Problems
Lecture 20 Solving Dual Problems We consider a constrained problem where, in addition to the constraint set X, there are also inequality and linear equality constraints. Specifically the minimization problem
More informationSensitivity Analysis
Dr. Maddah ENMG 500 /9/07 Sensitivity Analysis Changes in the RHS (b) Consider an optimal LP solution. Suppose that the original RHS (b) is changed from b 0 to b new. In the following, we study the affect
More information1. (7pts) Find the points of intersection, if any, of the following planes. 3x + 9y + 6z = 3 2x 6y 4z = 2 x + 3y + 2z = 1
Math 125 Exam 1 Version 1 February 20, 2006 1. (a) (7pts) Find the points of intersection, if any, of the following planes. Solution: augmented R 1 R 3 3x + 9y + 6z = 3 2x 6y 4z = 2 x + 3y + 2z = 1 3 9
More informationSpecial cases of linear programming
Special cases of linear programming Infeasible solution Multiple solution (infinitely many solution) Unbounded solution Degenerated solution Notes on the Simplex tableau 1. The intersection of any basic
More informationChapter 4 The Simplex Algorithm Part I
Chapter 4 The Simplex Algorithm Part I Based on Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan Lewis Ntaimo 1 Modeling
More informationLecture 4: Algebra, Geometry, and Complexity of the Simplex Method. Reading: Sections 2.6.4, 3.5,
Lecture 4: Algebra, Geometry, and Complexity of the Simplex Method Reading: Sections 2.6.4, 3.5, 10.2 10.5 1 Summary of the Phase I/Phase II Simplex Method We write a typical simplex tableau as z x 1 x
More informationDistributed Real-Time Control Systems. Lecture Distributed Control Linear Programming
Distributed Real-Time Control Systems Lecture 13-14 Distributed Control Linear Programming 1 Linear Programs Optimize a linear function subject to a set of linear (affine) constraints. Many problems can
More informationTRINITY COLLEGE DUBLIN THE UNIVERSITY OF DUBLIN. School of Mathematics
JS and SS Mathematics JS and SS TSM Mathematics TRINITY COLLEGE DUBLIN THE UNIVERSITY OF DUBLIN School of Mathematics MA3484 Methods of Mathematical Economics Trinity Term 2015 Saturday GOLDHALL 09.30
More informationPart 1. The Review of Linear Programming
In the name of God Part 1. The Review of Linear Programming 1.2. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Basic Feasible Solutions Key to the Algebra of the The Simplex Algorithm
More informationSlide 1 Math 1520, Lecture 10
Slide 1 Math 1520, Lecture 10 In this lecture, we study the simplex method which is a powerful method for solving maximization/minimization problems developed in the late 1940 s. It has been implemented
More informationThe simplex algorithm
The simplex algorithm The simplex algorithm is the classical method for solving linear programs. Its running time is not polynomial in the worst case. It does yield insight into linear programs, however,
More informationOPERATIONS RESEARCH. Michał Kulej. Business Information Systems
OPERATIONS RESEARCH Michał Kulej Business Information Systems The development of the potential and academic programmes of Wrocław University of Technology Project co-financed by European Union within European
More informationMATH 445/545 Test 1 Spring 2016
MATH 445/545 Test Spring 06 Note the problems are separated into two sections a set for all students and an additional set for those taking the course at the 545 level. Please read and follow all of these
More information1. Algebraic and geometric treatments Consider an LP problem in the standard form. x 0. Solutions to the system of linear equations
The Simplex Method Most textbooks in mathematical optimization, especially linear programming, deal with the simplex method. In this note we study the simplex method. It requires basically elementary linear
More informationMulticommodity Flows and Column Generation
Lecture Notes Multicommodity Flows and Column Generation Marc Pfetsch Zuse Institute Berlin pfetsch@zib.de last change: 2/8/2006 Technische Universität Berlin Fakultät II, Institut für Mathematik WS 2006/07
More informationLinear and Combinatorial Optimization
Linear and Combinatorial Optimization The dual of an LP-problem. Connections between primal and dual. Duality theorems and complementary slack. Philipp Birken (Ctr. for the Math. Sc.) Lecture 3: Duality
More informationHomework Assignment 4 Solutions
MTAT.03.86: Advanced Methods in Algorithms Homework Assignment 4 Solutions University of Tartu 1 Probabilistic algorithm Let S = {x 1, x,, x n } be a set of binary variables of size n 1, x i {0, 1}. Consider
More information56:270 Final Exam - May
@ @ 56:270 Linear Programming @ @ Final Exam - May 4, 1989 @ @ @ @ @ @ @ @ @ @ @ @ @ @ Select any 7 of the 9 problems below: (1.) ANALYSIS OF MPSX OUTPUT: Please refer to the attached materials on the
More informationExample. 1 Rows 1,..., m of the simplex tableau remain lexicographically positive
3.4 Anticycling Lexicographic order In this section we discuss two pivoting rules that are guaranteed to avoid cycling. These are the lexicographic rule and Bland s rule. Definition A vector u R n is lexicographically
More informationThe Simplex Method. Standard form (max) z c T x = 0 such that Ax = b.
The Simplex Method Standard form (max) z c T x = 0 such that Ax = b. The Simplex Method Standard form (max) z c T x = 0 such that Ax = b. Build initial tableau. z c T 0 0 A b The Simplex Method Standard
More informationIntroduction to linear programming using LEGO.
Introduction to linear programming using LEGO. 1 The manufacturing problem. A manufacturer produces two pieces of furniture, tables and chairs. The production of the furniture requires the use of two different
More informationDecember 2014 MATH 340 Name Page 2 of 10 pages
December 2014 MATH 340 Name Page 2 of 10 pages Marks [8] 1. Find the value of Alice announces a pure strategy and Betty announces a pure strategy for the matrix game [ ] 1 4 A =. 5 2 Find the value of
More informationAM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1
AM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1 Prof. Yiling Chen Fall 2018 Here are some practice questions to help to prepare for the midterm. The midterm will
More informationGeneralization to inequality constrained problem. Maximize
Lecture 11. 26 September 2006 Review of Lecture #10: Second order optimality conditions necessary condition, sufficient condition. If the necessary condition is violated the point cannot be a local minimum
More informationLinear Programming, Lecture 4
Linear Programming, Lecture 4 Corbett Redden October 3, 2016 Simplex Form Conventions Examples Simplex Method To run the simplex method, we start from a Linear Program (LP) in the following standard simplex
More informationConvex Analysis 2013 Let f : Q R be a strongly convex function with convexity parameter µ>0, where Q R n is a bounded, closed, convex set, which contains the origin. Let Q =conv(q, Q) andconsiderthefunction
More informationΩ R n is called the constraint set or feasible set. x 1
1 Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize subject to f(x) x Ω Ω R n is called the constraint set or feasible set. any point x Ω is called a feasible point We
More information4. The Dual Simplex Method
4. The Dual Simplex Method Javier Larrosa Albert Oliveras Enric Rodríguez-Carbonell Problem Solving and Constraint Programming (RPAR) Session 4 p.1/34 Basic Idea (1) Algorithm as explained so far known
More informationAnswers to problems. Chapter 1. Chapter (0, 0) (3.5,0) (0,4.5) (2, 3) 2.1(a) Last tableau. (b) Last tableau /2 -3/ /4 3/4 1/4 2.
Answers to problems Chapter 1 1.1. (0, 0) (3.5,0) (0,4.5) (, 3) Chapter.1(a) Last tableau X4 X3 B /5 7/5 x -3/5 /5 Xl 4/5-1/5 8 3 Xl =,X =3,B=8 (b) Last tableau c Xl -19/ X3-3/ -7 3/4 1/4 4.5 5/4-1/4.5
More informationMath 210 Finite Mathematics Chapter 4.2 Linear Programming Problems Minimization - The Dual Problem
Math 2 Finite Mathematics Chapter 4.2 Linear Programming Problems Minimization - The Dual Problem Richard Blecksmith Dept. of Mathematical Sciences Northern Illinois University Math 2 Website: http://math.niu.edu/courses/math2.
More information56:171 Operations Research Midterm Exam--15 October 2002
Name 56:171 Operations Research Midterm Exam--15 October 2002 Possible Score 1. True/False 25 _ 2. LP sensitivity analysis 25 _ 3. Transportation problem 15 _ 4. LP tableaux 15 _ Total 80 _ Part I: True(+)
More informationLinear Programming Redux
Linear Programming Redux Jim Bremer May 12, 2008 The purpose of these notes is to review the basics of linear programming and the simplex method in a clear, concise, and comprehensive way. The book contains
More informationQuiz Discussion. IE417: Nonlinear Programming: Lecture 12. Motivation. Why do we care? Jeff Linderoth. 16th March 2006
Quiz Discussion IE417: Nonlinear Programming: Lecture 12 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 16th March 2006 Motivation Why do we care? We are interested in
More information