Chapter 1 Linear Programming. Paragraph 5 Duality
|
|
- Rosemary Cecily Hoover
- 5 years ago
- Views:
Transcription
1 Chapter 1 Linear Programming Paragraph 5 Duality
2 What we did so far We developed the 2-Phase Simplex Algorithm: Hop (reasonably) from basic solution (bs) to bs until you find a basic feasible solution (bfs) or until it becomes clear that none exists. Hop from bfs to bfs until you cannot find an improving solution or the improving trajectory is unbounded. We have seen that we can guarantee termination but that the worst-case runtime is exponential. Again: Note that LP is polynomial time solvable! CS Intro to CO 2
3 Lower Bounds on the Objective Let π Ñ m, π. Consider two sets: P 1 = {x Ñ n Ax b, x } and P 2 = {x Ñ n π T Ax π T b, x }. Since π, it is clear that P 1 Œ P 2, in other words: For all x P 1 we have that π T Ax π T b. Now assume that π and also π T A c T. Then, for all x P 1, we have that π T b π T Ax c T x. Therefore, every such π provides us with a lower bound on the objective! CS Intro to CO 3
4 Duality The problem of finding the best such lower bound π T b with π Ñ such that π T A c T, π is called the Dual Problem of (P 1,c). Formally, let P 3 = {π Ñ m A T π c, π }. Then, (P 3,b) is called the dual of the primal (P 1,c). Theorem The dual of the dual is the primal. For all primal feasible x and dual feasible π it holds that π T b c T x [weak duality]. CS Intro to CO 4
5 Example Minimize x 1 +3x 2 +x 3 such that 2x 1 +x 2 3 -x 1 +x 3 4 (P) x 1, x 2, x 3 π 1T = (1/2,1) Relax P 1 to.5x 2 +x x 1, x 2, x 3 Maximize 3π 1 +4π 2 such that 2π 1 - π 2 1 π 1 3 (D) π 2 1 π 1 is dual feasible and shows: every feasible x (if any exists!) has an objective value of at least 5.5! CS Intro to CO 5
6 Example Minimize x 1 +3x 2 +x 3 such that 2x 1 +x 2 3 -x 1 +x 3 4 (P) x 1, x 2, x 3 π T = (1,1) Relax P 1 to x1+x2+x3 7 x 1, x 2, x 3 Maximize 3π 1 +4π 2 such that 2π 1 - π 2 1 π 1 3 (D) π 2 1 π is dual optimal and shows: every feasible x (if any exists!) has an objective value of at least 7! CS Intro to CO 6
7 The Dual of the Standard Form P 1 = {x Ñ n Ax = b, x } (LP) S = (P 1,c) P 2 = {x Ñ n Ax b, -Ax -b, x } (LP) C = (P 2,c) P 3 = {(η,μ) Ñ m A T η -A T μ c, η,μ P 4 = {π Ñ m A T π c} [π not non-negative anymore!] The dual of (LP) S is therefore (P 4,b). CS Intro to CO 7
8 The Dual of Unspecified Forms Min c T x A M+ x b + A M x = b A M- x b - x N+ x N unrestricted x N- Max b T π π M+ π M unrestricted π M- (A T ) N+ π c N+ (A T ) N π = c N (A T ) N- π c N- M+ M M- = {1,..,m} N+ N N- = {1,..,n} CS Intro to CO 8
9 Strong Duality c T = c T c BT A B -1 A -c T x A B -1 A A B -1 b Given is an LP in standard form. We know: The current bfs x in the tableau is optimal when c T c BT A B -1 A ñ (c BT A B -1 ) A c. Therefore, π T := c BT A B -1 is dual feasible! But: π T b = c BT A B -1 b= c BT x B = c T x! CS Intro to CO 9
10 Duality The problem of finding the best such lower bound π T b with π Ñ such that π T A c T, π is called the Dual Problem of (P 1,c). Formally, let P 3 = {π Ñ m A T π c, π }. Then, (P 3,b) is called the dual of the primal (P 1,c). Theorem The dual of the dual is the primal. For all primal feasible x and dual feasible π it holds that π T b c T x [weak duality]. For primal optimal x and dual optimal π it holds that π T b = c T x [strong duality]. CS Intro to CO 1
11 Complementary Slackness Consider an LP in canonical form with x primal optimal, and π dual optimal. c T x = π T b π T Ax c T x π T (Ax -b) =. π T b = c T x π T Ax π T b (c T - π T A) x =. Consequently, since π, Ax -b, π can only have non-zero components where there is no slack in the corresponding primal constraint. Analogously, since x, c T - π T A, x can only have non-zero components where there is no slack in the corresponding dual constraint. CS Intro to CO 11
12 Relation of Primal and Dual Primal Dual Finite Optimum Unbounded Infeasible Finite Optimum [sd] [wd] [sd] Unbounded [wd] [wd] [wd] Infeasible [sd] [wd] wd weak duality sd strong duality CS Intro to CO 12
13 Farkas Lemma Given are a i Ñ n,1 i m, and a vector c Ñ n. The cone of the a i is defined as the set C(a 1,..,a m ) := {x Ñ n x = Σπ i a i, π i }. Farkas Lemma (For all y with y T a i for all i it holds y T c ) c C(a 1,..,a m ). Proof: Exercise! CS Intro to CO 13
14 The Dual Simplex Algorithm We have seen how we compute a dual optimal solution after having solved the primal with the simplex algorithm. Consequently, we can solve the primal by solving the dual! In case that c, π = is dual feasible so we may save the effort for solving phase I when focusing on the dual right away. Example: Transportation Problem, Diet Problem, Shortest Path, CS Intro to CO 14
15 The Dual Simplex Algorithm Recall that the dual works by looking at A T instead of A in the tableau. Consequently, we can simply exchange the roles of rows and columns and consider matrix entries with opposite sign. Assume we are dual feasible. Then: Search for a row with negative b i. (no such i dual optimal primal feasible) Find a column j such that a ij < and c j / -a ij is minimal! (no such j dual unbounded primal infeasible) Pivot around a ij. CS Intro to CO 15
16 The Dual Simplex Algorithm Consider our instance of the diet problem: CS Intro to CO 16
17 Examples Diet Problem M B Amount needed [g] Cost/kg [$] 2 15 Minimize 25M + 15B Carbs [%] M + 25B 5 Fat [%] M + 5B 25 Protein [%] M + 2B 6 CS Intro to CO 17
18 The Dual Simplex Algorithm Consider our instance of the diet problem: B 5 4 $ M CS Intro to CO 18
19 The Dual Simplex Algorithm Consider our instance of the diet problem: B 5 4 $ 1 3/4-5/4 55/4 35/ /4-25/ M -1/2 3/4 1 3 CS Intro to CO 19
20 The Dual Simplex Algorithm Consider our instance of the diet problem: B 5 4 $ 1 11/5 7/5 1/5-8/ /25 1/25 1 8/ M 3/25-4/25 1 9/5 CS Intro to CO 2
21 Shadow Prices Consider production planning. In the optimal tableau, the relative costs of the slack variables are exactly -c BT A B -1. Note that the objective in the simplex was to minimize. Thus, the relative costs of slack variables give us the dual solution. If a slack variable is not in the basis, it is in the optimal solution. I.e. the corresponding constraint is tight for optimality. Now, the relative costs tell us, how much we could gain if the constraint was not tight. Consequently, the optimal duals are called shadow prices of their corresponding constraints. CS Intro to CO 21
22 Thank you!
Farkas 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 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 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 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 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 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 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 informationLecture 10: Linear programming duality and sensitivity 0-0
Lecture 10: Linear programming duality and sensitivity 0-0 The canonical primal dual pair 1 A R m n, b R m, and c R n maximize z = c T x (1) subject to Ax b, x 0 n and minimize w = b T y (2) subject to
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 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 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 informationAlgorithms and Theory of Computation. Lecture 13: Linear Programming (2)
Algorithms and Theory of Computation Lecture 13: Linear Programming (2) Xiaohui Bei MAS 714 September 25, 2018 Nanyang Technological University MAS 714 September 25, 2018 1 / 15 LP Duality Primal problem
More information21. 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 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 informationEND3033 Operations Research I Sensitivity Analysis & Duality. to accompany Operations Research: Applications and Algorithms Fatih Cavdur
END3033 Operations Research I Sensitivity Analysis & Duality to accompany Operations Research: Applications and Algorithms Fatih Cavdur Introduction Consider the following problem where x 1 and x 2 corresponds
More informationThe Simplex Algorithm
8.433 Combinatorial Optimization The Simplex Algorithm October 6, 8 Lecturer: Santosh Vempala We proved the following: Lemma (Farkas). Let A R m n, b R m. Exactly one of the following conditions is true:.
More informationLinear Programming Duality
Summer 2011 Optimization I Lecture 8 1 Duality recap Linear Programming Duality We motivated the dual of a linear program by thinking about the best possible lower bound on the optimal value we can achieve
More information4. Duality and Sensitivity
4. Duality and Sensitivity For every instance of an LP, there is an associated LP known as the dual problem. The original problem is known as the primal problem. There are two de nitions of the dual pair
More informationLecture 10: Linear programming. duality. and. The dual of the LP in standard form. maximize w = b T y (D) subject to A T y c, minimize z = c T x (P)
Lecture 10: Linear programming duality Michael Patriksson 19 February 2004 0-0 The dual of the LP in standard form minimize z = c T x (P) subject to Ax = b, x 0 n, and maximize w = b T y (D) subject to
More informationDuality of LPs and Applications
Lecture 6 Duality of LPs and Applications Last lecture we introduced duality of linear programs. We saw how to form duals, and proved both the weak and strong duality theorems. In this lecture we will
More informationDuality Theory, Optimality Conditions
5.1 Duality Theory, Optimality Conditions Katta G. Murty, IOE 510, LP, U. Of Michigan, Ann Arbor We only consider single objective LPs here. Concept of duality not defined for multiobjective LPs. Every
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 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 informationSimplex Algorithm Using Canonical Tableaus
41 Simplex Algorithm Using Canonical Tableaus Consider LP in standard form: Min z = cx + α subject to Ax = b where A m n has rank m and α is a constant In tableau form we record it as below Original Tableau
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 informationDiscrete Optimization
Prof. Friedrich Eisenbrand Martin Niemeier Due Date: April 15, 2010 Discussions: March 25, April 01 Discrete Optimization Spring 2010 s 3 You can hand in written solutions for up to two of the exercises
More information"SYMMETRIC" PRIMAL-DUAL PAIR
"SYMMETRIC" PRIMAL-DUAL PAIR PRIMAL Minimize cx DUAL Maximize y T b st Ax b st A T y c T x y Here c 1 n, x n 1, b m 1, A m n, y m 1, WITH THE PRIMAL IN STANDARD FORM... Minimize cx Maximize y T b st Ax
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 informationLectures 6, 7 and part of 8
Lectures 6, 7 and part of 8 Uriel Feige April 26, May 3, May 10, 2015 1 Linear programming duality 1.1 The diet problem revisited Recall the diet problem from Lecture 1. There are n foods, m nutrients,
More information4. Duality Duality 4.1 Duality of LPs and the duality theorem. min c T x x R n, c R n. s.t. ai Tx = b i i M a i R n
2 4. Duality of LPs and the duality theorem... 22 4.2 Complementary slackness... 23 4.3 The shortest path problem and its dual... 24 4.4 Farkas' Lemma... 25 4.5 Dual information in the tableau... 26 4.6
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 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 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 informationCSCI5654 (Linear Programming, Fall 2013) Lecture-8. Lecture 8 Slide# 1
CSCI5654 (Linear Programming, Fall 2013) Lecture-8 Lecture 8 Slide# 1 Today s Lecture 1. Recap of dual variables and strong duality. 2. Complementary Slackness Theorem. 3. Interpretation of dual variables.
More informationThe Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006
The Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006 1 Simplex solves LP by starting at a Basic Feasible Solution (BFS) and moving from BFS to BFS, always improving the objective function,
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 informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 7: Duality and applications Prof. John Gunnar Carlsson September 29, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 29, 2010 1
More information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP Different spaces and objective functions but in general same optimal
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 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 informationFoundations of Operations Research
Solved exercises for the course of Foundations of Operations Research Roberto Cordone The dual simplex method Given the following LP problem: maxz = 5x 1 +8x 2 x 1 +x 2 6 5x 1 +9x 2 45 x 1,x 2 0 1. solve
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 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 informationCO350 Linear Programming Chapter 8: Degeneracy and Finite Termination
CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination 27th June 2005 Chapter 8: Finite Termination 1 The perturbation method Recap max c T x (P ) s.t. Ax = b x 0 Assumption: B is a feasible
More informationWeek 3 Linear programming duality
Week 3 Linear programming duality This week we cover the fascinating topic of linear programming duality. We will learn that every minimization program has associated a maximization program that has the
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 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 informationBBM402-Lecture 20: LP Duality
BBM402-Lecture 20: LP Duality Lecturer: Lale Özkahya Resources for the presentation: https://courses.engr.illinois.edu/cs473/fa2016/lectures.html An easy LP? which is compact form for max cx subject to
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 informationLP Duality: outline. Duality theory for Linear Programming. alternatives. optimization I Idea: polyhedra
LP Duality: outline I Motivation and definition of a dual LP I Weak duality I Separating hyperplane theorem and theorems of the alternatives I Strong duality and complementary slackness I Using duality
More informationLinear Programming Inverse Projection Theory Chapter 3
1 Linear Programming Inverse Projection Theory Chapter 3 University of Chicago Booth School of Business Kipp Martin September 26, 2017 2 Where We Are Headed We want to solve problems with special structure!
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 informationLecture 5. x 1,x 2,x 3 0 (1)
Computational Intractability Revised 2011/6/6 Lecture 5 Professor: David Avis Scribe:Ma Jiangbo, Atsuki Nagao 1 Duality The purpose of this lecture is to introduce duality, which is an important concept
More informationAlgorithmic Game Theory and Applications. Lecture 7: The LP Duality Theorem
Algorithmic Game Theory and Applications Lecture 7: The LP Duality Theorem Kousha Etessami recall LP s in Primal Form 1 Maximize c 1 x 1 + c 2 x 2 +... + c n x n a 1,1 x 1 + a 1,2 x 2 +... + a 1,n x n
More informationMotivating examples Introduction to algorithms Simplex algorithm. On a particular example General algorithm. Duality An application to game theory
Instructor: Shengyu Zhang 1 LP Motivating examples Introduction to algorithms Simplex algorithm On a particular example General algorithm Duality An application to game theory 2 Example 1: profit maximization
More informationAM 121: Intro to Optimization
AM 121: Intro to Optimization Models and Methods Lecture 6: Phase I, degeneracy, smallest subscript rule. Yiling Chen SEAS Lesson Plan Phase 1 (initialization) Degeneracy and cycling Smallest subscript
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 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 informationOptimization 4. GAME THEORY
Optimization GAME THEORY DPK Easter Term Saddle points of two-person zero-sum games We consider a game with two players Player I can choose one of m strategies, indexed by i =,, m and Player II can choose
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 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 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 informationToday: Linear Programming (con t.)
Today: Linear Programming (con t.) COSC 581, Algorithms April 10, 2014 Many of these slides are adapted from several online sources Reading Assignments Today s class: Chapter 29.4 Reading assignment for
More informationEE364a Review Session 5
EE364a Review Session 5 EE364a Review announcements: homeworks 1 and 2 graded homework 4 solutions (check solution to additional problem 1) scpd phone-in office hours: tuesdays 6-7pm (650-723-1156) 1 Complementary
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 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 informationLecture Notes 3: Duality
Algorithmic Methods 1/11/21 Professor: Yossi Azar Lecture Notes 3: Duality Scribe:Moran Bar-Gat 1 Introduction In this lecture we will present the dual concept, Farkas s Lema and their relation to the
More informationIn the original knapsack problem, the value of the contents of the knapsack is maximized subject to a single capacity constraint, for example weight.
In the original knapsack problem, the value of the contents of the knapsack is maximized subject to a single capacity constraint, for example weight. In the multi-dimensional knapsack problem, additional
More informationOptimisation and Operations Research
Optimisation and Operations Research Lecture 9: Duality and Complementary Slackness Matthew Roughan http://www.maths.adelaide.edu.au/matthew.roughan/ Lecture_notes/OORII/
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 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 informationIntroduction to Mathematical Programming
Introduction to Mathematical Programming Ming Zhong Lecture 22 October 22, 2018 Ming Zhong (JHU) AMS Fall 2018 1 / 16 Table of Contents 1 The Simplex Method, Part II Ming Zhong (JHU) AMS Fall 2018 2 /
More informationLinear Programming: Chapter 5 Duality
Linear Programming: Chapter 5 Duality Robert J. Vanderbei September 30, 2010 Slides last edited on October 5, 2010 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544
More information3. Duality: What is duality? Why does it matter? Sensitivity through duality.
1 Overview of lecture (10/5/10) 1. Review Simplex Method 2. Sensitivity Analysis: How does solution change as parameters change? How much is the optimal solution effected by changing A, b, or c? How much
More informationExample Problem. Linear Program (standard form) CSCI5654 (Linear Programming, Fall 2013) Lecture-7. Duality
CSCI5654 (Linear Programming, Fall 013) Lecture-7 Duality Lecture 7 Slide# 1 Lecture 7 Slide# Linear Program (standard form) Example Problem maximize c 1 x 1 + + c n x n s.t. a j1 x 1 + + a jn x n b j
More informationA Review of Linear Programming
A Review of Linear Programming Instructor: Farid Alizadeh IEOR 4600y Spring 2001 February 14, 2001 1 Overview In this note we review the basic properties of linear programming including the primal simplex
More informationTopic: Primal-Dual Algorithms Date: We finished our discussion of randomized rounding and began talking about LP Duality.
CS787: Advanced Algorithms Scribe: Amanda Burton, Leah Kluegel Lecturer: Shuchi Chawla Topic: Primal-Dual Algorithms Date: 10-17-07 14.1 Last Time We finished our discussion of randomized rounding and
More informationOPERATIONS RESEARCH. Linear Programming Problem
OPERATIONS RESEARCH Chapter 1 Linear Programming Problem Prof. Bibhas C. Giri Department of Mathematics Jadavpur University Kolkata, India Email: bcgiri.jumath@gmail.com MODULE - 2: Simplex Method for
More informationCO350 Linear Programming Chapter 6: The Simplex Method
CO50 Linear Programming Chapter 6: The Simplex Method rd June 2005 Chapter 6: The Simplex Method 1 Recap Suppose A is an m-by-n matrix with rank m. max. c T x (P ) s.t. Ax = b x 0 On Wednesday, we learned
More information3. THE SIMPLEX ALGORITHM
Optimization. THE SIMPLEX ALGORITHM DPK Easter Term. Introduction We know that, if a linear programming problem has a finite optimal solution, it has an optimal solution at a basic feasible solution (b.f.s.).
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 informationCSC373: Algorithm Design, Analysis and Complexity Fall 2017 DENIS PANKRATOV NOVEMBER 1, 2017
CSC373: Algorithm Design, Analysis and Complexity Fall 2017 DENIS PANKRATOV NOVEMBER 1, 2017 Linear Function f: R n R is linear if it can be written as f x = a T x for some a R n Example: f x 1, x 2 =
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 informationLINEAR PROGRAMMING. Relation to the Text (cont.) Relation to Material in Text. Relation to the Text. Relation to the Text (cont.
LINEAR PROGRAMMING Relation to Material in Text After a brief introduction to linear programming on p. 3, Cornuejols and Tϋtϋncϋ give a theoretical discussion including duality, and the simplex solution
More information56:171 Operations Research Fall 1998
56:171 Operations Research Fall 1998 Quiz Solutions D.L.Bricker Dept of Mechanical & Industrial Engineering University of Iowa 56:171 Operations Research Quiz
More informationChapter 3, Operations Research (OR)
Chapter 3, Operations Research (OR) Kent Andersen February 7, 2007 1 Linear Programs (continued) In the last chapter, we introduced the general form of a linear program, which we denote (P) Minimize Z
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 informationAnswer the following questions: Q1: Choose the correct answer ( 20 Points ):
Benha University Final Exam. (ختلفات) Class: 2 rd Year Students Subject: Operations Research Faculty of Computers & Informatics Date: - / 5 / 2017 Time: 3 hours Examiner: Dr. El-Sayed Badr Answer the following
More informationOptimization (168) Lecture 7-8-9
Optimization (168) Lecture 7-8-9 Jesús De Loera UC Davis, Mathematics Wednesday, April 2, 2012 1 DEGENERACY IN THE SIMPLEX METHOD 2 DEGENERACY z =2x 1 x 2 + 8x 3 x 4 =1 2x 3 x 5 =3 2x 1 + 4x 2 6x 3 x 6
More informationSupplementary lecture notes on linear programming. We will present an algorithm to solve linear programs of the form. maximize.
Cornell University, Fall 2016 Supplementary lecture notes on linear programming CS 6820: Algorithms 26 Sep 28 Sep 1 The Simplex Method We will present an algorithm to solve linear programs of the form
More informationDuality in LPP Every LPP called the primal is associated with another LPP called dual. Either of the problems is primal with the other one as dual. The optimal solution of either problem reveals the information
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 12 Luca Trevisan October 3, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analysis Handout 1 Luca Trevisan October 3, 017 Scribed by Maxim Rabinovich Lecture 1 In which we begin to prove that the SDP relaxation exactly recovers communities
More informationCS261: A Second Course in Algorithms Lecture #9: Linear Programming Duality (Part 2)
CS261: A Second Course in Algorithms Lecture #9: Linear Programming Duality (Part 2) Tim Roughgarden February 2, 2016 1 Recap This is our third lecture on linear programming, and the second on linear programming
More informationLinear Programming and the Simplex method
Linear Programming and the Simplex method Harald Enzinger, Michael Rath Signal Processing and Speech Communication Laboratory Jan 9, 2012 Harald Enzinger, Michael Rath Jan 9, 2012 page 1/37 Outline Introduction
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 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 informationAppendix PRELIMINARIES 1. THEOREMS OF ALTERNATIVES FOR SYSTEMS OF LINEAR CONSTRAINTS
Appendix PRELIMINARIES 1. THEOREMS OF ALTERNATIVES FOR SYSTEMS OF LINEAR CONSTRAINTS Here we consider systems of linear constraints, consisting of equations or inequalities or both. A feasible solution
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 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 informationThe Strong Duality Theorem 1
1/39 The Strong Duality Theorem 1 Adrian Vetta 1 This presentation is based upon the book Linear Programming by Vasek Chvatal 2/39 Part I Weak Duality 3/39 Primal and Dual Recall we have a primal linear
More information