# Paper Name: Linear Programming & Theory of Games. Lesson Name: Duality in Linear Programing Problem

Save this PDF as:

Size: px
Start display at page:

Download "Paper Name: Linear Programming & Theory of Games. Lesson Name: Duality in Linear Programing Problem"

## Transcription

1 Paper Name: Linear Programming & Theory of Games Lesson Name: Duality in Linear Programing Problem Lesson Developers: DR. VAJALA RAVI, Dr. Manoj Kumar Varshney College/Department: Department of Statistics, Lady Shri Ram College College, & Department of Statistics, Hindu College, University of Delhi

2 DUALITY IN LINEAR PROGRAMMING PROBLEM Table of Contents:. OBJECTIVES. INTRODUCTION 3. TO DERIVE DUAL FROM PRIMAL 3. Thumb rules for converting Primal to Dual 3. Primal and Dual 3.3 Eamples 4. IMPORTANT THEOREMS ON DUALITY 5. SOLUTION TO DUAL LPP USING SIMPLEX METHOD 6. PRACTICE PROBLEMS 7. REFERENCES

3 . OBJECTIVES: After studying this chapter, you should be able to : Understand the Linear programing problem from two perspectives. Formulate such problems from both these perspectives Obtain solutions for both the problems simultaneously by solving only one of them. Identify various types of solutions

4 . INTRODUCTION: Theory of Economics suggests that the resources are scarce and hence allocation of these should be optimal to make best use of them. In today's world, every situation can be viewed from two perspectives, either a protagonist or an antagonist. In the same manner, a given LPP problem is seen from either a profiteer's or a loser's viewpoint. The amount of gain that a person makes is eactly equal to the amount of loss that the opponent incurs. Hence for every LPP which is of maimization (or minimization) type, there always eist another LPP which is of minimization (or maimization). Such a set of two problems constitute the Primal-Dual pair. The original problem is called primal while the associated one is called dual. Whenever we solve an LPP, we implicitly solve two problems viz., primal resource allocation problem and the dual resource valuation problem. It is important to note that if we have primal solution then the solution of dual form can be easily obtained. The final simple table giving the solution to primal yields the solution to the dual problem too. This fact is important because the situation can arise where the dual is easier to solve than the primal. 3. HOW TO DERIVE DUAL FROM PRIMAL? In order to obtain dual from the primal, we first need to convert the primal into standard form i.e., where every linear constraint has an equality (=) sign. For this, we add (if necessary) the slack and surplus (and artificial) variables to the constraints and also allocate a zero (large) cost to these in the objective function. The number of variables in the dual is eactly equal to the number of constraints in the primal. The primal-dual transformation can be clearly understood using the following table:

5 Dual variables Primal Variables n Maimize Z = w a a a n b w a a a n b : : : : : w m a m a m a mn b m Minimize Z* = c c c n Primal Problem Dual problem Ma z = c + c c n n s.t a + + a n n b a + + a n n b and : : : a m + + a mn n b m,,, n Min z = b w + b w b m w m s.t a w + + a m w m c a w + + a m w m c and : : : a n w + + a mn w m c n w, w,, w m

6 Eample : Consider the following primal Maimize z and, 3 The primal is then reduced to standard form as follows: Maimize z 6 7 s s 3 s 5 s 4 3 and,, s, s where s and s are the slack variables The coefficients can be arranged in following table: s s b j 's Constraint 3 4..multiply by w Constraint 5 3..multiply by w c j 's 6 7 The dual is then written as follows:

7 Minimize z* 4w 3w w w 3w 5w 6 7 w w w w w, w Eample : Minimize z and, The primal is reduced to standard form as follows: Minimize z 3 s s s3 3 s 3 s s3 4 3

8 and,, s, s where s and s are surplus while 3 s is slack variable The coefficients can be arranged in following table: s s s 3 b j 's Constraint -..multiply by w Constraint 3 3..multiply by w Constraint 3-4..multiply by w 3 c j 's 3 - The dual is then written as follows: Maimize z* w 3w 4w3 w w w 3 3w w3 w w w3 w w w w w and w 3 w w w3, 3

9 Eample 3: Maimize z and, unrestrict ed The primal is reduced to standard form as follows: ' '' Maimize z 8 8 s s ' '' ' '' s ' '' ' '' 3 3 s 6 such that ' '' ' '' and ' '' ' '' and,,,, s, s where s, s are surplus variables The coefficients can be arranged in following table: ' '' '' '' s s b j 's Constraint multiply by w Constraint multiply by w c j 's 8-8 The dual is then written as follows:

10 Minimize z* 6w w w w w w w w w w w 3w 3w w w w w 8 8 w 3w 8 w w 3w, w 8 Eample 4: Maimize z and,, 3 The primal is reduced to standard form as follows: Maimize z 53 s s 3 s 6

11 63 s 3 3 and,, s, s where s is surplus while s is slack variable The coefficients can be arranged in following table: 6 3 s s b j 's Constraint -..multiply by w Constraint 6 6..multiply by w Constraint 3-3..multiply by w 3 c j 's -5 The dual is then written as follows: Minimize z* w 6w w3 w w w w3 w3 w 6w 3w3 5 w w w3 w w w3 w and w and w3 unrestricted

12 3.. Thumb Rules (made easy!!) for converting Primal into its dual The following steps to obtain dual of a given primal LPP Step I First convert the objective function to maimization form, if not. Step II If a constraint has inequality sign then multiply both sides by and make the inequality sign. Step III If a constraint has an equality sign =, then it is replaced by two constraints involving the inequalities going in opposite directions simultaneously. for eample 8 then replaced by two inequalities 8 and 8. Step IV Every unrestricted variable is replaced by the difference of two non-negative variables. Step V We get the standard primal form of given LPP in which - (i) All the constraints have sign when objective function is maimize type. (ii) All the constraints have sign when objective function is minimize type. Step VI Finally, Dual of the given LPP is as follows (i) Transposes the row and column of constraints coefficient matri. (ii) Transpose the coefficient function and the right side constants c, c,..., c n of the objective b, b,..., b m. (iii) Change the inequalities from to form.

13 (iv) Change the maimize type into minimize type LPP. 3.. Primal and dual If the Primal is as - Maimize Z n c j j constraints n j Then its dual would be j aij ij bi i,,..., m Minimize Z m biw i i m a w c i,,..., n i ij i j

14 S. No. Primal Dual. Maimization Problem Minimization Problem. Minimization Problem Maimization Problem 3. R.H.S. constants of the b i Coefficient of the objective function c j 4. Constraints Constraints 5. Constraints Constraints 6. i th constraint with = sign i th variable unrestricted in sign 7. Decision variable j Slack or surplus variables 8. Slack or surplus variable s Decision variable w j 9. i th row of coefficients i th column of coefficient. j th column of coefficient j th row of coefficient. j th variable unrestricted in sign j th constraint as equality. i th constraint as equality i th variable unrestricted in sign

15 3.3. Eamples Find the dual of the LPP Maimize Z 5 (a) S.t , Maimize Z (b) 3 S.t ,, 3 Minimize Z 5 (c) 3 S.t. Solutions: ,, 3 Minimize Z 5w w w (a) 3 3w w 5w 5 S.t 3

16 4w w 3w 3 and w, w, w3 Minimize Z 4w 9w (b) S.t. 5w6 w 7 w w 4 w w 3 w, w Minimize Z w 6w 4w (c) w w w w w w 6w 3w 5 w, w & w unrestricted in sign. 4. IMPORTANT THEOREMS ON DUALITY: Theorem : The dual of dual is primal. Theorem : If the primal or the dual has a finite optimum solution, then the other problem also possesses a finite optimum solution and the optimum values of the objective functions of the two problems are equal. Theorem 3: If either the primal or the dual problem has an unbounded objective function value, then the other problem has no feasible solution.

17 Theorem 4: The value of the objective function for any feasible solution of the primal is less than the value of the objective function for any feasible solution of the dual. Theorem 5: Each basic feasible solution in primal problem is related to a complementary basic feasible solution in dual problem. 5. SOLUTION TO THE DUAL LPP USING SIMPLEX METHOD: The two constituents of a Linear Programming problem are the primal and dual. Since simple method provides a solution to the primal, it eventually provides a solution for the dual too. In fact, the final simple table provides not only, a solution to the primal problem, but also the dual one. Hence, it is imperative to choose the easiest of the primal-dual pair, solve it using simple method and hence find a solution to both the problems using the optimal solution table. Once the primal problem is solved, the solution of dual problem can be obtained from the optimal simple table as follows:. The slack/artificial variables of primal problem correspond to the basic variables of dual problem in optimal solution.. The z j - c j values under these slack/artificial variables give the optimal values of basic dual variables. 3. The z j - c j values under these non-basic variables of primal give the optimal values of slack dual variables. 4. The value of objective function is same for primal and dual problems. 5. If the primal is a maimization problem, then, Rule : Corresponding net evaluations of the starting primal variables = Difference between the left and right sides of the dual constraints associated with the starting primal variables Rule : Negative of the corresponding net evaluations of the starting dual variables = Difference between the left and right sides of the primal constraints associated with dual starting variables (if the objective is

18 minimization, then first change it to maimization by using the relation ma z = -min (-z)) Rule 3: If the primal (dual) problem is unbounded, the dual (primal) problem does not have a feasible solution. ( Kantiswarup, Gupta and Manmohan) Eample (contd): The optimal solution table for primal is given below: 6 7 Basic Variables C b b s s s 5/ / -/ 6 3/ 5/ / z j -c j So the solution of primal is = 3/, = and Ma. Z = 9 The primal slack variables correspond to the dual basic variables, and hence the solution is given in the marked region of above table. Hence the dual solution is given by w =, w = 3 and Min Z * = 9 Eample (contd): The optimal solution table for primal is given below: -3 -M -M Basic Variables C b b s s s 3 A A -3 3/5-3/5 -/5 3/5 4/5 /5 /5 -/5 A -M 6/5 -/5 -/5 - /5 z j -c j 6/5 M/5+ M/5+ M 4M/5-

19 Since the final simple table comprises of artificial variable at positive level, therefore the given primal problem has no solution. Hence, the solution for dual is unbounded. Eample 3 (contd): The optimal solution table for primal is given below: Basic Variables C b b 8-8 -M ' '' ' '' s s A '' 6/5-3/5 -/5 /5 '' -8 6/5 - -/5 -/5 /5 z j -c j -48/5 6/5 8/5 M-8/5 So the solution of primal is 48/5 ' '' = -6/5, ' '' = -6/5and Ma. Z = - The primal slack/artificial variables correspond to the dual basic variables i.e., w s, w A. The solution of dual problem is given in the marked region of above table. Hence the dual solution is given by w = -6/5, w = -8/5and Min Z * = -48/5. Eample 4 (contd): The optimal solution table for primal is given below: -5 -M -M Basic Variables C b b 3 s s A A 4/3 /3 -/3 s /3 /3 - /3 7/3 3 /3 /3 z j -c j 8/3 5 /3 M M-4/3 So the solution of primal is = 7/3, = 4/3and Ma. Z = 8/3

20 The primal slack/artificial variables correspond to the dual basic variables i.e., w A, w s, w 3 A. The solution of dual problem is given in the marked region of above table. Hence the dual solution is given by w =, w =, w 3 = 4/3 and Min Z * = 8/3. 6. PRACTICE PROBLEMS:. Minimize z 5 3 subject to 4,, 5 and, (Sol: Primal: = 4, = and minimum Z = 3; Dual: w = -, w = 7/, w 3 =, maimum Z* = 3). Minimize z subject to 3 3, 4 3 6, 3 and, (Sol: Primal: =, = 3 and minimum Z = 3; Dual: w =, w =, w 3 =, maimum Z* = 3) 3. Maimize z 3 subject to 4 3, and,, 3 (Sol: Primal: unbounded solution; Dual: infeasible solution) 7 7. REFERENCES: Chawla, K. K.; Gupta, Vijay; Sharma, Bhushan K. (9): Operations Research, Kalyani Publishers, New Delhi. F.S.Hillier and G.J.Lieberman(9): Introduction to Operations Research( 9 th edition), Tata McGraw Hill, Singapore. G. Hadley ( ):Linear Programming, Narosa Publishing House,New Delhi.

21 Hamdy A. Taha (6) Operations Research, An Introdution ( 8 th edition), Prentice- Hall,India. Mahajan, Manohar (9): Operations Research, Dhanpat Rai &Co. (P) Limited, Second Edition,Delhi. Mokhtar S., Bazaraa, John.Jarvis and D. Sherali (4): Linear Programming and Network Flaws( nd edition), John Wiley and Sons, India,4. Prakash, R. Hari; Prasad, B. Durga; Sreenivashul P.(): Operations Research, Scitech Publication (India) Pvt. Ltd., Hyderabad. Saifi, Sharif; Rajput, Kamini(): Operations Research, Sun India Publications, first Edition, New Delhi. Sharma, S. D. (4): Operation Research Theory, Methods & Applications, Kedar Nath Ram Nath, Seventeenth Edition, Meerut, India. Sharma, J.K. (9): "Operations Research: Theory and Applications", 4th edition, Macmillan India Ltd. Swarup, Kanti; Gupta, P. K.; Man Mohan(4): Operations Research, Sultan Chand & Sons publishers, XVII th Edition,New Delhi, India.

### A = Chapter 6. Linear Programming: The Simplex Method. + 21x 3 x x 2. C = 16x 1. + x x x 1. + x 3. 16,x 2.

Chapter 6 Linear rogramming: The Simple Method Section The Dual roblem: Minimization with roblem Constraints of the Form Learning Objectives for Section 6. Dual roblem: Minimization with roblem Constraints

### OPERATIONS 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

### Game Theory- Normal Form Games

Chapter 6 Game Theory- Normal Form Games Key words: Game theory, strategy, finite game, zero-sum game, pay-off matrix, dominant strategy, value of the game, fair game, stable solution, saddle point, pure

### OPERATIONS 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

### Chap6 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

### Duality 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

### CSCI5654 (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.

### The 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

### Summary 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:

### Lecture 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

### IE 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

Essential Microeconomics -- CHAPTER -: SHADOW PRICES An intuitive approach: profit maimizing firm with a fied supply of an input Shadow prices 5 Concave maimization problem 7 Constraint qualifications

### Linear Programming Applications. Transportation Problem

Linear Programming Applications Transportation Problem 1 Introduction Transportation problem is a special problem of its own structure. Planning model that allocates resources, machines, materials, capital

### Summary 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

### Lecture 11 Linear programming : The Revised Simplex Method

Lecture 11 Linear programming : The Revised Simplex Method 11.1 The Revised Simplex Method While solving linear programming problem on a digital computer by regular simplex method, it requires storing

### Simplex 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

### c) 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

### Chapter 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

### In Chapters 3 and 4 we introduced linear programming

SUPPLEMENT The Simplex Method CD3 In Chapters 3 and 4 we introduced linear programming and showed how models with two variables can be solved graphically. We relied on computer programs (WINQSB, Excel,

### The 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

### The 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

### Optimality, Duality, Complementarity for Constrained Optimization

Optimality, Duality, Complementarity for Constrained Optimization Stephen Wright University of Wisconsin-Madison May 2014 Wright (UW-Madison) Optimality, Duality, Complementarity May 2014 1 / 41 Linear

### MVE165/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

### Relation 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

### Linear Programming. Linear Programming I. Lecture 1. Linear Programming. Linear Programming

Linear Programming Linear Programming Lecture Linear programming. Optimize a linear function subject to linear inequalities. (P) max " c j x j n j= n s. t. " a ij x j = b i # i # m j= x j 0 # j # n (P)

### Convex Optimization Overview (cnt d)

Conve Optimization Overview (cnt d) Chuong B. Do November 29, 2009 During last week s section, we began our study of conve optimization, the study of mathematical optimization problems of the form, minimize

### Sensitivity Analysis and Duality

Sensitivity Analysis and Duality Part II Duality Based on Chapter 6 Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan

### UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION

MATHEMATICAL ECONOMICS COMPLEMENTARY COURSE B.Sc. Mathematics II SEMESTER UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION Calicut University P. O. Malappuram, Kerala, India 67 65 40 UNIVERSITY OF CALICUT

### Operations Research. Duality in linear programming.

Operations Research Duality in linear programming Duality in linear programming As we have seen in past lessons, linear programming are either maximization or minimization type, containing m conditions

### Overview of course. Introduction to Optimization, DIKU Monday 12 November David Pisinger

Introduction to Optimization, DIKU 007-08 Monday November David Pisinger Lecture What is OR, linear models, standard form, slack form, simplex repetition, graphical interpretation, extreme points, basic

### Introduction to Linear Programming

Nanjing University October 27, 2011 What is LP The Linear Programming Problem Definition Decision variables Objective Function x j, j = 1, 2,..., n ζ = n c i x i i=1 We will primarily discuss maxizming

### Paul E. Fishback. Instructors Solutions Manual for Linear and Nonlinear Programming with Maple: An Interactive, Applications-Based Approach

Paul E. Fishback Instructors Solutions Manual for Linear and Nonlinear Programming with Maple: An Interactive, Applications-Based Approach ii Contents I Linear Programming An Introduction to Linear Programming

### 4. 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

### LINEAR PROGRAMMING I. a refreshing example standard form fundamental questions geometry linear algebra simplex algorithm

Linear programming Linear programming. Optimize a linear function subject to linear inequalities. (P) max c j x j n j= n s. t. a ij x j = b i i m j= x j 0 j n (P) max c T x s. t. Ax = b Lecture slides

### B.SC. III YEAR MATHEMATICS. Semester V & VI. Syllabus of. [ Effective from & onwards ]

S-[F] FACULTY OF SCIENCE[ NC] B.Sc. III Yr. Mathematics Semester-V & VI.doc - 1 - Syllabus of B.SC. III YEAR MATHEMATICS Semester V & VI [ Effective from 2011-12 & onwards ] 1 S-[F] FACULTY OF SCIENCE[

### Worked 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

### STATIC LECTURE 4: CONSTRAINED OPTIMIZATION II - KUHN TUCKER THEORY

STATIC LECTURE 4: CONSTRAINED OPTIMIZATION II - KUHN TUCKER THEORY UNIVERSITY OF MARYLAND: ECON 600 1. Some Eamples 1 A general problem that arises countless times in economics takes the form: (Verbally):

### (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

### Lecture 7: Weak Duality

EE 227A: Conve Optimization and Applications February 7, 2012 Lecture 7: Weak Duality Lecturer: Laurent El Ghaoui 7.1 Lagrange Dual problem 7.1.1 Primal problem In this section, we consider a possibly

### Simplex 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:

### SOLVING TRANSPORTATION PROBLEM WITH THE HELP OF REVISED SIMPLEX METHOD

SOLVING TRANSPORTATION PROBLEM WITH THE HELP OF REVISED SIMPLEX METHOD Gaurav Sharma 1, S. H. Abbas 2, Vijay Kumar Gupta 3 Abstract: The purpose of this paper is establishing usefulness of newly revised

### The Kuhn-Tucker and Envelope Theorems

The Kuhn-Tucker and Envelope Theorems Peter Ireland EC720.01 - Math for Economists Boston College, Department of Economics Fall 2010 The Kuhn-Tucker and envelope theorems can be used to characterize the

### 14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness.

CS/ECE/ISyE 524 Introduction to Optimization Spring 2016 17 14. Duality ˆ Upper and lower bounds ˆ General duality ˆ Constraint qualifications ˆ Counterexample ˆ Complementary slackness ˆ Examples ˆ Sensitivity

### LP 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

### Yinyu 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

### II BSc(Information Technology)-[ ] Semester-III Allied:Computer Based Optimization Techniques-312C Multiple Choice Questions.

Dr.G.R.Damodaran College of Science (Autonomous, affiliated to the Bharathiar University, recognized by the UGC)Re-accredited at the 'A' Grade Level by the NAAC and ISO 9001:2008 Certified CRISL rated

### Duality in Linear Programming

Duality in Linear Programming Gary D. Knott Civilized Software Inc. 1219 Heritage Park Circle Silver Spring MD 296 phone:31-962-3711 email:knott@civilized.com URL:www.civilized.com May 1, 213.1 Duality

### The 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

### 9.1 Linear Programs in canonical form

9.1 Linear Programs in canonical form LP in standard form: max (LP) s.t. where b i R, i = 1,..., m z = j c jx j j a ijx j b i i = 1,..., m x j 0 j = 1,..., n But the Simplex method works only on systems

### Algebraic Simplex Active Learning Module 4

Algebraic Simplex Active Learning Module 4 J. René Villalobos and Gary L. Hogg Arizona State University Paul M. Griffin Georgia Institute of Technology Time required for the module: 50 Min. Reading Most

### 3 Development of the Simplex Method Constructing Basic Solution Optimality Conditions The Simplex Method...

Contents Introduction to Linear Programming Problem. 2. General Linear Programming problems.............. 2.2 Formulation of LP problems.................... 8.3 Compact form and Standard form of a general

### Duality 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

### OPRE 6201 : 3. Special Cases

OPRE 6201 : 3. Special Cases 1 Initialization: The Big-M Formulation Consider the linear program: Minimize 4x 1 +x 2 3x 1 +x 2 = 3 (1) 4x 1 +3x 2 6 (2) x 1 +2x 2 3 (3) x 1, x 2 0. Notice that there are

### The Kuhn-Tucker and Envelope Theorems

The Kuhn-Tucker and Envelope Theorems Peter Ireland ECON 77200 - Math for Economists Boston College, Department of Economics Fall 207 The Kuhn-Tucker and envelope theorems can be used to characterize the

### Benders Decomposition

Benders Decomposition Yuping Huang, Dr. Qipeng Phil Zheng Department of Industrial and Management Systems Engineering West Virginia University IENG 593G Nonlinear Programg, Spring 2012 Yuping Huang (IMSE@WVU)

### Introduction to Operations Research

Introduction to Operations Research (Week 5: Linear Programming: More on Simplex) José Rui Figueira Instituto Superior Técnico Universidade de Lisboa (figueira@tecnico.ulisboa.pt) March 14-15, 2016 This

### SELECT TWO PROBLEMS (OF A POSSIBLE FOUR) FROM PART ONE, AND FOUR PROBLEMS (OF A POSSIBLE FIVE) FROM PART TWO. PART ONE: TOTAL GRAND

1 56:270 LINEAR PROGRAMMING FINAL EXAMINATION - MAY 17, 1985 SELECT TWO PROBLEMS (OF A POSSIBLE FOUR) FROM PART ONE, AND FOUR PROBLEMS (OF A POSSIBLE FIVE) FROM PART TWO. PART ONE: 1 2 3 4 TOTAL GRAND

### ENM 202 OPERATIONS RESEARCH (I) OR (I) 2 LECTURE NOTES. Solution Cases:

ENM 202 OPERATIONS RESEARCH (I) OR (I) 2 LECTURE NOTES Solution Cases: 1. Unique Optimal Solution Case 2. Alternative Optimal Solution Case 3. Infeasible Solution Case 4. Unbounded Solution Case 5. Degenerate

### Benders' Method Paul A Jensen

Benders' Method Paul A Jensen The mixed integer programming model has some variables, x, identified as real variables and some variables, y, identified as integer variables. Except for the integrality

### Chapter 4 The Simplex Algorithm Part II

Chapter 4 The Simple Algorithm Part II Based on Introduction to Mathematical Programming: Operations Research, Volume 4th edition, by Wayne L Winston and Munirpallam Venkataramanan Lewis Ntaimo L Ntaimo

### that nds a basis which is optimal for both the primal and the dual problems, given

On Finding Primal- and Dual-Optimal Bases Nimrod Megiddo (revised June 1990) Abstract. We show that if there exists a strongly polynomial time algorithm that nds a basis which is optimal for both the primal

### 10701 Recitation 5 Duality and SVM. Ahmed Hefny

10701 Recitation 5 Duality and SVM Ahmed Hefny Outline Langrangian and Duality The Lagrangian Duality Eamples Support Vector Machines Primal Formulation Dual Formulation Soft Margin and Hinge Loss Lagrangian

### Math 354 Summer 2004 Solutions to review problems for Midterm #1

Solutions to review problems for Midterm #1 First: Midterm #1 covers Chapter 1 and 2. In particular, this means that it does not explicitly cover linear algebra. Also, I promise there will not be any proofs.

### Linear Programming: The Simplex Method

7206 CH09 GGS /0/05 :5 PM Page 09 9 C H A P T E R Linear Programming: The Simplex Method TEACHING SUGGESTIONS Teaching Suggestion 9.: Meaning of Slack Variables. Slack variables have an important physical

### IP 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 +

### Lagrangian Duality for Dummies

Lagrangian Duality for Dummies David Knowles November 13, 2010 We want to solve the following optimisation problem: f 0 () (1) such that f i () 0 i 1,..., m (2) For now we do not need to assume conveity.

### Minimum cost transportation problem

Minimum cost transportation problem Complements of Operations Research Giovanni Righini Università degli Studi di Milano Definitions The minimum cost transportation problem is a special case of the minimum

### An Integer Solution of Fractional Programming Problem

Gen. Math. Notes, Vol. 4, No., June 0, pp. -9 ISSN 9-784; Copyright ICSRS Publiation, 0 www.i-srs.org Available free online at http://www.geman.in An Integer Solution of Frational Programming Problem S.C.

### Math 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.

### Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras

Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture - 3 Simplex Method for Bounded Variables We discuss the simplex algorithm

### Linear Programming in Matrix Form

Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,

### - 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

### Dr. Maddah ENMG 500 Engineering Management I 10/21/07

Dr. Maddah ENMG 500 Engineering Management I 10/21/07 Computational Procedure of the Simplex Method The optimal solution of a general LP problem is obtained in the following steps: Step 1. Express the

### Lecture 1 Introduction

L. Vandenberghe EE236A (Fall 2013-14) Lecture 1 Introduction course overview linear optimization examples history approximate syllabus basic definitions linear optimization in vector and matrix notation

### U.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

### B.A./B. Sc. Mathematics Course Structure (Semester System)

KUMAUN UNIVERSITY, NAINITAL Department of Mathematics B.A./B. Sc. Mathematics SEMESTERWISE COURSE STRUCTURE AND DETAILED SYLLABUS: 1. There shall be six semesters in the three- years B.A./B.Sc. Programme.

### Computational Optimization. Constrained Optimization Part 2

Computational Optimization Constrained Optimization Part Optimality Conditions Unconstrained Case X* is global min Conve f X* is local min SOSC f ( *) = SONC Easiest Problem Linear equality constraints

### Week 3: Simplex Method I

Week 3: Simplex Method I 1 1. Introduction The simplex method computations are particularly tedious and repetitive. It attempts to move from one corner point of the solution space to a better corner point

### Interior Point Methods for LP

11.1 Interior Point Methods for LP Katta G. Murty, IOE 510, LP, U. Of Michigan, Ann Arbor, Winter 1997. Simplex Method - A Boundary Method: Starting at an extreme point of the feasible set, the simplex

### The Simplex Method for Solving a Linear Program Prof. Stephen Graves

The Simplex Method for Solving a Linear Program Prof. Stephen Graves Observations from Geometry feasible region is a convex polyhedron an optimum occurs at a corner point possible algorithm - search over

### 1 Kernel methods & optimization

Machine Learning Class Notes 9-26-13 Prof. David Sontag 1 Kernel methods & optimization One eample of a kernel that is frequently used in practice and which allows for highly non-linear discriminant functions

### Module 13. Working Stress Method. Version 2 CE IIT, Kharagpur

Module 13 Working Stress Method Lesson 35 Numerical Problems Instructional Objectives: At the end of this lesson, the student should be able to: employ the equations established for the analysis and design

### Lecture 6 Simplex method for linear programming

Lecture 6 Simplex method for linear programming Weinan E 1,2 and Tiejun Li 2 1 Department of Mathematics, Princeton University, weinan@princeton.edu 2 School of Mathematical Sciences, Peking University,

### Structured Problems and Algorithms

Integer and quadratic optimization problems Dept. of Engg. and Comp. Sci., Univ. of Cal., Davis Aug. 13, 2010 Table of contents Outline 1 2 3 Benefits of Structured Problems Optimization problems may become

### Prelude 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

### Gujarat University Choice Based Credit System (CBCS) Syllabus for B. Sc. Semester V (Mathematics) MAT 301: Linear Algebra II (Theory) Hours: 4 /week

MAT 301: Linear Algebra II (Theory) Hours: 4 /week Credits: 4 Composition of Linear Maps, The Space L(U,V), Operator Equation, Linear Functional, Dual Spaces, Dual of Dual, Dual Basis Existence Theorem,

### February 17, Simplex Method Continued

15.053 February 17, 2005 Simplex Method Continued 1 Today s Lecture Review of the simplex algorithm. Formalizing the approach Alternative Optimal Solutions Obtaining an initial bfs Is the simplex algorithm

### A NEW APPROACH TO SOLVE AN UNBALANCED ASSIGNMENT PROBLEM

A NEW APPROACH TO SOLVE AN UNBALANCED ASSIGNMENT PROBLEM *Kore B. G. Departmet Of Statistics, Balwat College, VITA - 415 311, Dist.: Sagli (M. S.). Idia *Author for Correspodece ABSTRACT I this paper I

### CO350 Linear Programming Chapter 6: The Simplex Method

CO350 Linear Programming Chapter 6: The Simplex Method 8th June 2005 Chapter 6: The Simplex Method 1 Minimization Problem ( 6.5) We can solve minimization problems by transforming it into a maximization

### Optimeringslära för F (SF1811) / Optimization (SF1841)

Optimeringslära för F (SF1811) / Optimization (SF1841) 1. Information about the course 2. Examples of optimization problems 3. Introduction to linear programming Introduction - Per Enqvist 1 Linear programming

### Chapter 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

### Content Pricing in Peer-to-Peer Networks

Content Pricing in Peer-to-Peer Networks Jaeok Park and Mihaela van der Schaar Electrical Engineering Department, UCLA 21 Workshop on the Economics of Networks, Systems, and Computation (NetEcon 1) October

### ECONOMIC OPTIMALITY. Date: October 10, 2005.

ECONOMIC OPTIMALITY 1. FORMAL STATEMENT OF THE DECISION PROBLEM 1.1. Statement of the problem. ma h(, a) (1) such that G(a) This says that the problem is to maimize the function h which depends on and

### Equations and Inequalities

Equations and Inequalities Figure 1 CHAPTER OUTLINE.1 The Rectangular Coordinate Systems and Graphs. Linear Equations in One Variable.3 Models and Applications. Comple Numbers.5 Quadratic Equations.6 Other

### 16.1 L.P. Duality Applied to the Minimax Theorem

CS787: Advanced Algorithms Scribe: David Malec and Xiaoyong Chai Lecturer: Shuchi Chawla Topic: Minimax Theorem and Semi-Definite Programming Date: October 22 2007 In this lecture, we first conclude our

### A Strongly Polynomial Simplex Method for Totally Unimodular LP

A Strongly Polynomial Simplex Method for Totally Unimodular LP Shinji Mizuno July 19, 2014 Abstract Kitahara and Mizuno get new bounds for the number of distinct solutions generated by the simplex method

### Finite Pivot Algorithms and Feasibility. Bohdan Lubomyr Kaluzny School of Computer Science, McGill University Montreal, Quebec, Canada May 2001

Finite Pivot Algorithms and Feasibility Bohdan Lubomyr Kaluzny School of Computer Science, McGill University Montreal, Quebec, Canada May A thesis submitted to the Faculty of Graduate Studies and Research

### Dual methods and ADMM. Barnabas Poczos & Ryan Tibshirani Convex Optimization /36-725

Dual methods and ADMM Barnabas Poczos & Ryan Tibshirani Convex Optimization 10-725/36-725 1 Given f : R n R, the function is called its conjugate Recall conjugate functions f (y) = max x R n yt x f(x)