# VARIANTS OF THE SIMPLEX METHOD

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 C H A P T E R 6 VARIANTS OF THE SIMPLEX METHOD By a variant of the Simplex Method (in this chapter) we mean an algorithm consisting of a sequence of pivot steps in the primal system using alternative rules for the selection of the pivot Historically these variants were developed to take advantage of a situation where an infeasible basic solution of the primal is available In other applications there often occurs a set of problems differing from one another only in their constant terms and cost factors In such cases, it is convenient to omit Phase I and to use the optimal basis of one problem as the initial basis for the next 61 INTRODUCTION Several methods have been proposed for varying the Simplex Algorithm to reduce the number of iterations This is especially needed for problems involving many equations in order to reduce the computation time It is also needed for problems involving a large number of variables n, for the number of iterations in practice appears to grow roughly proportional to n For example, instead of using the selection rule c s =min c j, one could select j = s such that introducing x s into the basic set gives the largest decrease in the value of z in the next basic solution This requires computing, for c j < 0, the largest in absolute value of c j θ j,whereθ j is determined so that if x j replaced x jr then the solution will remain feasible This rule is obviously not practical when using the revised Simplex Method with multipliers Even using the standard canonical form, considerably more computations would be required per iteration It is possible, however, in the nondegenerate case, to develop a modification of the canonical form in which the coefficient of the ith basic variable is allowed to be different from unity 173

2 174 VARIANTS OF THE SIMPLEX METHOD in the ith equation but b i = 1 In this form the selection of s by the steepest descent criterion would require a little more effort; moreover (by means of a special device), no more effort than that for the standard Simplex Algorithm would be required to maintain the tableau in proper modified form from iteration to iteration (see Section 62 for details) The simplest variant occurs when the new problem differs from the original in the cost coefficients alone In this case, the cost coefficients are replaced by the new ones, and Phase II of the Revised Simplex Method is applied Another possibility is to use the parametric objective method to be described in Section 64 An important variant occurs when the new problem differs from the original in the constant terms only In this case the optimal basis of the first problem will still price out dual feasible, ie, c j 0, for the second, but the associated primal solution may not be feasible For this situation, we could use the Dual-Simplex Algorithm, which is the variant of the standard Primal-Simplex Algorithm, to be discussed in Section 63, or the paramteric right-hand-side method to be described in Section 64, or the Primal-Dual method of Section 66 However, when the problems differ by more than either the constant terms or the cost coefficient terms, the old basis may be neither primal feasible nor dual feasible When neither the basic solution nor the dual solution generated by its simplex multipliers remains feasible, the corresponding algorithm is called composite The Self-Dual parametric algorithm discussed in Section 65 is an example of such a composite algorithm Correspondence of Primal and Dual Bases In 1954, Lemke discovered a certain correspondence between the bases of the primal and dual systems that made it possible to interpret the Simplex Algorithm as applied to the dual as a sequence of basis changes in the primal; this interpretation is called the Dual-Simplex Algorithm From a computational point of view, the Dual- Simplex Algorithm is advantageous because the size of the basis being manipulated in the computer is m m instead of n n In this case, however, the associated basic solutions of the primal are not feasible, but the simplex multipliers continue to price out optimal (hence, yield a basic feasible solution to the dual) It is good to understand the details of this correspondence, for it provides a means of easily dualizing a problem without transposing the constraint matrix Consider the standard form Minimize c T x = z subject to Ax = b, A : m n, x 0, (61) and the dual of the standard form: Maximize b T π = v subject to A T π c, A : m n, (62)

3 61 INTRODUCTION 175 PRIMAL-DUAL CORRESPONDENCES Primal Dual ( ) B Basis B B T 0 = N T I Basic variables x B π, y N = c N Nonbasic variables x N y B = c B Feasibility condition Ax = b, x 0 c 0 Table 6-1: Primal-Dual Correspondences where π i is unrestricted in sign We assume A is full rank Adding slack variables y 0 to the dual problem (62) we get: Maximize b T π = v subject to A T π + Iy = c, A : m n, y 0 (63) Clearly the dual has n basic variables and m nonbasic variables The variables π areunrestrictedinsignand,whena is full rank, always constitute m out of the n basic variables of the dual The basis for the primal is denoted by B and the nonbasic columns are denoted by N Note that y = c A T π = c 0 when the dual is feasible and that c B =0and c N 0 when the primal is optimal The basic and nonbasic columns for the dual are denoted by B, ann n matrix, and N, an n m matrix ( ) ( B T 0 Im B = N T, N = I n m 0 ), (64) where I n m is an (n m)-dimensional identity matrix and I m is an m-dimensional identity matrix Thus, (π, y N ) as basic variables and y B as nonbasic variables constitute a basic solution to the dual if and only if y B =0 Exercise 61 Show that the determinant of B has the same value as that of B Also show that if B 1 exists, then B 1 exists It is now clear that there is a correspondence between primal and dual bases These correspondences are shown in Table 6-1 With these correspondences in mind, we shall discuss variations of the Simplex Method Exercise 62 How is the concept of complementary primal and dual variables related to the correspondence of primal and dual bases? Definition (Dual Degeneracy): We have already defined degeneracy and nondegeneracy with respect to the primal A basis is said to be dual degenerate

4 176 VARIANTS OF THE SIMPLEX METHOD if one or more of the c j corresponding to nonbasic variables x j are zero and to be dual nondegenerate otherwise 62 MAX IMPROVEMENT PER ITERATION In this section we present an alternative canonical form for efficiently determining the incoming column that yields the maximum improvement per iteration It has been observed on many test problems that this often leads to fewer iterations We will assume, to simplify the discussion, that all basic feasible solutions that are generated are nondegenerate Assume x j 0forj =1,,n and x j for j =1,,m are basic variables, in the standard canonical form: z + c m+1 x m c j x j + + c n x n = z 0 x 1 + ā 1,m+1 x m ā 1j x j + + ā 1n x n = b 1 x p + ā p,m+1 x m ā pj x j + + ā pn x n = b p x m +ā m,m+1 x m+1 + +ā mj x j + +ā mn x n = b m (65) where ā ij, c j, b i,and z 0 are constants and we assume that b i > 0fori =1,,mIf there are two or more c j < 0, our problem is to find which j = s among them has the property that putting column s into the basis and driving some column j = r out of the basis gives the greatest decrease in z while preserving feasibility Using the standard canonical form (65), the main work is thatof performing ratiotests for the several columns j such that c j < 0, plus the update work of pivoting, which takes mn operations where each operation consists of one addition (or subtraction) plus one multiplication To do this more efficiently, consider the alternative canonical form: z + c m+1 x m c n x n = z 0 α 11 x 1 + α 1,m+1 x m α 1n x n = 1 α pp x p + α p,m+1 x m α pn x n = 1 α mm x m + α m,m+1 x m α mn x n = 1 (66) formed by rescaling the rows by dividing them by b i > 0 for all i =1,,m In this format we scan column j such that c j < 0 and find for each such j the row r j =argmax α ij

5

### Section Notes 5. Review of Linear Programming. Applied Math / Engineering Sciences 121. Week of October 15, 2017

Section Notes 5 Review of Linear Programming Applied Math / Engineering Sciences 121 Week of October 15, 2017 The following list of topics is an overview of the material that was covered in the lectures

### 4.1 Graphical solution of a linear program and standard form

4.1 Graphical solution of a linear program and standard form Consider the problem min c T x Ax b x where x = ( x1 x ) ( 16, c = 5 ), b = 4 5 9, A = 1 7 1 5 1. Solve the problem graphically and determine

### 5.4 Pure Minimal Cost Flow

Pure Minimal Cost Flow Problem. Pure Minimal Cost Flow Networks are especially convenient for modeling because of their simple nonmathematical structure that can be easily portrayed with a graph. This

### 5. DUAL LP, SOLUTION INTERPRETATION, AND POST-OPTIMALITY

5. DUAL LP, SOLUTION INTERPRETATION, AND POST-OPTIMALITY 5.1 DUALITY Associated with every linear programming problem (the primal) is another linear programming problem called its dual. If the primal involves

### AM 121: Intro to Optimization Models and Methods Fall 2017

AM 121: Intro to Optimization Models and Methods Fall 2017 Lecture 10: Dual Simplex Yiling Chen SEAS Lesson Plan Interpret primal simplex in terms of pivots on the corresponding dual tableau Dictionaries

### The Simplex Algorithm

The Simplex Algorithm Uri Feige November 2011 1 The simplex algorithm The simplex algorithm was designed by Danzig in 1947. This write-up presents the main ideas involved. It is a slight update (mostly

### BCN Decision and Risk Analysis. Syed M. Ahmed, Ph.D.

Linear Programming Module Outline Introduction The Linear Programming Model Examples of Linear Programming Problems Developing Linear Programming Models Graphical Solution to LP Problems The Simplex Method

### Graphs that have the feasible bases of a given linear

Algorithmic Operations Research Vol.1 (2006) 46 51 Simplex Adjacency Graphs in Linear Optimization Gerard Sierksma and Gert A. Tijssen University of Groningen, Faculty of Economics, P.O. Box 800, 9700

### Optimization of Design. Lecturer:Dung-An Wang Lecture 8

Optimization of Design Lecturer:Dung-An Wang Lecture 8 Lecture outline Reading: Ch8 of text Today s lecture 2 8.1 LINEAR FUNCTIONS Cost Function Constraints 3 8.2 The standard LP problem Only equality

### CASE STUDY. fourteen. Animating The Simplex Method. case study OVERVIEW. Application Overview and Model Development.

CASE STUDY fourteen Animating The Simplex Method case study OVERVIEW CS14.1 CS14.2 CS14.3 CS14.4 CS14.5 CS14.6 CS14.7 Application Overview and Model Development Worksheets User Interface Procedures Re-solve

### MATLAB Solution of Linear Programming Problems

MATLAB Solution of Linear Programming Problems The simplex method is included in MATLAB using linprog function. All is needed is to have the problem expressed in the terms of MATLAB definitions. Appendix

### 3 INTEGER LINEAR PROGRAMMING

3 INTEGER LINEAR PROGRAMMING PROBLEM DEFINITION Integer linear programming problem (ILP) of the decision variables x 1,..,x n : (ILP) subject to minimize c x j j n j= 1 a ij x j x j 0 x j integer n j=

### Generalized Network Flow Programming

Appendix C Page Generalized Network Flow Programming This chapter adapts the bounded variable primal simplex method to the generalized minimum cost flow problem. Generalized networks are far more useful

### Discrete Optimization. Lecture Notes 2

Discrete Optimization. Lecture Notes 2 Disjunctive Constraints Defining variables and formulating linear constraints can be straightforward or more sophisticated, depending on the problem structure. The

### 5 The Theory of the Simplex Method

5 The Theory of the Simplex Method Chapter 4 introduced the basic mechanics of the simplex method. Now we shall delve a little more deeply into this algorithm by examining some of its underlying theory.

### Linear Programming. Revised Simplex Method, Duality of LP problems and Sensitivity analysis

Linear Programming Revised Simple Method, Dualit of LP problems and Sensitivit analsis Introduction Revised simple method is an improvement over simple method. It is computationall more efficient and accurate.

### maximize minimize b T y subject to A T y c, 0 apple y. The problem on the right is in standard form so we can take its dual to get the LP c T x

4 Duality Theory Recall from Section that the dual to an LP in standard form (P) is the LP maximize subject to c T x Ax apple b, apple x (D) minimize b T y subject to A T y c, apple y. Since the problem

Iterative Improvement Algorithm design technique for solving optimization problems Start with a feasible solution Repeat the following step until no improvement can be found: change the current feasible

### 16.410/413 Principles of Autonomy and Decision Making

16.410/413 Principles of Autonomy and Decision Making Lecture 17: The Simplex Method Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 10, 2010 Frazzoli (MIT)

### Math Introduction to Operations Research

Math 300 Introduction to Operations Research Examination (50 points total) Solutions. (6 pt total) Consider the following linear programming problem: Maximize subject to and x, x, x 3 0. 3x + x + 5x 3

### Introductory Operations Research

Introductory Operations Research Theory and Applications Bearbeitet von Harvir Singh Kasana, Krishna Dev Kumar 1. Auflage 2004. Buch. XI, 581 S. Hardcover ISBN 978 3 540 40138 4 Format (B x L): 15,5 x

### A Generalized Model for Fuzzy Linear Programs with Trapezoidal Fuzzy Numbers

J. Appl. Res. Ind. Eng. Vol. 4, No. 1 (017) 4 38 Journal of Applied Research on Industrial Engineering www.journal-aprie.com A Generalized Model for Fuzzy Linear Programs with Trapezoidal Fuzzy Numbers

### Tuesday, April 10. The Network Simplex Method for Solving the Minimum Cost Flow Problem

. Tuesday, April The Network Simplex Method for Solving the Minimum Cost Flow Problem Quotes of the day I think that I shall never see A poem lovely as a tree. -- Joyce Kilmer Knowing trees, I understand

### Linear programming and duality theory

Linear programming and duality theory Complements of Operations Research Giovanni Righini Linear Programming (LP) A linear program is defined by linear constraints, a linear objective function. Its variables

### How to Solve a Standard Maximization Problem Using the Simplex Method and the Rowops Program

How to Solve a Standard Maximization Problem Using the Simplex Method and the Rowops Program Problem: Maximize z = x + 0x subject to x + x 6 x + x 00 with x 0 y 0 I. Setting Up the Problem. Rewrite each

### Part 1. The Review of Linear Programming The Revised Simplex Method

In the name of God Part 1. The Review of Linear Programming 1.4. Spring 2010 Instructor: Dr. Masoud Yaghini Introduction Outline in Tableau Format Comparison Between the Simplex and the Revised Simplex

### David G. Luenberger Yinyu Ye. Linear and Nonlinear. Programming. Fourth Edition. ö Springer

David G. Luenberger Yinyu Ye Linear and Nonlinear Programming Fourth Edition ö Springer Contents 1 Introduction 1 1.1 Optimization 1 1.2 Types of Problems 2 1.3 Size of Problems 5 1.4 Iterative Algorithms

### A Simplex Based Parametric Programming Method for the Large Linear Programming Problem

A Simplex Based Parametric Programming Method for the Large Linear Programming Problem Huang, Rujun, Lou, Xinyuan Abstract We present a methodology of parametric objective function coefficient programming

### MA4254: Discrete Optimization. Defeng Sun. Department of Mathematics National University of Singapore Office: S Telephone:

MA4254: Discrete Optimization Defeng Sun Department of Mathematics National University of Singapore Office: S14-04-25 Telephone: 6516 3343 Aims/Objectives: Discrete optimization deals with problems of

### COLUMN GENERATION IN LINEAR PROGRAMMING

COLUMN GENERATION IN LINEAR PROGRAMMING EXAMPLE: THE CUTTING STOCK PROBLEM A certain material (e.g. lumber) is stocked in lengths of 9, 4, and 6 feet, with respective costs of \$5, \$9, and \$. An order for

### Lecture 9: Linear Programming

Lecture 9: Linear Programming A common optimization problem involves finding the maximum of a linear function of N variables N Z = a i x i i= 1 (the objective function ) where the x i are all non-negative

### Math 5593 Linear Programming Lecture Notes

Math 5593 Linear Programming Lecture Notes Unit II: Theory & Foundations (Convex Analysis) University of Colorado Denver, Fall 2013 Topics 1 Convex Sets 1 1.1 Basic Properties (Luenberger-Ye Appendix B.1).........................

### Linear Programming. Larry Blume. Cornell University & The Santa Fe Institute & IHS

Linear Programming Larry Blume Cornell University & The Santa Fe Institute & IHS Linear Programs The general linear program is a constrained optimization problem where objectives and constraints are all

### Lecture notes on Transportation and Assignment Problem (BBE (H) QTM paper of Delhi University)

Transportation and Assignment Problems The transportation model is a special class of linear programs. It received this name because many of its applications involve determining how to optimally transport

### Pivot Rules for Linear Programming: A Survey on Recent Theoretical Developments

Pivot Rules for Linear Programming: A Survey on Recent Theoretical Developments T. Terlaky and S. Zhang November 1991 1 Address of the authors: Tamás Terlaky 1 Faculty of Technical Mathematics and Computer

### 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 1.0 Introduction Linear programming

### Discrete Optimization 2010 Lecture 5 Min-Cost Flows & Total Unimodularity

Discrete Optimization 2010 Lecture 5 Min-Cost Flows & Total Unimodularity Marc Uetz University of Twente m.uetz@utwente.nl Lecture 5: sheet 1 / 26 Marc Uetz Discrete Optimization Outline 1 Min-Cost Flows

### CSE 40/60236 Sam Bailey

CSE 40/60236 Sam Bailey Solution: any point in the variable space (both feasible and infeasible) Cornerpoint solution: anywhere two or more constraints intersect; could be feasible or infeasible Feasible

### DEGENERACY AND THE FUNDAMENTAL THEOREM

DEGENERACY AND THE FUNDAMENTAL THEOREM The Standard Simplex Method in Matrix Notation: we start with the standard form of the linear program in matrix notation: (SLP) m n we assume (SLP) is feasible, and

### TRANSPORTATION AND ASSIGNMENT PROBLEMS

TRANSPORTATION AND ASSIGNMENT PROBLEMS Transportation problem Example P&T Company produces canned peas. Peas are prepared at three canneries (Bellingham, Eugene and Albert Lea). Shipped by truck to four

### A modification of the simplex method reducing roundoff errors

A modification of the simplex method reducing roundoff errors Ralf Ulrich Kern Georg-Simon-Ohm-Hochschule Nürnberg Fakultät Informatik Postfach 21 03 20 90121 Nürnberg Ralf-Ulrich.Kern@ohm-hochschule.de

### Recap, and outline of Lecture 18

Recap, and outline of Lecture 18 Previously Applications of duality: Farkas lemma (example of theorems of alternative) A geometric view of duality Degeneracy and multiple solutions: a duality connection

### 2 Computation with Floating-Point Numbers

2 Computation with Floating-Point Numbers 2.1 Floating-Point Representation The notion of real numbers in mathematics is convenient for hand computations and formula manipulations. However, real numbers

### A PARAMETRIC SIMPLEX METHOD FOR OPTIMIZING A LINEAR FUNCTION OVER THE EFFICIENT SET OF A BICRITERIA LINEAR PROBLEM. 1.

ACTA MATHEMATICA VIETNAMICA Volume 21, Number 1, 1996, pp. 59 67 59 A PARAMETRIC SIMPLEX METHOD FOR OPTIMIZING A LINEAR FUNCTION OVER THE EFFICIENT SET OF A BICRITERIA LINEAR PROBLEM NGUYEN DINH DAN AND

### Balinski-Tucker Simplex Tableaus: Dimensions, Degeneracy Degrees, and Interior Points of Optimal Faces

Balinski-Tucker Simplex Tableaus: Dimensions, Degeneracy Degrees, and Interior Points of Optimal Faces Gert A Tijssen and Gerard Sierksma Department of Econometrics University of Groningen, The Netherlands

### Column Generation: Cutting Stock

Column Generation: Cutting Stock A very applied method thst@man.dtu.dk DTU-Management Technical University of Denmark 1 Outline History The Simplex algorithm (re-visited) Column Generation as an extension

### MLR Institute of Technology

Course Name : Engineering Optimization Course Code : 56021 Class : III Year Branch : Aeronautical Engineering Year : 2014-15 Course Faculty : Mr Vamsi Krishna Chowduru, Assistant Professor Course Objective

### The simplex method and the diameter of a 0-1 polytope

The simplex method and the diameter of a 0-1 polytope Tomonari Kitahara and Shinji Mizuno May 2012 Abstract We will derive two main results related to the primal simplex method for an LP on a 0-1 polytope.

### Chap5 The Theory of the Simplex Method

College of Management, NCTU Operation Research I Fall, Chap The Theory of the Simplex Method Terminology Constraint oundary equation For any constraint (functional and nonnegativity), replace its,, sign

### Selected Topics in Column Generation

Selected Topics in Column Generation February 1, 2007 Choosing a solver for the Master Solve in the dual space(kelly s method) by applying a cutting plane algorithm In the bundle method(lemarechal), a

### Mid-term Exam of Operations Research

Mid-term Exam of Operations Research Economics and Management School, Wuhan University November 10, 2016 ******************************************************************************** Rules: 1. No electronic

### Programs. Introduction

16 Interior Point I: Linear Programs Lab Objective: For decades after its invention, the Simplex algorithm was the only competitive method for linear programming. The past 30 years, however, have seen

### Journal of Business & Economics Research November, 2009 Volume 7, Number 11

Alternate Solutions Analysis For Transportation Problems Veena Adlakha, University of Baltimore, USA Krzysztof Kowalski, Connecticut Department of Transportation, USA ABSTRACT The constraint structure

### Using Ones Assignment Method and. Robust s Ranking Technique

Applied Mathematical Sciences, Vol. 7, 2013, no. 113, 5607-5619 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.37381 Method for Solving Fuzzy Assignment Problem Using Ones Assignment

### CSc 545 Lecture topic: The Criss-Cross method of Linear Programming

CSc 545 Lecture topic: The Criss-Cross method of Linear Programming Wanda B. Boyer University of Victoria November 21, 2012 Presentation Outline 1 Outline 2 3 4 Please note: I would be extremely grateful

### Other Algorithms for Linear Programming Chapter 7

Other Algorithms for Linear Programming Chapter 7 Introduction The Simplex method is only a part of the arsenal of algorithms regularly used by LP practitioners. Variants of the simplex method the dual

### Ellipsoid Algorithm :Algorithms in the Real World. Ellipsoid Algorithm. Reduction from general case

Ellipsoid Algorithm 15-853:Algorithms in the Real World Linear and Integer Programming II Ellipsoid algorithm Interior point methods First polynomial-time algorithm for linear programming (Khachian 79)

### LECTURE 6: INTERIOR POINT METHOD. 1. Motivation 2. Basic concepts 3. Primal affine scaling algorithm 4. Dual affine scaling algorithm

LECTURE 6: INTERIOR POINT METHOD 1. Motivation 2. Basic concepts 3. Primal affine scaling algorithm 4. Dual affine scaling algorithm Motivation Simplex method works well in general, but suffers from exponential-time

### Chapter 4: The Mechanics of the Simplex Method

Chapter 4: The Mechanics of the Simplex Method The simplex method is a remarkably simple and elegant algorithmic engine for solving linear programs. In this chapter we will examine the internal mechanics

### J Linear Programming Algorithms

Simplicibus itaque verbis gaudet Mathematica Veritas, cum etiam per se simplex sit Veritatis oratio. [And thus Mathematical Truth prefers simple words, because the language of Truth is itself simple.]

### Cluster Analysis. Mu-Chun Su. Department of Computer Science and Information Engineering National Central University 2003/3/11 1

Cluster Analysis Mu-Chun Su Department of Computer Science and Information Engineering National Central University 2003/3/11 1 Introduction Cluster analysis is the formal study of algorithms and methods

### Application of Bounded Variable Simplex Algorithm in Solving Maximal Flow Model

Dhaka Univ. J. Sci. (): 9-, 3 (January) Application of Bounded Variable Simplex Algorithm in Solving Maximal Flow Model Sohana Jahan, Marzia Yesmin and Fatima Tuj Jahra Department of Mathematics,University

### CSCI 5654 (Fall 2013): Network Simplex Algorithm for Transshipment Problems.

CSCI 5654 (Fall 21): Network Simplex Algorithm for Transshipment Problems. Sriram Sankaranarayanan November 14, 21 1 Introduction e will present the basic ideas behind the network Simplex algorithm for

### Addressing degeneracy in the dual simplex algorithm using a decompositon approach

Addressing degeneracy in the dual simplex algorithm using a decompositon approach Ambros Gleixner, Stephen J Maher, Matthias Miltenberger Zuse Institute Berlin Berlin, Germany 16th July 2015 @sj_maher

### Unit.9 Integer Programming

Unit.9 Integer Programming Xiaoxi Li EMS & IAS, Wuhan University Dec. 22-29, 2016 (revised) Operations Research (Li, X.) Unit.9 Integer Programming Dec. 22-29, 2016 (revised) 1 / 58 Organization of this

### PRIMAL-DUAL INTERIOR POINT METHOD FOR LINEAR PROGRAMMING. 1. Introduction

PRIMAL-DUAL INTERIOR POINT METHOD FOR LINEAR PROGRAMMING KELLER VANDEBOGERT AND CHARLES LANNING 1. Introduction Interior point methods are, put simply, a technique of optimization where, given a problem

### Introduction to Homogeneous coordinates

Last class we considered smooth translations and rotations of the camera coordinate system and the resulting motions of points in the image projection plane. These two transformations were expressed mathematically

### Approximation Algorithms

Approximation Algorithms Group Members: 1. Geng Xue (A0095628R) 2. Cai Jingli (A0095623B) 3. Xing Zhe (A0095644W) 4. Zhu Xiaolu (A0109657W) 5. Wang Zixiao (A0095670X) 6. Jiao Qing (A0095637R) 7. Zhang

### 6. Lecture notes on matroid intersection

Massachusetts Institute of Technology 18.453: Combinatorial Optimization Michel X. Goemans May 2, 2017 6. Lecture notes on matroid intersection One nice feature about matroids is that a simple greedy algorithm

### 5. Consider the tree solution for a minimum cost network flow problem shown in Figure 5.

FIGURE 1. Data for a network flow problem. As usual the numbers above the nodes are supplies (negative values represent demands) and numbers shown above the arcs are unit shipping costs. The darkened arcs

### Linear Programming has been used to:

Linear Programming Linear programming became important during World War II: used to solve logistics problems for the military. Linear Programming (LP) was the first widely used form of optimization in

### Linear Programming MT 4.0

Linear Programming MT 4.0 for GAUSS TM Mathematical and Statistical System Aptech Systems, Inc. Information in this document is subject to change without notice and does not represent a commitment on the

### A Comparative study on Algorithms for Shortest-Route Problem and Some Extensions

International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: No: 0 A Comparative study on Algorithms for Shortest-Route Problem and Some Extensions Sohana Jahan, Md. Sazib Hasan Abstract-- The shortest-route

### Linear Optimization and Extensions: Theory and Algorithms

AT&T Linear Optimization and Extensions: Theory and Algorithms Shu-Cherng Fang North Carolina State University Sarai Puthenpura AT&T Bell Labs Prentice Hall, Englewood Cliffs, New Jersey 07632 Contents

### SUBSTITUTING GOMORY CUTTING PLANE METHOD TOWARDS BALAS ALGORITHM FOR SOLVING BINARY LINEAR PROGRAMMING

Bulletin of Mathematics Vol. 06, No. 0 (20), pp.. SUBSTITUTING GOMORY CUTTING PLANE METHOD TOWARDS BALAS ALGORITHM FOR SOLVING BINARY LINEAR PROGRAMMING Eddy Roflin, Sisca Octarina, Putra B. J Bangun,

### COVERING POINTS WITH AXIS PARALLEL LINES. KAWSAR JAHAN Bachelor of Science, Bangladesh University of Professionals, 2009

COVERING POINTS WITH AXIS PARALLEL LINES KAWSAR JAHAN Bachelor of Science, Bangladesh University of Professionals, 2009 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge

### Floating-point representation

Lecture 3-4: Floating-point representation and arithmetic Floating-point representation The notion of real numbers in mathematics is convenient for hand computations and formula manipulations. However,

### Ahigh school curriculum in Algebra 2 contains both solving systems of linear equations,

The Simplex Method for Systems of Linear Inequalities Todd O. Moyer, Towson University Abstract: This article details the application of the Simplex Method for an Algebra 2 class. Students typically learn

### AH Matrices.notebook November 28, 2016

Matrices Numbers are put into arrays to help with multiplication, division etc. A Matrix (matrices pl.) is a rectangular array of numbers arranged in rows and columns. Matrices If there are m rows and

### x = 12 x = 12 1x = 16

2.2 - The Inverse of a Matrix We've seen how to add matrices, multiply them by scalars, subtract them, and multiply one matrix by another. The question naturally arises: Can we divide one matrix by another?

### A method for unbalanced transportation problems in fuzzy environment

Sādhanā Vol. 39, Part 3, June 2014, pp. 573 581. c Indian Academy of Sciences A method for unbalanced transportation problems in fuzzy environment 1. Introduction DEEPIKA RANI 1,, T R GULATI 1 and AMIT

### Outline: Finish uncapacitated simplex method Negative cost cycle algorithm The max-flow problem Max-flow min-cut theorem

Outline: Finish uncapacitated simplex method Negative cost cycle algorithm The max-flow problem Max-flow min-cut theorem Uncapacitated Networks: Basic primal and dual solutions Flow conservation constraints

### What is the Worst Case Behavior of the Simplex Algorithm?

Centre de Recherches Mathématiques CRM Proceedings and Lecture Notes Volume, 28 What is the Worst Case Behavior of the Simplex Algorithm? Norman Zadeh Abstract. The examples published by Klee and Minty

### Problem set 2. Problem 1. Problem 2. Problem 3. CS261, Winter Instructor: Ashish Goel.

CS261, Winter 2017. Instructor: Ashish Goel. Problem set 2 Electronic submission to Gradescope due 11:59pm Thursday 2/16. Form a group of 2-3 students that is, submit one homework with all of your names.

### Approximation Algorithms: The Primal-Dual Method. My T. Thai

Approximation Algorithms: The Primal-Dual Method My T. Thai 1 Overview of the Primal-Dual Method Consider the following primal program, called P: min st n c j x j j=1 n a ij x j b i j=1 x j 0 Then the

### MATH 890 HOMEWORK 2 DAVID MEREDITH

MATH 890 HOMEWORK 2 DAVID MEREDITH (1) Suppose P and Q are polyhedra. Then P Q is a polyhedron. Moreover if P and Q are polytopes then P Q is a polytope. The facets of P Q are either F Q where F is a facet

### Solving the Linear Transportation Problem by Modified Vogel Method

Solving the Linear Transportation Problem by Modified Vogel Method D. Almaatani, S.G. Diagne, Y. Gningue and P. M. Takouda Abstract In this chapter, we propose a modification of the Vogel Approximation

### One-mode Additive Clustering of Multiway Data

One-mode Additive Clustering of Multiway Data Dirk Depril and Iven Van Mechelen KULeuven Tiensestraat 103 3000 Leuven, Belgium (e-mail: dirk.depril@psy.kuleuven.ac.be iven.vanmechelen@psy.kuleuven.ac.be)

### Solving Linear and Integer Programs

Solving Linear and Integer Programs Robert E. Bixby Gurobi Optimization, Inc. and Rice University Overview Linear Programming: Example and introduction to basic LP, including duality Primal and dual simplex

### Operations Research. Lecture Notes By Prof A K Saxena Professor and Head Dept of CSIT G G Vishwavidyalaya, Bilaspur-India

Lecture Notes By Prof A K Saxena Professor and Head Dept of CSIT G G Vishwavidyalaya, Bilaspur-India Some important tips before start of course material to students Mostly we followed Book by S D Sharma,

### MINET: Fast Network LP Solver. Description and User's Guide for V2.00

MINET: Fast Network LP Solver. Description and User's Guide for V2.00 Maros, I. IIASA Working Paper WP-88-006 January 1988 Maros, I. (1988) MINET: Fast Network LP Solver. Description and User's Guide for

### Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood

PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720-E October 1998 \$3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced

### Math Week in Review #5

Math 141 Spring 2006 c Heather Ramsey Page 1 Math 141 - Week in Review #5 Section 4.1 - Simplex Method for Standard Maximization Problems A standard maximization problem is a linear programming problem

### Lecture 16 October 23, 2014

CS 224: Advanced Algorithms Fall 2014 Prof. Jelani Nelson Lecture 16 October 23, 2014 Scribe: Colin Lu 1 Overview In the last lecture we explored the simplex algorithm for solving linear programs. While

### A New Product Form of the Inverse

Applied Mathematical Sciences, Vol 6, 2012, no 93, 4641-4650 A New Product Form of the Inverse Souleymane Sarr Department of mathematics and computer science University Cheikh Anta Diop Dakar, Senegal

### Algorithmic Game Theory and Applications. Lecture 6: The Simplex Algorithm

Algorithmic Game Theory and Applications Lecture 6: The Simplex Algorithm Kousha Etessami Recall our example 1 x + y

### HW 1: Project Report (Camera Calibration)

HW 1: Project Report (Camera Calibration) ABHISHEK KUMAR (abhik@sci.utah.edu) 1 Problem The problem is to calibrate a camera for a fixed focal length using two orthogonal checkerboard planes, and to find

### 22. LECTURE 22. I can define critical points. I know the difference between local and absolute minimums/maximums.

. LECTURE Objectives I can define critical points. I know the difference between local and absolute minimums/maximums. In many physical problems, we re interested in finding the values (x, y) that maximize