Optimization in R n Introduction

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Optimization in R n Introduction"

Transcription

1 Optimization in R n Introduction Rudi Pendavingh Eindhoven Technical University Optimization in R n, lecture Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4

2 Some optimization problems designing a cylindrical can containing liter, with minimum surface area designing a closed spatial object of volume, with minimum surface area finding the shortest path between two cities in a road network finding the shortest path between two cities in a landscape finding a maximum number of cards without a set in the SET game max{n Z x n + y n = z n for some x, y, z Z \ {,, } } Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 2 / 4

3 Optimization An optimization problem is of the form min{f (x) x S} where S is a set and f : S R is a function. S is the feasible set f : S R is the objective function if S =, then the problem is infeasible if inf{f (x) x S} =, the problem is unbounded if y S is such that f (y) = inf{f (x) x S}, then y is an optimal solution, f (y) is the optimum, then the optimum is attained by y. Solving the optimization problem means finding an optimal solution y, or deciding that the problem is infeasible, unbounded or that the optimum is not attained. Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 3 / 4

4 Constrained optimization in R n These problems are of the form So here the feasible set is Typical constraints are min{f (x) x R n satisfies constraints} S := {x R n x satisfies constraints }. inequality constraints: g(x) for some g : R n R. In particular, linear inequalities like 3x + 5x 2 + 2x n 7 equality constraints: h(x) = for some h : R n R. In particular, linear equations like 3x + 5x 2 + 2x n = 7 integrality constraints: x Z n. The entries x,..., x n R of x are called decision variables Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 4 / 4

5 Linear optimization A linear optimization problem has a linear objective and linear inequality constraints, e.g. max{cx Ax b, x R n } where A is a matrix, b a column vector, c a row vector. cx=d a a 2 c a 5 a 3 a4 Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 5 / 4

6 The Diet problem given: n foods items, m nutrients, with the j-th food item containing a ij units of nutrient i per unit. Required units of nutrient i: b i, Costs per unit of food j: c j. find: amounts x j of each food j so that the intake of each nutrient is sufficient and so that total costs are minimal. This is a linear optimization problem min{ j c j x j j a ij x j b i for all i, x } or min{cx Ax b, x }. Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 6 / 4

7 Integer linear optimization Integer linear optimization is like linear optimization, but with integrality constraints: max{cx Ax b, x Z n } a 3 a4 a 5 c cx=d a 2 a Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 7 / 4

8 The Knapsack problem given: n items; weight of i-th item :a i R; maximum weight b R; value of i-th item c i R find: set of items X {,..., n} of total weight b and maximizing the total value. This is an integer linear optimization problem max{ c i x i a i x i b, x i for each i, x Z n } i i or max{cx ax b, x, x Z n }. Note: i X x i = Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 8 / 4

9 Convex optimization A convex optimization problem is a problem of the form min{f (x) g (x),..., g n (x), x R n } where f, g i : R n R are convex functions f(x)=d S Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 9 / 4

10 Certifying optimality Consider the linear optimization problem max{cx Ax b, x R n }. If ˆx R n is a feasible solution, then ˆx is optimal max{cx Ax b, x R n } cˆx there is no x R n such that Ax b, cx > cˆx How to show that a system of linear inequalities has no solution? Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4

11 Fredholms Alternative Theorem Let A be an m n matrix, and let b R m. Either: (i) there exists an column vector x R n so that Ax = b; or (ii) there exists a row vector y R m so that ya = and yb =. In other words, there are x,... x n R such that a x + + a n x n = b... a m x + + a mn x n = b m or there exist y,..., y m R such that y (a x + + a n x n = b ).... y m (a m x + + a mn x n = b m ) + x + + x n =. Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4

12 Farkas Lemma Theorem Let A be an m n matrix, and let b R m. Either: (i) there exists a column vector x R n so that Ax b; or (ii) there exists a row vector y R m so that y, ya = and yb <. A variant of this theorem is Theorem Let A be an m n matrix, and let b R m. Either: (i) there exists a column vector x R n so that Ax = b and x ; or (ii) there exists a row vector y R m so that ya and yb <. Farkas Lemma is the key to understanding linear optimization problems. Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 2 / 4

13 Kroneckers Approximation Theorem The following theorem deals with diophantine linear equations. Theorem Let A be an m n rational matrix, and let b Q m. Then either (i) there exists a column vector x Z n such that Ax = b; or (ii) there exists a row vector y Q m such that ya Z n and yb Z. To handle integer linear optimization problems, we would need a similar theorem for diophantine linear inequalities. Such a theorem is known only for very special matrices A. Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 3 / 4

14 Homework, AIMMS Weekly schedule, homework: rudi/2wo.html To download AIMMS: login: aimms password: AIMMS for TU/e Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 4 / 4

4.6 Linear Programming duality

4.6 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 information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

More information

Definition of a Linear Program

Definition of a Linear Program Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

More information

International Doctoral School Algorithmic Decision Theory: MCDA and MOO

International Doctoral School Algorithmic Decision Theory: MCDA and MOO International Doctoral School Algorithmic Decision Theory: MCDA and MOO Lecture 2: Multiobjective Linear Programming Department of Engineering Science, The University of Auckland, New Zealand Laboratoire

More information

The Equivalence of Linear Programs and Zero-Sum Games

The Equivalence of Linear Programs and Zero-Sum Games The Equivalence of Linear Programs and Zero-Sum Games Ilan Adler, IEOR Dep, UC Berkeley adler@ieor.berkeley.edu Abstract In 1951, Dantzig showed the equivalence of linear programming problems and two-person

More information

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming

More information

Chapter 4. Duality. 4.1 A Graphical Example

Chapter 4. Duality. 4.1 A Graphical Example Chapter 4 Duality Given any linear program, there is another related linear program called the dual. In this chapter, we will develop an understanding of the dual linear program. This understanding translates

More information

IEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2

IEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2 IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3

More information

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where. Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

More information

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013 Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

More information

LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method

LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right

More information

Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

More information

Proximal mapping via network optimization

Proximal mapping via network optimization L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

More information

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued. Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.

More information

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

More information

Actually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York.

Actually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York. 1: 1. Compute a random 4-dimensional polytope P as the convex hull of 10 random points using rand sphere(4,10). Run VISUAL to see a Schlegel diagram. How many 3-dimensional polytopes do you see? How many

More information

Inverse Optimization by James Orlin

Inverse Optimization by James Orlin Inverse Optimization by James Orlin based on research that is joint with Ravi Ahuja Jeopardy 000 -- the Math Programming Edition The category is linear objective functions The answer: When you maximize

More information

What is Linear Programming?

What is Linear Programming? Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

More information

Linear Programming. March 14, 2014

Linear Programming. March 14, 2014 Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

More information

NP-Hardness Results Related to PPAD

NP-Hardness Results Related to PPAD NP-Hardness Results Related to PPAD Chuangyin Dang Dept. of Manufacturing Engineering & Engineering Management City University of Hong Kong Kowloon, Hong Kong SAR, China E-Mail: mecdang@cityu.edu.hk Yinyu

More information

3. Linear Programming and Polyhedral Combinatorics

3. Linear Programming and Polyhedral Combinatorics Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

More information

Some representability and duality results for convex mixed-integer programs.

Some representability and duality results for convex mixed-integer programs. Some representability and duality results for convex mixed-integer programs. Santanu S. Dey Joint work with Diego Morán and Juan Pablo Vielma December 17, 2012. Introduction About Motivation Mixed integer

More information

Lecture 1: Systems of Linear Equations

Lecture 1: Systems of Linear Equations MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

More information

Nonlinear Optimization: Algorithms 3: Interior-point methods

Nonlinear Optimization: Algorithms 3: Interior-point methods Nonlinear Optimization: Algorithms 3: Interior-point methods INSEAD, Spring 2006 Jean-Philippe Vert Ecole des Mines de Paris Jean-Philippe.Vert@mines.org Nonlinear optimization c 2006 Jean-Philippe Vert,

More information

Two-Stage Stochastic Linear Programs

Two-Stage Stochastic Linear Programs Two-Stage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 Two-Stage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic

More information

This exposition of linear programming

This exposition of linear programming Linear Programming and the Simplex Method David Gale This exposition of linear programming and the simplex method is intended as a companion piece to the article in this issue on the life and work of George

More information

. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2

. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2 4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the

More information

Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints

Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and

More information

24. The Branch and Bound Method

24. The Branch and Bound Method 24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

More information

Contents. What is Wirtschaftsmathematik?

Contents. What is Wirtschaftsmathematik? Contents. Introduction Modeling cycle SchokoLeb example Graphical procedure Standard-Form of Linear Program Vorlesung, Lineare Optimierung, Sommersemester 04 Page What is Wirtschaftsmathematik? Using mathematical

More information

constraint. Let us penalize ourselves for making the constraint too big. We end up with a

constraint. Let us penalize ourselves for making the constraint too big. We end up with a Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the

More information

Lecture 5/6. Enrico Bini

Lecture 5/6. Enrico Bini Advanced Lecture 5/6 November 8, 212 Outline 1 2 3 Theorem (Lemma 3 in [?]) Space of feasible C i N is EDF-schedulable if and only if i U i 1 and: t D n i=1 { max, t + Ti D i T i } C i t with D = {d i,k

More information

Linear Programming I

Linear Programming I Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

More information

Linear Programming: Chapter 11 Game Theory

Linear Programming: Chapter 11 Game Theory Linear Programming: Chapter 11 Game Theory Robert J. Vanderbei October 17, 2007 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/ rvdb Rock-Paper-Scissors

More information

Some Optimization Fundamentals

Some Optimization Fundamentals ISyE 3133B Engineering Optimization Some Optimization Fundamentals Shabbir Ahmed E-mail: sahmed@isye.gatech.edu Homepage: www.isye.gatech.edu/~sahmed Basic Building Blocks min or max s.t. objective as

More information

Approximation Algorithms

Approximation Algorithms Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

More information

26 Linear Programming

26 Linear Programming The greatest flood has the soonest ebb; the sorest tempest the most sudden calm; the hottest love the coldest end; and from the deepest desire oftentimes ensues the deadliest hate. Th extremes of glory

More information

Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

More information

1 Norms and Vector Spaces

1 Norms and Vector Spaces 008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

More information

Date: April 12, 2001. Contents

Date: April 12, 2001. Contents 2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........

More information

Week 5 Integral Polyhedra

Week 5 Integral Polyhedra Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

More information

LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005

LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005 LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005 DAVID L. BERNICK dbernick@soe.ucsc.edu 1. Overview Typical Linear Programming problems Standard form and converting

More information

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

More information

First Welfare Theorem

First Welfare Theorem First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto

More information

Nonlinear Programming Methods.S2 Quadratic Programming

Nonlinear Programming Methods.S2 Quadratic Programming Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective

More information

Optimization Modeling for Mining Engineers

Optimization Modeling for Mining Engineers Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2

More information

Absolute Value Programming

Absolute Value Programming Computational Optimization and Aplications,, 1 11 (2006) c 2006 Springer Verlag, Boston. Manufactured in The Netherlands. Absolute Value Programming O. L. MANGASARIAN olvi@cs.wisc.edu Computer Sciences

More information

Warshall s Algorithm: Transitive Closure

Warshall s Algorithm: Transitive Closure CS 0 Theory of Algorithms / CS 68 Algorithms in Bioinformaticsi Dynamic Programming Part II. Warshall s Algorithm: Transitive Closure Computes the transitive closure of a relation (Alternatively: all paths

More information

Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

Lecture 4: Equality Constrained Optimization. Tianxi Wang

Lecture 4: Equality Constrained Optimization. Tianxi Wang Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x

More information

Lecture 2: August 29. Linear Programming (part I)

Lecture 2: August 29. Linear Programming (part I) 10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.

More information

Inverses and powers: Rules of Matrix Arithmetic

Inverses and powers: Rules of Matrix Arithmetic Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

More information

9.4 THE SIMPLEX METHOD: MINIMIZATION

9.4 THE SIMPLEX METHOD: MINIMIZATION SECTION 9 THE SIMPLEX METHOD: MINIMIZATION 59 The accounting firm in Exercise raises its charge for an audit to $5 What number of audits and tax returns will bring in a maximum revenue? In the simplex

More information

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

More information

Jianlin Cheng, PhD Computer Science Department University of Missouri, Columbia Fall, 2013

Jianlin Cheng, PhD Computer Science Department University of Missouri, Columbia Fall, 2013 Jianlin Cheng, PhD Computer Science Department University of Missouri, Columbia Fall, 2013 Princeton s class notes on linear programming MIT s class notes on linear programming Xian Jiaotong University

More information

Math 315: Linear Algebra Solutions to Midterm Exam I

Math 315: Linear Algebra Solutions to Midterm Exam I Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =

More information

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

More information

Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725

Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725 Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T

More information

Minimally Infeasible Set Partitioning Problems with Balanced Constraints

Minimally Infeasible Set Partitioning Problems with Balanced Constraints Minimally Infeasible Set Partitioning Problems with alanced Constraints Michele Conforti, Marco Di Summa, Giacomo Zambelli January, 2005 Revised February, 2006 Abstract We study properties of systems of

More information

1 Determinants. Definition 1

1 Determinants. Definition 1 Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described

More information

Lecture 5 Principal Minors and the Hessian

Lecture 5 Principal Minors and the Hessian Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and

More information

5.1 Bipartite Matching

5.1 Bipartite Matching CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

More information

1 Linear Programming. 1.1 Introduction. Problem description: motivate by min-cost flow. bit of history. everything is LP. NP and conp. P breakthrough.

1 Linear Programming. 1.1 Introduction. Problem description: motivate by min-cost flow. bit of history. everything is LP. NP and conp. P breakthrough. 1 Linear Programming 1.1 Introduction Problem description: motivate by min-cost flow bit of history everything is LP NP and conp. P breakthrough. general form: variables constraints: linear equalities

More information

Transportation Polytopes: a Twenty year Update

Transportation Polytopes: a Twenty year Update Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,

More information

Cost Minimization and the Cost Function

Cost Minimization and the Cost Function Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This is

More information

3.1 Solving Systems Using Tables and Graphs

3.1 Solving Systems Using Tables and Graphs Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

More information

Homework # 3 Solutions

Homework # 3 Solutions Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8

More information

We shall turn our attention to solving linear systems of equations. Ax = b

We shall turn our attention to solving linear systems of equations. Ax = b 59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system

More information

3 Does the Simplex Algorithm Work?

3 Does the Simplex Algorithm Work? Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances

More information

Dantzig-Wolfe bound and Dantzig-Wolfe cookbook

Dantzig-Wolfe bound and Dantzig-Wolfe cookbook Dantzig-Wolfe bound and Dantzig-Wolfe cookbook thst@man.dtu.dk DTU-Management Technical University of Denmark 1 Outline LP strength of the Dantzig-Wolfe The exercise from last week... The Dantzig-Wolfe

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued). MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

More information

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the

More information

Numerisches Rechnen. (für Informatiker) M. Grepl J. Berger & J.T. Frings. Institut für Geometrie und Praktische Mathematik RWTH Aachen

Numerisches Rechnen. (für Informatiker) M. Grepl J. Berger & J.T. Frings. Institut für Geometrie und Praktische Mathematik RWTH Aachen (für Informatiker) M. Grepl J. Berger & J.T. Frings Institut für Geometrie und Praktische Mathematik RWTH Aachen Wintersemester 2010/11 Problem Statement Unconstrained Optimality Conditions Constrained

More information

Solutions Of Some Non-Linear Programming Problems BIJAN KUMAR PATEL. Master of Science in Mathematics. Prof. ANIL KUMAR

Solutions Of Some Non-Linear Programming Problems BIJAN KUMAR PATEL. Master of Science in Mathematics. Prof. ANIL KUMAR Solutions Of Some Non-Linear Programming Problems A PROJECT REPORT submitted by BIJAN KUMAR PATEL for the partial fulfilment for the award of the degree of Master of Science in Mathematics under the supervision

More information

160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

More information

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6 Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a

More information

Chapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach

Chapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach Chapter 5 Linear Programming in Two Dimensions: A Geometric Approach Linear Inequalities and Linear Programming Section 3 Linear Programming gin Two Dimensions: A Geometric Approach In this section, we

More information

FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM

FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM International Journal of Innovative Computing, Information and Control ICIC International c 0 ISSN 34-48 Volume 8, Number 8, August 0 pp. 4 FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT

More information

The Multiplicative Weights Update method

The Multiplicative Weights Update method Chapter 2 The Multiplicative Weights Update method The Multiplicative Weights method is a simple idea which has been repeatedly discovered in fields as diverse as Machine Learning, Optimization, and Game

More information

Duality in Linear Programming

Duality in Linear Programming Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow

More information

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Facts About Eigenvalues

Facts About Eigenvalues Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

More information

Lecture 2 Matrix Operations

Lecture 2 Matrix Operations Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or

More information

Integer Programming: Algorithms - 3

Integer Programming: Algorithms - 3 Week 9 Integer Programming: Algorithms - 3 OPR 992 Applied Mathematical Programming OPR 992 - Applied Mathematical Programming - p. 1/12 Dantzig-Wolfe Reformulation Example Strength of the Linear Programming

More information

Linear Programming in Matrix Form

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,

More information

Operation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1

Operation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1 Operation Research Module 1 Unit 1 1.1 Origin of Operations Research 1.2 Concept and Definition of OR 1.3 Characteristics of OR 1.4 Applications of OR 1.5 Phases of OR Unit 2 2.1 Introduction to Linear

More information

Discuss the size of the instance for the minimum spanning tree problem.

Discuss the size of the instance for the minimum spanning tree problem. 3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

More information

1 Solving LPs: The Simplex Algorithm of George Dantzig

1 Solving LPs: The Simplex Algorithm of George Dantzig Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

More information

(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7

(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7 (67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition

More information

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

More information

Jeffrey Kantor. August 28, 2006

Jeffrey Kantor. August 28, 2006 Mosel August 28, 2006 Announcements Mosel Enrollment and Registration - Last Day for Class Change is August 30th. Homework set 1 was handed out today. A copy is available on the course web page http://jkantor.blogspot.com

More information

3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max

3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,

More information

Discrete Optimization

Discrete Optimization Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

More information

Linear programming and reductions

Linear programming and reductions Chapter 7 Linear programming and reductions Many of the problems for which we want algorithms are optimization tasks: the shortest path, the cheapest spanning tree, the longest increasing subsequence,

More information

Lecture 5: Conic Optimization: Overview

Lecture 5: Conic Optimization: Overview EE 227A: Conve Optimization and Applications January 31, 2012 Lecture 5: Conic Optimization: Overview Lecturer: Laurent El Ghaoui Reading assignment: Chapter 4 of BV. Sections 3.1-3.6 of WTB. 5.1 Linear

More information

Derivative Free Optimization

Derivative Free Optimization Department of Mathematics Derivative Free Optimization M.J.D. Powell LiTH-MAT-R--2014/02--SE Department of Mathematics Linköping University S-581 83 Linköping, Sweden. Three lectures 1 on Derivative Free

More information