Optimization in R n Introduction
|
|
- Gordon Walters
- 8 years ago
- Views:
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 To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More information2.3 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 information5 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 informationInternational 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 informationThe 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 informationLecture 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 informationIEOR 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 informationChapter 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 information1 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 informationProximal 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 informationLECTURE 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 informationMathematical 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 informationLinear 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 informationActually 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 informationWhat 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 informationLinear 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 information3. 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 informationNP-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 informationSome 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 informationTwo-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 informationNonlinear 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 informationLecture 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 informationThis 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 informationLinear 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 information24. 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. 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 informationconstraint. 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 informationLinear 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 informationChapter 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 informationContents. 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 informationApproximation 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 informationLecture 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 information26 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 informationDate: 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 information1 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 informationLECTURE: 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 informationMATH 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 informationNonlinear 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 informationOptimization 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 informationWarshall 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 informationa 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 informationLecture 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 information9.4 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 informationLinear 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 informationDuality 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 informationMinimally 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 informationLecture 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 information1 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 information5.1 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 informationTransportation 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 informationCost 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 informationWe 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 informationHomework # 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 informationDantzig-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 informationSimilarity 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 information3.1 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 informationMATH 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 informationNumerisches 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 informationChapter 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 informationSolutions 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 informationSECOND 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 information160 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 informationFUZZY 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 informationModern 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 informationChapter 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 informationMathematics 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 informationThe 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 informationDuality 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 informationInteger 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 informationLinear 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 informationLecture 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 informationDiscuss 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 informationOperation 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 informationLinear 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(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 informationDiscrete 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 information3. 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 information1 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 informationDerivative 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 informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More informationA NEW LOOK AT CONVEX ANALYSIS AND OPTIMIZATION
1 A NEW LOOK AT CONVEX ANALYSIS AND OPTIMIZATION Dimitri Bertsekas M.I.T. FEBRUARY 2003 2 OUTLINE Convexity issues in optimization Historical remarks Our treatment of the subject Three unifying lines of
More informationBX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationModule1. x 1000. y 800.
Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationEXCEL SOLVER TUTORIAL
ENGR62/MS&E111 Autumn 2003 2004 Prof. Ben Van Roy October 1, 2003 EXCEL SOLVER TUTORIAL This tutorial will introduce you to some essential features of Excel and its plug-in, Solver, that we will be using
More informationLecture 4: AC 0 lower bounds and pseudorandomness
Lecture 4: AC 0 lower bounds and pseudorandomness Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: Jason Perry and Brian Garnett In this lecture,
More informationInner products on R n, and more
Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal
More informationA Lagrangian-DNN Relaxation: a Fast Method for Computing Tight Lower Bounds for a Class of Quadratic Optimization Problems
A Lagrangian-DNN Relaxation: a Fast Method for Computing Tight Lower Bounds for a Class of Quadratic Optimization Problems Sunyoung Kim, Masakazu Kojima and Kim-Chuan Toh October 2013 Abstract. We propose
More informationCyber-Security Analysis of State Estimators in Power Systems
Cyber-Security Analysis of State Estimators in Electric Power Systems André Teixeira 1, Saurabh Amin 2, Henrik Sandberg 1, Karl H. Johansson 1, and Shankar Sastry 2 ACCESS Linnaeus Centre, KTH-Royal Institute
More informationPermutation Betting Markets: Singleton Betting with Extra Information
Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu
More informationLinear Programming. April 12, 2005
Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More informationLecture 4: Partitioned Matrices and Determinants
Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying
More information