An example of LP problem: Political Elections
|
|
- Elwin Atkinson
- 5 years ago
- Views:
Transcription
1 Linear Programming
2 An example of LP problem: Political Elections Suppose that you are a politician trying to win an election. Your district has three different types of areas: urban, suburban, and rural. These areas have, respectively, 100,000, 200,000,and 50,000 registered voters. To govern effectively, you would like to win a majority of the votes in each of the three regions. you can estimate how many votes you win or lose from each population segment by spending $1,000 on advertising on each issue.
3 policy urban suburban rural build roads gun control farm subsidies gasoline tax Figure 1: The effects of policies on voters. Each entry describes the number of thousands of urban, suburban, or rural voters who could be won over by spending $1,000 on advertising support of a policy on a particular issue. Negative entries denote votes that would be lost.
4 x1 is the number of thousands of dollars spent on advertising on building roads, x2 is the number of thousands of dollars spent on advertising on gun control, x3 is the number of thousands of dollars spent on advertising on farm subsidies, and x4 is the number of thousands of dollars spent on advertising on a gasoline tax. We can write the requirement that we win at least 50,000 urban votes as
5 Similarly, we can write the requirements that we win at least 100,000 suburban votes and 25,000 rural votes as And To minimize the expression
6 there is no such thing as negative-cost advertising. We format this problem as
7
8 General linear programs In the general linear-programming problem, we wish to optimize a linear function subject to a set of linear inequalities. Given a set of real numbers a1, a2,..., an and a set of variables x1, x2,..., xn, a linear function f on those variables is defined by
9 are linear inequalities. We use the term linear constraints to denote either linear equalities or linear inequalities. In linear programming, we do not allow strict inequalities.
10 A linear programming in standard form is the maximization of a linear function subject to linear inequalitis. A linear program in slack form is the maximization of a linear function subject to linear equalities.
11 Let us first consider the following linear program with two variables (example 2): maximize x 1 + x 2 subject to 4x 1 - x 2 < 8 2x 1 + x 2 < 10 5x 1 - x 2 > -2 x 1, x 2 > 0.
12 We call any setting of the variables x 1 and x 2 that satisfies all the constraints a feasible solution to the linear program.
13 Figure 2: (a) The linear program given in example 2. Each constraint is represented by a line and a direction. The intersection of the constraints, which is the feasible region, is shaded. (b) The dotted lines show, respectively, the points for which the objective value is 0, 4, and 8. The optimal solution to the linear program is x1 = 2 and x2 = 6 with objective value 8.
14 Because the feasible region in Figure 2 is bounded, there must be some maximum value z for which the intersection of the line x 1 + x 2 = z and the feasible region is nonempty. An optimal solution to the linear program must be on the boundary of the bounded feasible region. In this case, the point is x 1 = 2 and x 2 = 6 with objective value 8. An optimal solution to the linear program occurred at a vertex or a line segment of the feasible region for two variables.
15 This is because of the convexity of bounded feasible regions. Similarly, if the LP has n variables, each constraint defines a half-space in n-dimensional space. The feasible region formed by the intersection of these halfspaces is called a simplex. The objective function is now a hyper-plane and, because of the convexity, an optimal solution will still occur at a vertex of the simplex.
16 Algorithms for LP Simplex algorithm worst case exponentialtime. Practical very simple and performance is good. Ellipsoid algorithm -- polynomial-time Interior-point method polynomial-time.
17 The basic idea of simplex algorithm The simplex algorithm takes as input a linear program and returns an optimal solution. It starts at some vertex of the simplex and performs a sequence of iterations. In each iteration, it moves along an edge of the simplex from a current vertex to a neighboring vertex whose objective value is no smaller than that of the current vertex (and usually is larger.)
18 The simplex algorithm terminates when it reaches a local maximum, which is a vertex from which all neighboring vertices have a smaller objective value. Because the feasible region is convex and the objective function is linear, this local optimum is actually a global optimum. (What if it is non-convex or non-linear)
19 If we add to a linear program the additional requirement that all variables take on integer values, we have an integer linear program. It has been proven that even finding the feasible region of an integer program is NPhard.
20 Standard form In standard form, we are given n real numbers c1, c2,..., cn; m real numbers b1, b2,..., bm; and mn real numbers aij for i = 1, 2,..., m and j = 1, 2,..., n. We wish to find n real numbers for n variables: x1, x2,..., xn that maximize the objective function.
21 We call the maximize expression above, the objective function and the n + m inequalities, the constraints, among them the n constraints, are called the non-negativity constraints.
22 In a more compact form for LP, let A = (a ij ) be an m x n matrix b = (b i ) be an m dimensional vector c = (c j ) be an n dimensional vector x = (x j ) be an n dimensional vector We can rewrite the LP as follows:
23 Where c T x is inner product of two n dimensional vectors; Ax is a matrix-vector Product; x must be non-negative.
24 A linear program may not be in standard form for one of four possible reasons: 1. The objective function may be a minimization rather than a maximization. 2. There may be variables without non-negativity constraints. 3. There may be equality constraints, which have an equal sign rather than a less-than or -equal-to sign. 4. There may be inequality constraints, but instead of having a less-than-or-equal-to sign, they have a greater-than-or-equal-to sign.
25 Method to convert to Standard form: To convert a minimization linear program L into an equivalent maximization linear program L, we simply negate the coefficients in the objective function. For example, if we have the linear program
26 max
27 To convert a linear program in which some of the variables do not have non-negativity constraints into one in which each variable has a non-negativity constraint. Suppose that some variable xj does not have a nonnegativity constraint. Then we replace each occurrence of xj by x' j - x" j and add the nonnegativity constraints x' j > 0 and x" j > 0.
28 Thus, if the objective function has a term cjxj, it is replaced by c j x' j - c j x" j. Any feasible solution x to the new linear program corresponds to a feasible solution to the original linear program with x j = x' j - x" j and with the same objective value, and thus the two solutions are equivalent. We apply this conversion scheme to each variable that does not have a non-negativity constraint to yield an equivalent linear program in which all variables have nonnegativity constraints.
29 In our previous example, Variable x1 has a nonnegative constraint, but variable x2 does not. To ensure that each variable has a corresponding non-negativity constraint. we replace x2 by two variables, x 2 x 2 then we solve the modified linear program to obtain x2 = x 2 x 2.
30
31 To convert equality constraints into inequality constraints, we can replace this equality constraint by the pair of inequality constraints. Suppose that a linear program has an equality constraint f (x1, x2,..., xn) = b. Since x = y if and only if both x y and x y, we can replace f (x1, x2,..., xn) = b by f (x1, x2,...,xn) b and f (x1, x2,..., xn) b.
32 Finally, we can convert the greaterthan-or-equal-to constraints to lessthan-or-equal-to constraints by multiplying these constraints through by -1. That is, any inequality of the form
33 Thus, by replacing each coefficient aij by -aij and each value bi by -bi, we obtain an equivalent lessthan-or-equal-to constraint. Finishing our example, we replace the equality in constraint by two inequalities, obtaining
34 Finally, we negate constraint. For consistency in variable names, we rename X' 2 to x2 and X" 2 to x3, obtaining the standard form as follows:
35
36 Converting linear programs into slack form To efficiently solve a linear program with the simplex algorithm, we prefer to express it in a form in which some of the constraints are equality constraints. Let be an inequality constraint. We introduce a new variable s and rewrite inequality as the two constraints:
37 We call s a slack variable because it measures the slack, or difference, between the left-hand and right-hand sides of equation.
38 we shall use xn+i (instead of s) to denote the slack variable associated with the i th inequality. The i th constraint is therefore along with the non-negativity constraint xn+i 0.
39 For example, we introduce slack variables x4, x5, and x6, obtaining maximize The variables on the left-hand side of the equalities are called basic variables, and those on the right-hand side are called non-basic variables.
40 We shall also use the variable z to denote the value of the objective function. Thus we can concisely define a slack form by a 5-tuple (N, B, A, b, c, v), denoting the slack form
41 N set of indices of nonbasic variables B set of indices of basic variables N = n; B =m; N U B = {1,2,, n+m} A coefficient matrix of variables b,c vectors; v -- constant
42 In this slack form, we have that B= {4,5,6}; N={1,2,3}; A = (a 41, a 42, a 43, a 51, a 52, a 53, a 61, a 62, a 63 ) = (-1, -1, 1, 1, 1, -1, ) b = (b 4, b 5, b 6 ) = (7, -7, 4) c = (c 1, c 2, c 3 ) = (2, -3, 3) v = 0
43 For more example, the slack form below (
44 Examples for formulation of problems into LP Shortest paths Maximum flow Minimum-cost flow Multi-commodity flow
45 Shortest paths The single-pair shortest path problem: Given a weighted directed graph G = (V,E) weight function w: E R, a source vertex s and destination vertex t, compute the value d[t], the weight of the shortest path from s to t. Note that for each edge (u,v) in E, d[v] < _ d[u] + w(u,v); d[s] = 0
46 Shortest paths We obtain the following linear program to compute the shortest-path weight from nodes s to t: In this linear program, there are V variables d[v] s, one for each vertex v in V. There are E + 1 constraints, one for each edge plus the additional constraint that the source vertex always has the value 0.
47 Maximum flow A flow network G=(V,E) is a directed graph s.t. each edge (u,v) in E has non-negative capacity c(u,v) > _ 0. If (u,v) not in E, then c(u,v)=0. Designate a source s and a sink t in V. G is a connected graph. A flow in G is a real-valued function f: V X V R satisfies three properties:
48 Capacity constraint: for all u,v in V, f(u,v) < _ c(u,v) Skew symmetry: for all u,v in V, f(u,v) < _ - f(u,v) Flow conservation: for all u,v in V-{s,t}, SUM v in V f(u,v) = 0 The value of a flow f is f = SUM v in V f(s,v) That is the total flow out of the source s.
49 The maximum flow problem is defined as Given a flow network G with source s and sink t, find a flow of maximum value. Ford-Fulkerson-method initialize flow f to 0 while there exists an augmenting path p do augment flow f along p return f O(E f* ) time, f* the maximum value found by algorithm
50 Maximum flow we can express the maximum-flow problem as following linear program: This linear program has V -2 variables, corresponding to the flow between each pair of vertices, and it has 2 V 2 + V - 2 constraints.
51 Minimum-cost flow Figure.3: (a) An example of a minimum-cost-flow problem. We denote the capacities by c and the costs by a. Vertex s is the source and vertex t is the sink, and we wish to send 4 units of flow from s to t. (b) A solution to the minimum-cost flow problem in which 4 units of flow are sent from s to t. For each edge, the flow and capacity are written as flow/capacity.
52
53 Multi-commodity flow
54 The real power of linear programming comes from the ability to solve new problems (not the above shortest path, maximum flow etc, which already had efficient algorithms). Such as political vote problem is new one. It is also useful to solve these problems do not have a known efficient algorithms.
55
Chapter 29 Linear Programming 771. policy urban suburban rural build roads gun control farm subsidies 0 10 gasoline tax
29 LinearProgramming Many problems can be formulated as maximizing or minimizing an objective, given limited resources and competing constraints. If we can specify the objective as a linear function of
More informationWhat s Linear Programming? Often your try is to maximize or minimize an objective within given constraints
Linear Programming What s Linear Programming? Often your try is to maximize or minimize an objective within given constraints A linear programming problem can be expressed as a linear function of certain
More informationLinear Programming. Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015
Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015 Linear Programming 2015 Goodrich and Tamassia 1 Formulating the Problem q The function
More informationCOT 6936: Topics in Algorithms! Giri Narasimhan. ECS 254A / EC 2443; Phone: x3748
COT 6936: Topics in Algorithms! Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 giri@cs.fiu.edu http://www.cs.fiu.edu/~giri/teach/cot6936_s12.html https://moodle.cis.fiu.edu/v2.1/course/view.php?id=174
More informationCSC 8301 Design & Analysis of Algorithms: Linear Programming
CSC 8301 Design & Analysis of Algorithms: Linear Programming Professor Henry Carter Fall 2016 Iterative Improvement Start with a feasible solution Improve some part of the solution Repeat until the solution
More informationLinear Programming. Readings: Read text section 11.6, and sections 1 and 2 of Tom Ferguson s notes (see course homepage).
Linear Programming Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory: Feasible Set, Vertices, Existence of Solutions. Equivalent formulations. Outline
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 informationHeuristic Optimization Today: Linear Programming. Tobias Friedrich Chair for Algorithm Engineering Hasso Plattner Institute, Potsdam
Heuristic Optimization Today: Linear Programming Chair for Algorithm Engineering Hasso Plattner Institute, Potsdam Linear programming Let s first define it formally: A linear program is an optimization
More informationCopyright 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin Introduction to the Design & Analysis of Algorithms, 2 nd ed., Ch.
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
More informationBCN 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
More informationLinear and Integer Programming :Algorithms in the Real World. Related Optimization Problems. How important is optimization?
Linear and Integer Programming 15-853:Algorithms in the Real World Linear and Integer Programming I Introduction Geometric Interpretation Simplex Method Linear or Integer programming maximize z = c T x
More informationLinear Programming. them such that they
Linear Programming l Another "Sledgehammer" in our toolkit l Many problems fit into the Linear Programming approach l These are optimization tasks where both the constraints and the objective are linear
More informationLinear Programming CISC4080, Computer Algorithms CIS, Fordham Univ. Linear Programming
Linear Programming CISC4080, Computer Algorithms CIS, Fordham Univ. Instructor: X. Zhang! Linear Programming In a linear programming problem, there is a set of variables, and we want to assign real values
More informationLecture 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
More informationSimulation. Lecture O1 Optimization: Linear Programming. Saeed Bastani April 2016
Simulation Lecture O Optimization: Linear Programming Saeed Bastani April 06 Outline of the course Linear Programming ( lecture) Integer Programming ( lecture) Heuristics and Metaheursitics (3 lectures)
More informationDiscrete 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
More informationSolutions for Operations Research Final Exam
Solutions for Operations Research Final Exam. (a) The buffer stock is B = i a i = a + a + a + a + a + a 6 + a 7 = + + + + + + =. And the transportation tableau corresponding to the transshipment problem
More informationLinear Programming. Linear Programming. Linear Programming. Example: Profit Maximization (1/4) Iris Hui-Ru Jiang Fall Linear programming
Linear Programming 3 describes a broad class of optimization tasks in which both the optimization criterion and the constraints are linear functions. Linear Programming consists of three parts: A set of
More informationCMPSCI611: The Simplex Algorithm Lecture 24
CMPSCI611: The Simplex Algorithm Lecture 24 Let s first review the general situation for linear programming problems. Our problem in standard form is to choose a vector x R n, such that x 0 and Ax = b,
More informationSome Advanced Topics in Linear Programming
Some Advanced Topics in Linear Programming Matthew J. Saltzman July 2, 995 Connections with Algebra and Geometry In this section, we will explore how some of the ideas in linear programming, duality theory,
More informationNotes for Lecture 18
U.C. Berkeley CS17: Intro to CS Theory Handout N18 Professor Luca Trevisan November 6, 21 Notes for Lecture 18 1 Algorithms for Linear Programming Linear programming was first solved by the simplex method
More informationThe 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
More information4.1 The original problem and the optimal tableau
Chapter 4 Sensitivity analysis The sensitivity analysis is performed after a given linear problem has been solved, with the aim of studying how changes to the problem affect the optimal solution In particular,
More information11 Linear Programming
11 Linear Programming 11.1 Definition and Importance The final topic in this course is Linear Programming. We say that a problem is an instance of linear programming when it can be effectively expressed
More informationDesign and Analysis of Algorithms (V)
Design and Analysis of Algorithms (V) An Introduction to Linear Programming Guoqiang Li School of Software, Shanghai Jiao Tong University Homework Assignment 2 is announced! (deadline Apr. 10) Linear Programming
More informationArtificial Intelligence
Artificial Intelligence Combinatorial Optimization G. Guérard Department of Nouvelles Energies Ecole Supérieur d Ingénieurs Léonard de Vinci Lecture 1 GG A.I. 1/34 Outline 1 Motivation 2 Geometric resolution
More informationOptimization 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
More informationMathematical and Algorithmic Foundations Linear Programming and Matchings
Adavnced Algorithms Lectures Mathematical and Algorithmic Foundations Linear Programming and Matchings Paul G. Spirakis Department of Computer Science University of Patras and Liverpool Paul G. Spirakis
More informationAdvanced Algorithms Linear Programming
Reading: Advanced Algorithms Linear Programming CLRS, Chapter29 (2 nd ed. onward). Linear Algebra and Its Applications, by Gilbert Strang, chapter 8 Linear Programming, by Vasek Chvatal Introduction to
More information16.410/413 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)
More informationDiscrete 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
More informationChapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2015 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of
More informationAdvanced Operations Research Techniques IE316. Quiz 1 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 1 Review Dr. Ted Ralphs IE316 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material
More informationLinear Programming. Course review MS-E2140. v. 1.1
Linear Programming MS-E2140 Course review v. 1.1 Course structure Modeling techniques Linear programming theory and the Simplex method Duality theory Dual Simplex algorithm and sensitivity analysis Integer
More information3 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=
More informationUNIT 2 LINEAR PROGRAMMING PROBLEMS
UNIT 2 LINEAR PROGRAMMING PROBLEMS Structure 2.1 Introduction Objectives 2.2 Linear Programming Problem (LPP) 2.3 Mathematical Formulation of LPP 2.4 Graphical Solution of Linear Programming Problems 2.5
More information15-451/651: Design & Analysis of Algorithms October 11, 2018 Lecture #13: Linear Programming I last changed: October 9, 2018
15-451/651: Design & Analysis of Algorithms October 11, 2018 Lecture #13: Linear Programming I last changed: October 9, 2018 In this lecture, we describe a very general problem called linear programming
More informationLecture 4: Linear Programming
COMP36111: Advanced Algorithms I Lecture 4: Linear Programming Ian Pratt-Hartmann Room KB2.38: email: ipratt@cs.man.ac.uk 2017 18 Outline The Linear Programming Problem Geometrical analysis The Simplex
More informationDM545 Linear and Integer Programming. Lecture 2. The Simplex Method. Marco Chiarandini
DM545 Linear and Integer Programming Lecture 2 The Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 3. 4. Standard Form Basic Feasible Solutions
More informationIntroduction to Linear Programming
Introduction to Linear Programming Eric Feron (updated Sommer Gentry) (updated by Paul Robertson) 16.410/16.413 Historical aspects Examples of Linear programs Historical contributor: G. Dantzig, late 1940
More informationThe Simplex Algorithm for LP, and an Open Problem
The Simplex Algorithm for LP, and an Open Problem Linear Programming: General Formulation Inputs: real-valued m x n matrix A, and vectors c in R n and b in R m Output: n-dimensional vector x There is one
More informationChapter 4 Concepts from Geometry
Chapter 4 Concepts from Geometry An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Line Segments The line segment between two points and in R n is the set of points on the straight line joining
More informationOutline. CS38 Introduction to Algorithms. Linear programming 5/21/2014. Linear programming. Lecture 15 May 20, 2014
5/2/24 Outline CS38 Introduction to Algorithms Lecture 5 May 2, 24 Linear programming simplex algorithm LP duality ellipsoid algorithm * slides from Kevin Wayne May 2, 24 CS38 Lecture 5 May 2, 24 CS38
More information4 LINEAR PROGRAMMING (LP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
4 LINEAR PROGRAMMING (LP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 Mathematical programming (optimization) problem: min f (x) s.t. x X R n set of feasible solutions with linear objective function
More information4 Integer Linear Programming (ILP)
TDA6/DIT37 DISCRETE OPTIMIZATION 17 PERIOD 3 WEEK III 4 Integer Linear Programg (ILP) 14 An integer linear program, ILP for short, has the same form as a linear program (LP). The only difference is that
More informationLinear Programming. Slides by Carl Kingsford. Apr. 14, 2014
Linear Programming Slides by Carl Kingsford Apr. 14, 2014 Linear Programming Suppose you are given: A matrix A with m rows and n columns. A vector b of length m. A vector c of length n. Find a length-n
More information16.410/413 Principles of Autonomy and Decision Making
16.410/413 Principles of Autonomy and Decision Making Lecture 16: Mathematical Programming I Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 8, 2010 E. Frazzoli
More informationLinear programming II João Carlos Lourenço
Decision Support Models Linear programming II João Carlos Lourenço joao.lourenco@ist.utl.pt Academic year 2012/2013 Readings: Hillier, F.S., Lieberman, G.J., 2010. Introduction to Operations Research,
More informationLinear Programming Motivation: The Diet Problem
Agenda We ve done Greedy Method Divide and Conquer Dynamic Programming Network Flows & Applications NP-completeness Now Linear Programming and the Simplex Method Hung Q. Ngo (SUNY at Buffalo) CSE 531 1
More informationLecture 3: Totally Unimodularity and Network Flows
Lecture 3: Totally Unimodularity and Network Flows (3 units) Outline Properties of Easy Problems Totally Unimodular Matrix Minimum Cost Network Flows Dijkstra Algorithm for Shortest Path Problem Ford-Fulkerson
More informationIntroduction. Linear because it requires linear functions. Programming as synonymous of planning.
LINEAR PROGRAMMING Introduction Development of linear programming was among the most important scientific advances of mid-20th cent. Most common type of applications: allocate limited resources to competing
More informationLecture notes on the simplex method September We will present an algorithm to solve linear programs of the form. maximize.
Cornell University, Fall 2017 CS 6820: Algorithms Lecture notes on the simplex method September 2017 1 The Simplex Method We will present an algorithm to solve linear programs of the form maximize subject
More informationMA4254: 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
More informationLinear Programming in Small Dimensions
Linear Programming in Small Dimensions Lekcija 7 sergio.cabello@fmf.uni-lj.si FMF Univerza v Ljubljani Edited from slides by Antoine Vigneron Outline linear programming, motivation and definition one dimensional
More informationLP-Modelling. dr.ir. C.A.J. Hurkens Technische Universiteit Eindhoven. January 30, 2008
LP-Modelling dr.ir. C.A.J. Hurkens Technische Universiteit Eindhoven January 30, 2008 1 Linear and Integer Programming After a brief check with the backgrounds of the participants it seems that the following
More information(67686) Mathematical Foundations of AI July 30, Lecture 11
(67686) Mathematical Foundations of AI July 30, 2008 Lecturer: Ariel D. Procaccia Lecture 11 Scribe: Michael Zuckerman and Na ama Zohary 1 Cooperative Games N = {1,...,n} is the set of players (agents).
More informationMATH 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
More informationISE203 Optimization 1 Linear Models. Dr. Arslan Örnek Chapter 4 Solving LP problems: The Simplex Method SIMPLEX
ISE203 Optimization 1 Linear Models Dr. Arslan Örnek Chapter 4 Solving LP problems: The Simplex Method SIMPLEX Simplex method is an algebraic procedure However, its underlying concepts are geometric Understanding
More informationLinear Programming 1
Linear Programming 1 Fei Li March 5, 2012 1 With references of Algorithms by S. Dasgupta, C. H. Papadimitriou, and U. V. Vazirani. Many of the problems for which we want algorithms are optimization tasks.
More informationLinear Programming Duality and Algorithms
COMPSCI 330: Design and Analysis of Algorithms 4/5/2016 and 4/7/2016 Linear Programming Duality and Algorithms Lecturer: Debmalya Panigrahi Scribe: Tianqi Song 1 Overview In this lecture, we will cover
More informationOPERATIONS RESEARCH. Linear Programming Problem
OPERATIONS RESEARCH Chapter 1 Linear Programming Problem Prof. Bibhas C. Giri Department of Mathematics Jadavpur University Kolkata, India Email: bcgiri.jumath@gmail.com 1.0 Introduction Linear programming
More information5 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.
More informationReview for Mastery Using Graphs and Tables to Solve Linear Systems
3-1 Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations
More informationPart 4. Decomposition Algorithms Dantzig-Wolf Decomposition Algorithm
In the name of God Part 4. 4.1. Dantzig-Wolf Decomposition Algorithm Spring 2010 Instructor: Dr. Masoud Yaghini Introduction Introduction Real world linear programs having thousands of rows and columns.
More informationCombinatorial Optimization
Combinatorial Optimization Frank de Zeeuw EPFL 2012 Today Introduction Graph problems - What combinatorial things will we be optimizing? Algorithms - What kind of solution are we looking for? Linear Programming
More informationChapter 3 Linear Programming: A Geometric Approach
Chapter 3 Linear Programming: A Geometric Approach Section 3.1 Graphing Systems of Linear Inequalities in Two Variables y 4x + 3y = 12 4 3 4 x 3 y 12 x y 0 x y = 0 2 1 P(, ) 12 12 7 7 1 1 2 3 x We ve seen
More informationLinear Programming and its Applications
Linear Programming and its Applications Outline for Today What is linear programming (LP)? Examples Formal definition Geometric intuition Why is LP useful? A first look at LP algorithms Duality Linear
More informationCS 473: Algorithms. Ruta Mehta. Spring University of Illinois, Urbana-Champaign. Ruta (UIUC) CS473 1 Spring / 36
CS 473: Algorithms Ruta Mehta University of Illinois, Urbana-Champaign Spring 2018 Ruta (UIUC) CS473 1 Spring 2018 1 / 36 CS 473: Algorithms, Spring 2018 LP Duality Lecture 20 April 3, 2018 Some of the
More informationIntroduction to Mathematical Programming IE496. Final Review. Dr. Ted Ralphs
Introduction to Mathematical Programming IE496 Final Review Dr. Ted Ralphs IE496 Final Review 1 Course Wrap-up: Chapter 2 In the introduction, we discussed the general framework of mathematical modeling
More informationLinear Programming Problems
Linear Programming Problems Two common formulations of linear programming (LP) problems are: min Subject to: 1,,, 1,2,,;, max Subject to: 1,,, 1,2,,;, Linear Programming Problems The standard LP problem
More informationInteger Programming Theory
Integer Programming Theory Laura Galli October 24, 2016 In the following we assume all functions are linear, hence we often drop the term linear. In discrete optimization, we seek to find a solution x
More informationLecture 10,11: General Matching Polytope, Maximum Flow. 1 Perfect Matching and Matching Polytope on General Graphs
CMPUT 675: Topics in Algorithms and Combinatorial Optimization (Fall 2009) Lecture 10,11: General Matching Polytope, Maximum Flow Lecturer: Mohammad R Salavatipour Date: Oct 6 and 8, 2009 Scriber: Mohammad
More informationCS 580: Algorithm Design and Analysis. Jeremiah Blocki Purdue University Spring 2018
CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 2018 Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved.
More informationEaster Term OPTIMIZATION
DPK OPTIMIZATION Easter Term Example Sheet It is recommended that you attempt about the first half of this sheet for your first supervision and the remainder for your second supervision An additional example
More informationCS 372: Computational Geometry Lecture 10 Linear Programming in Fixed Dimension
CS 372: Computational Geometry Lecture 10 Linear Programming in Fixed Dimension Antoine Vigneron King Abdullah University of Science and Technology November 7, 2012 Antoine Vigneron (KAUST) CS 372 Lecture
More informationII. Linear Programming
II. Linear Programming A Quick Example Suppose we own and manage a small manufacturing facility that produced television sets. - What would be our organization s immediate goal? - On what would our relative
More informationProgramming, numerics and optimization
Programming, numerics and optimization Lecture C-4: Constrained optimization Łukasz Jankowski ljank@ippt.pan.pl Institute of Fundamental Technological Research Room 4.32, Phone +22.8261281 ext. 428 June
More informationSystems of Equations and Inequalities. Copyright Cengage Learning. All rights reserved.
5 Systems of Equations and Inequalities Copyright Cengage Learning. All rights reserved. 5.5 Systems of Inequalities Copyright Cengage Learning. All rights reserved. Objectives Graphing an Inequality Systems
More informationThree Dimensional Geometry. Linear Programming
Three Dimensional Geometry Linear Programming A plane is determined uniquely if any one of the following is known: The normal to the plane and its distance from the origin is given, i.e. equation of a
More informationIntroduction to Operations Research
- Introduction to Operations Research Peng Zhang April, 5 School of Computer Science and Technology, Shandong University, Ji nan 5, China. Email: algzhang@sdu.edu.cn. Introduction Overview of the Operations
More informationConvex Optimization CMU-10725
Convex Optimization CMU-10725 Ellipsoid Methods Barnabás Póczos & Ryan Tibshirani Outline Linear programs Simplex algorithm Running time: Polynomial or Exponential? Cutting planes & Ellipsoid methods for
More informationMATHEMATICS II: COLLECTION OF EXERCISES AND PROBLEMS
MATHEMATICS II: COLLECTION OF EXERCISES AND PROBLEMS GRADO EN A.D.E. GRADO EN ECONOMÍA GRADO EN F.Y.C. ACADEMIC YEAR 2011-12 INDEX UNIT 1.- AN INTRODUCCTION TO OPTIMIZATION 2 UNIT 2.- NONLINEAR PROGRAMMING
More informationMaximum Flow. Flow Networks. Flow Networks Ford-Fulkerson Method Edmonds-Karp Algorithm Push-Relabel Algorithms. Example Flow Network
Flow Networks Ford-Fulkerson Method Edmonds-Karp Algorithm Push-Relabel Algorithms Maximum Flow Flow Networks A flow network is a directed graph where: Each edge (u,v) has a capacity c(u,v) 0. If (u,v)
More informationChap5 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
More informationSystems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses
5 5 Systems and Matrices Systems and Matrices 5.6 Systems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses Sections 5.6 5.8 2008 Pearson Addison-Wesley. All rights
More informationThe Simplex Algorithm
The Simplex Algorithm April 25, 2005 We seek x 1,..., x n 0 which mini- Problem. mizes C(x 1,..., x n ) = c 1 x 1 + + c n x n, subject to the constraint Ax b, where A is m n, b = m 1. Through the introduction
More informationCS 473: Algorithms. Ruta Mehta. Spring University of Illinois, Urbana-Champaign. Ruta (UIUC) CS473 1 Spring / 50
CS 473: Algorithms Ruta Mehta University of Illinois, Urbana-Champaign Spring 2018 Ruta (UIUC) CS473 1 Spring 2018 1 / 50 CS 473: Algorithms, Spring 2018 Introduction to Linear Programming Lecture 18 March
More informationAMATH 383 Lecture Notes Linear Programming
AMATH 8 Lecture Notes Linear Programming Jakob Kotas (jkotas@uw.edu) University of Washington February 4, 014 Based on lecture notes for IND E 51 by Zelda Zabinsky, available from http://courses.washington.edu/inde51/notesindex.htm.
More informationMathematics. Linear Programming
Mathematics Linear Programming Table of Content 1. Linear inequations. 2. Terms of Linear Programming. 3. Mathematical formulation of a linear programming problem. 4. Graphical solution of two variable
More informationOutline: 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
More informationMathematics for Business and Economics - I. Chapter7 Linear Inequality Systems and Linear Programming (Lecture11)
Mathematics for Business and Economics - I Chapter7 Linear Inequality Systems and Linear Programming (Lecture11) A linear inequality in two variables is an inequality that can be written in the form Ax
More information56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998
56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998 Part A: Answer any four of the five problems. (15 points each) 1. Transportation problem 2. Integer LP Model Formulation
More informationDecision Aid Methodologies In Transportation Lecture 1: Polyhedra and Simplex method
Decision Aid Methodologies In Transportation Lecture 1: Polyhedra and Simplex method Chen Jiang Hang Transportation and Mobility Laboratory April 15, 2013 Chen Jiang Hang (Transportation and Mobility Decision
More informationLecture 3. Corner Polyhedron, Intersection Cuts, Maximal Lattice-Free Convex Sets. Tepper School of Business Carnegie Mellon University, Pittsburgh
Lecture 3 Corner Polyhedron, Intersection Cuts, Maximal Lattice-Free Convex Sets Gérard Cornuéjols Tepper School of Business Carnegie Mellon University, Pittsburgh January 2016 Mixed Integer Linear Programming
More informationLinear Programming Terminology
Linear Programming Terminology The carpenter problem is an example of a linear program. T and B (the number of tables and bookcases to produce weekly) are decision variables. The profit function is an
More informationCS261: Problem Set #2
CS261: Problem Set #2 Due by 11:59 PM on Tuesday, February 9, 2016 Instructions: (1) Form a group of 1-3 students. You should turn in only one write-up for your entire group. (2) Submission instructions:
More information4.1 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
More informationComputer Science & Engineering 423/823 Design and Analysis of Algorithms
Computer Science & Engineering 423/823 Design and Analysis of Algorithms Lecture 07 Single-Source Shortest Paths (Chapter 24) Stephen Scott and Vinodchandran N. Variyam sscott@cse.unl.edu 1/36 Introduction
More informationOutline. Linear Programming (LP): Principles and Concepts. Need for Optimization to Get the Best Solution. Linear Programming
Outline Linear Programming (LP): Principles and oncepts Motivation enoît hachuat Key Geometric Interpretation McMaster University Department of hemical Engineering 3 LP Standard Form
More informationMinimum Cost Edge Disjoint Paths
Minimum Cost Edge Disjoint Paths Theodor Mader 15.4.2008 1 Introduction Finding paths in networks and graphs constitutes an area of theoretical computer science which has been highly researched during
More information