R n a T i x = b i} is a Hyperplane.
|
|
- Logan Lyons
- 5 years ago
- Views:
Transcription
1 Geometry of LPs Consider the following LP : min {c T x a T i x b i The feasible region is i =1,...,m}. X := {x R n a T i x b i i =1,...,m} = m i=1 {x Rn a T i x b i} }{{} X i The set X i is a Half-space. The set H i = {x R n a T i x = b i} is a Hyperplane. The feasible region X is given by the intersection of m half-spaces and is known as a Polyhedron. A Polyhedron is a closed convexset (verify?). If it is bounded it is called a Polytope. LP: Minimize a linear function over a polyhedral set. Move in the direction c as far as possible while staying within the feasible region X. Verify that any LP can be written in this form 1
2 Example min c 1 x 1 + c 2 x 2 s.t. x 1 + x 2 1, x 1 0, x 2 0. X2 X2 X1 + X2 = 1 c =(1, 1) T,x =(0, 0) T X1 X1 + X2 = 1 c =(1, 0) T,x =(0,x 2 ) T for 0 x 2 1 X1 X2 X2 X1 + X2 = 1 c =(0, 1) T,x =(x 1, 0) T for 0 x 1 X1 X1 + X2 = 1 c =( 1, 1) T Unbounded. X1 2
3 Extreme Point Optimality An optimal solution (if it exists) always lies on a boundary of the feasible region X If there is an optimal solution, then there is one at a corner or a vertex or an extreme point of the polyhedron X. A point x X is an extreme point of X if it cannot be expressed as a convex combination of some other points x 1,...,x m X. A polyhedron has a finite number of extreme points. Representation theorem: When X is a polytope, any point x X can be expressed as a convex combination of the extreme points of X, i.e. x = Ii=1 λ i x i where x i vert(x) for all i =1,...,I, Ii=1 λ i =1,andλ i 0. 3
4 Proof of Extreme Point Optimality Consider the LP min{c T x x X}, where X is a polytope. Let x be an optimal solution, and suppose that there are no extreme points of X that are optimal. Then c T x <c T x i for all x i vert(x). By the representation thm: x = I i=1 λ i x i,then c T x = > I i=1 I i=1 = c T x and we have a contradiction. λ i c T x i λ i c T x 4
5 Unbounded Polyhedra A feasible direction of an unbounded polyhedra X R n is a (non-zero) vector d R n, such that if x 0 X then (x 0 + λd) X for all λ 0. An extreme direction of an unbounded polyhedra X R n is a direction d R n that cannot be expressed as a convexcombination of other directions of X. A polyhedron has a finite number of extreme directions. Representation theorem for general polyhedra: Any point x in a non-empty polyhedron can be expressed as a convexcombination of the extreme points of X, and a linear combination of the extreme directions of X, i.e. x = I i=1 λ i x i + J j=1 µ j d j where x i vert(x) foralli =1,...,I, I i=1 λ i =1, λ i 0, d j are the extreme directions of X for all j =1,J,andµ j 0. Extreme point optimality for general polyhedra can be proven using the above result. 5
6 Characterizing Extreme Points Consider an LP in standard form: min{c T x Ax = b, x 0}, where c R n, A R m n, and b R m. that rank(a) =m n. Assume Let A B =[a i 1,...,a i m ] be a matrix formed by m l.i. columns of A (called a Basis). Let A N be the matrixof the remaining columns, i.e. A = [A B A N ]. Partition the vector x = [ xb x N ] where x B =(x i1,..., x im ) T (Basic variables), and x N is the vector of remaining components of x (Non-basic variables). Set x B = A 1 B b and x N =0. Notex = [ A 1 B b 0 ( n m a solution to Ax = b. Known as a Basic solution. How many basic solutions? Ans: At most ] ) is. If x B 0, i.e. x 0, then the solution is a Basic Feasible Solution (BFS). 6
7 Example Consider the following polyhedral set: X := {x R 2 x 1 + x 2 6, x 2 3, x 1 0 x 2 0}. In standard form: x 1 + x 2 + x 3 = 6 x 2 + x 4 = 3 x 1, x 2, x 3, x 4, 0 [ ] Note A =[a 1,a 2,a 3,a 4 ]= Choose A B =[a 1,a 2 ]. Then x B =(x 1,x 2 ) T =(3, 3) T. The solution x =(3, 3, 0, 0) T is a basic solution (also BFS). Choose A B =[a 2,a 4 ]. Then x B =(x 2,x 4 ) T =(6, 3) T. The solution x =(0, 6, 0, 3) T is a basic solution (not BFS). Note that there are 5 = a 1 and a 3 are not l.i.). ( 4 2 ) 1 basic solutions (since 7
8 Example (contd) A x BFS [a 1,a 2 ] (3, 3, 0, 0) T Yes [a 1,a 4 ] (6, 0, 0, 3) T Yes [a 2,a 3 ] (0, 3, 3, 0) T Yes [a 2,a 4 ] (0, 6, 0, 3) T No [a 3,a 4 ] (0, 0, 6, 3) T Yes x1+x2 = 6 (0,6) (0,3) (3,3) x2 = 3 (0,0) (6,0) Each BFS correspond to an extreme point of the feasible region. Algorithm? Check all BFS (extreme points) and choose best. Exponential impractical for large problems. 8
9 Optimization Strategy Idea: Move from one extreme point to a better (adjacent) extreme point. Need to characterize adjacent extreme points. Need a strategy to move to a better adjacent extreme point. Need a stopping condition. Need to detect unboundedness and infeasibility. 9
10 Adjacent BFS Two extreme points are adjacent if they share an edge of the polyhedron. Algebraically, two BFS x 1 and x 2 are adjacent if (only) one of the basic variables in BFS x 1 is non-basic in BFS x 2, and vice versa. Moving from a BFS to an adjacent BFS one of the non-basic variables takes on a positive value (enters the basis), while one of the basic variables drops down to zero (leaves the basis). Values of other basic variables change too. (0, 3, 3, 0) (3, 3, 0, 0) (0, 0, 6, 3) (6, 0, 0, 4) 10
11 Improving Direction Which non-basic variable should enter the basis? Ans: One that improves the objective function value (z). Let N := {j x j is nonbasic, i.e. a j is a column of A N }. Also partition c = [ cb c N ]. Now, A B x B + A N x N = b x B = A 1 B b A 1 B A Nx N. The objective function value z = c T B x B + c T N x N = c T B A 1 B b + ( c T N ct B A 1 B A N ) xn = c T B A 1 B b + j N = c T B A 1 B b + j N (c j c T } B{{ A 1 B a j) x } j r j x j. Note for the current bfs the objective value is z = c T B x B = c T B A 1 B b. If a nonbasic variable x j becomes positive, the objective value increases by r j. Therefore choose j s.t. r j < 0 to reduce the obj. func. The quantity r j is known as the reduced cost for nonbasic variable x j. If r j 0 for all j N then the current solution is optimal. r j 11
12 Step Length How much should the value of the (chosen) nonbasic variable be increased? Ans: As long as all of the basic variables 0. Suppose we chose the nonbasic variable j to increase (enter into the basis). Then x B = A 1 B b A 1 B a jx j. Let B = {i x i is basic}, andlety ij =(A 1 B a j) i where i B and j N. Then, x i =(A 1 B b) i y ij x j i B. Suppose y ij > 0, then increasing x j would reduce x i. For feasibility, we must have x i 0foralli B,i.e. if y ij > 0 (A 1 B b) i y ij x j 0 i B, i.e. x j (A 1 B b) i y ij i B. Thus we can increase x j to: x j = min {i B y ij >0} (A 1 B b) i y ij. 12
13 Step Length (contd.) Suppose the minimization in the above expression is achieved for î B. Then we increase x j =(A 1 B b) î /y îj, and consequently the basic variable xî drops to zero (leaves the basis). Suppose y ij 0foralli B, then increasing x j would not cause any of the basic variables to decrease in value. We can then increase x j as much as we want without losing feasibility and keep on improving the objective. In this case the problem is unbounded. What if min {i B yij >0} { } (A 1 B b) i y ij = 0? This implies that one of the basic variables was at a value zero. Such a BFS is known as a degenerate BFS and may cause problems in the algorithm. However there are ways of recovering from degeneracy problems. 13
14 Finding an Initial BFS Consider an LP min{c T x Ax = b, x 0}. If A has an m m identity submatrix I, thenusei as the initial basis. Otherwise, suitably multiply the constraints by ±1 such that b 0. Introduce auxiliary variables y = (y 1,...,y m ) T, and construct the following LP: min e T y s.t. Ax+ Iy = b, x, y 0. Use I as the starting basis, and solve the above problem to optimality. If the optimal objective value is zero y =0,then we have a basis consisting of only the columns of A, which we can use as an initial BFS for the original problem. If the optimal objective value > 0, then the original problem is infeasible. 14
15 The Simplex Method - G. B. Dantzig (1947) 1. Find an initial feasible extreme point solution (BFS). If non exists, STOP, the LP is infeasible. 2. Find a direction (along an edge) in which the objective function improves (i.e. find j N such that r j < 0). If none exists, STOP, the current solution is optimal. 3. Move along the direction until you reach an adjacent feasible extreme point (BFS). If you can move along the edge of the feasible region without reaching an adjacent feasible extreme point (i.e. y ij 0), then STOP, the LP is unbounded. 4. Go to step 2 with the current feasible extreme point. 15
16 Remarks In the absence of degeneracy, the Simplexmethod terminates finitely with an optimal solution to the LP. In the worst case, it might require terminate. ( n m ) steps to In most practical problems the performance is much better. The performance is sensitive to the number of rows. Degeneracy might cause the method to cycle. However this can be avoided by some anti-cycling rules in choosing the variables leaving and entering the basis. The optimal solution produced by the simplexalgorithm is a BFS, therefore, (at most) m of the n variables are positive. 16
Advanced 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 informationLecture Notes 2: The Simplex Algorithm
Algorithmic Methods 25/10/2010 Lecture Notes 2: The Simplex Algorithm Professor: Yossi Azar Scribe:Kiril Solovey 1 Introduction In this lecture we will present the Simplex algorithm, finish some unresolved
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 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 informationCOMP331/557. Chapter 2: The Geometry of Linear Programming. (Bertsimas & Tsitsiklis, Chapter 2)
COMP331/557 Chapter 2: The Geometry of Linear Programming (Bertsimas & Tsitsiklis, Chapter 2) 49 Polyhedra and Polytopes Definition 2.1. Let A 2 R m n and b 2 R m. a set {x 2 R n A x b} is called polyhedron
More informationLesson 17. Geometry and Algebra of Corner Points
SA305 Linear Programming Spring 2016 Asst. Prof. Nelson Uhan 0 Warm up Lesson 17. Geometry and Algebra of Corner Points Example 1. Consider the system of equations 3 + 7x 3 = 17 + 5 = 1 2 + 11x 3 = 24
More informationCS 473: Algorithms. Ruta Mehta. Spring University of Illinois, Urbana-Champaign. Ruta (UIUC) CS473 1 Spring / 29
CS 473: Algorithms Ruta Mehta University of Illinois, Urbana-Champaign Spring 2018 Ruta (UIUC) CS473 1 Spring 2018 1 / 29 CS 473: Algorithms, Spring 2018 Simplex and LP Duality Lecture 19 March 29, 2018
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 informationSection Notes 5. Review of Linear Programming. Applied Math / Engineering Sciences 121. Week of October 15, 2017
Section Notes 5 Review of Linear Programming Applied Math / Engineering Sciences 121 Week of October 15, 2017 The following list of topics is an overview of the material that was covered in the lectures
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 informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 3: Linear Programming, Continued Prof. John Gunnar Carlsson September 15, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 15, 2010
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 informationLinear programming and duality theory
Linear programming and duality theory Complements of Operations Research Giovanni Righini Linear Programming (LP) A linear program is defined by linear constraints, a linear objective function. Its variables
More informationLP Geometry: outline. A general LP. minimize x c T x s.t. a T i. x b i, i 2 M 1 a T i x = b i, i 2 M 3 x j 0, j 2 N 1. where
LP Geometry: outline I Polyhedra I Extreme points, vertices, basic feasible solutions I Degeneracy I Existence of extreme points I Optimality of extreme points IOE 610: LP II, Fall 2013 Geometry of Linear
More informationAn iteration of the simplex method (a pivot )
Recap, and outline of Lecture 13 Previously Developed and justified all the steps in a typical iteration ( pivot ) of the Simplex Method (see next page). Today Simplex Method Initialization Start with
More informationAdvanced Operations Research Techniques IE316. Quiz 2 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 2 Review Dr. Ted Ralphs IE316 Quiz 2 Review 1 Reading for The Quiz Material covered in detail in lecture Bertsimas 4.1-4.5, 4.8, 5.1-5.5, 6.1-6.3 Material
More informationORIE 6300 Mathematical Programming I September 2, Lecture 3
ORIE 6300 Mathematical Programming I September 2, 2014 Lecturer: David P. Williamson Lecture 3 Scribe: Divya Singhvi Last time we discussed how to take dual of an LP in two different ways. Today we will
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 informationSubmodularity Reading Group. Matroid Polytopes, Polymatroid. M. Pawan Kumar
Submodularity Reading Group Matroid Polytopes, Polymatroid M. Pawan Kumar http://www.robots.ox.ac.uk/~oval/ Outline Linear Programming Matroid Polytopes Polymatroid Polyhedron Ax b A : m x n matrix b:
More information3. The Simplex algorithmn The Simplex algorithmn 3.1 Forms of linear programs
11 3.1 Forms of linear programs... 12 3.2 Basic feasible solutions... 13 3.3 The geometry of linear programs... 14 3.4 Local search among basic feasible solutions... 15 3.5 Organization in tableaus...
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 informationLinear Optimization. Andongwisye John. November 17, Linkoping University. Andongwisye John (Linkoping University) November 17, / 25
Linear Optimization Andongwisye John Linkoping University November 17, 2016 Andongwisye John (Linkoping University) November 17, 2016 1 / 25 Overview 1 Egdes, One-Dimensional Faces, Adjacency of Extreme
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 informationCS675: Convex and Combinatorial Optimization Spring 2018 The Simplex Algorithm. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Spring 2018 The Simplex Algorithm Instructor: Shaddin Dughmi Algorithms for Convex Optimization We will look at 2 algorithms in detail: Simplex and Ellipsoid.
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 informationmaximize c, x subject to Ax b,
Lecture 8 Linear programming is about problems of the form maximize c, x subject to Ax b, where A R m n, x R n, c R n, and b R m, and the inequality sign means inequality in each row. The feasible set
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 informationPOLYHEDRAL GEOMETRY. Convex functions and sets. Mathematical Programming Niels Lauritzen Recall that a subset C R n is convex if
POLYHEDRAL GEOMETRY Mathematical Programming Niels Lauritzen 7.9.2007 Convex functions and sets Recall that a subset C R n is convex if {λx + (1 λ)y 0 λ 1} C for every x, y C and 0 λ 1. A function f :
More informationIntroduction to Mathematical Programming IE406. Lecture 4. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 4 Dr. Ted Ralphs IE406 Lecture 4 1 Reading for This Lecture Bertsimas 2.2-2.4 IE406 Lecture 4 2 The Two Crude Petroleum Example Revisited Recall the
More informationMath 414 Lecture 2 Everyone have a laptop?
Math 44 Lecture 2 Everyone have a laptop? THEOREM. Let v,...,v k be k vectors in an n-dimensional space and A = [v ;...; v k ] v,..., v k independent v,..., v k span the space v,..., v k a basis v,...,
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 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 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 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 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 informationLecture 2 - Introduction to Polytopes
Lecture 2 - Introduction to Polytopes Optimization and Approximation - ENS M1 Nicolas Bousquet 1 Reminder of Linear Algebra definitions Let x 1,..., x m be points in R n and λ 1,..., λ m be real numbers.
More informationMath 5593 Linear Programming Lecture Notes
Math 5593 Linear Programming Lecture Notes Unit II: Theory & Foundations (Convex Analysis) University of Colorado Denver, Fall 2013 Topics 1 Convex Sets 1 1.1 Basic Properties (Luenberger-Ye Appendix B.1).........................
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 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 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 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 informationAMS : Combinatorial Optimization Homework Problems - Week V
AMS 553.766: Combinatorial Optimization Homework Problems - Week V For the following problems, A R m n will be m n matrices, and b R m. An affine subspace is the set of solutions to a a system of linear
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 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 Linear Programming. Organisation. Lecturers: Leen Stougie, CWI and Vrije Universiteit in Amsterdam
Advanced Linear Programming Organisation Lecturers: Leen Stougie, CWI and Vrije Universiteit in Amsterdam E-mail: stougie@cwi.nl Marjan van den Akker Universiteit Utrecht marjan@cs.uu.nl Advanced Linear
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 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 informationPolytopes Course Notes
Polytopes Course Notes Carl W. Lee Department of Mathematics University of Kentucky Lexington, KY 40506 lee@ms.uky.edu Fall 2013 i Contents 1 Polytopes 1 1.1 Convex Combinations and V-Polytopes.....................
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 informationNATCOR Convex Optimization Linear Programming 1
NATCOR Convex Optimization Linear Programming 1 Julian Hall School of Mathematics University of Edinburgh jajhall@ed.ac.uk 5 June 2018 What is linear programming (LP)? The most important model used in
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 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 informationWhat is linear programming (LP)? NATCOR Convex Optimization Linear Programming 1. Solving LP problems: The standard simplex method
NATCOR Convex Optimization Linear Programming 1 Julian Hall School of Mathematics University of Edinburgh jajhall@ed.ac.uk 14 June 2016 What is linear programming (LP)? The most important model used in
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 informationDuality. Primal program P: Maximize n. Dual program D: Minimize m. j=1 c jx j subject to n. j=1. i=1 b iy i subject to m. i=1
Duality Primal program P: Maximize n j=1 c jx j subject to n a ij x j b i, i = 1, 2,..., m j=1 x j 0, j = 1, 2,..., n Dual program D: Minimize m i=1 b iy i subject to m a ij x j c j, j = 1, 2,..., n i=1
More information15.082J and 6.855J. Lagrangian Relaxation 2 Algorithms Application to LPs
15.082J and 6.855J Lagrangian Relaxation 2 Algorithms Application to LPs 1 The Constrained Shortest Path Problem (1,10) 2 (1,1) 4 (2,3) (1,7) 1 (10,3) (1,2) (10,1) (5,7) 3 (12,3) 5 (2,2) 6 Find the shortest
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 informationCS522: Advanced Algorithms
Lecture 1 CS5: Advanced Algorithms October 4, 004 Lecturer: Kamal Jain Notes: Chris Re 1.1 Plan for the week Figure 1.1: Plan for the week The underlined tools, weak duality theorem and complimentary slackness,
More informationMATH 310 : Degeneracy and Geometry in the Simplex Method
MATH 310 : Degeneracy and Geometry in the Simplex Method Fayadhoi Ibrahima December 11, 2013 1 Introduction This project is exploring a bit deeper the study of the simplex method introduced in 1947 by
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 informationTheory of Linear Programming
62 Chapter 8 Theory of Linear Programming In this chapter the results of previous chapters will be applied to Linear programming problems (LP. We have already seen that the feasible set of a linear programming
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 informationSimplex Algorithm in 1 Slide
Administrivia 1 Canonical form: Simplex Algorithm in 1 Slide If we do pivot in A r,s >0, where c s
More informationApplied Integer Programming
Applied Integer Programming D.S. Chen; R.G. Batson; Y. Dang Fahimeh 8.2 8.7 April 21, 2015 Context 8.2. Convex sets 8.3. Describing a bounded polyhedron 8.4. Describing unbounded polyhedron 8.5. Faces,
More informationDual-fitting analysis of Greedy for Set Cover
Dual-fitting analysis of Greedy for Set Cover We showed earlier that the greedy algorithm for set cover gives a H n approximation We will show that greedy produces a solution of cost at most H n OPT LP
More informationIntroductory Operations Research
Introductory Operations Research Theory and Applications Bearbeitet von Harvir Singh Kasana, Krishna Dev Kumar 1. Auflage 2004. Buch. XI, 581 S. Hardcover ISBN 978 3 540 40138 4 Format (B x L): 15,5 x
More informationComputational Geometry
Casting a polyhedron CAD/CAM systems CAD/CAM systems allow you to design objects and test how they can be constructed Many objects are constructed used a mold Casting Casting A general question: Given
More informationA PRIMAL-DUAL EXTERIOR POINT ALGORITHM FOR LINEAR PROGRAMMING PROBLEMS
Yugoslav Journal of Operations Research Vol 19 (2009), Number 1, 123-132 DOI:10.2298/YUJOR0901123S A PRIMAL-DUAL EXTERIOR POINT ALGORITHM FOR LINEAR PROGRAMMING PROBLEMS Nikolaos SAMARAS Angelo SIFELARAS
More informationThe Simplex Method applies to linear programming problems in standard form :
Chapter 9 asic on the simplex method The Simplex Method is certainly the most famous and most used algorithm in optimization. Proposed in 1947 by G..Dantzig, he has undergone, in over 50 years of life,
More informationLinear Programming. Larry Blume. Cornell University & The Santa Fe Institute & IHS
Linear Programming Larry Blume Cornell University & The Santa Fe Institute & IHS Linear Programs The general linear program is a constrained optimization problem where objectives and constraints are all
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 informationModeling and Analysis of Hybrid Systems
Modeling and Analysis of Hybrid Systems Convex polyhedra Prof. Dr. Erika Ábrahám Informatik 2 - LuFG Theory of Hybrid Systems RWTH Aachen University Szeged, Hungary, 27 September - 06 October 2017 Ábrahám
More informationModeling and Analysis of Hybrid Systems
Modeling and Analysis of Hybrid Systems 6. Convex polyhedra Prof. Dr. Erika Ábrahám Informatik 2 - LuFG Theory of Hybrid Systems RWTH Aachen University Szeged, Hungary, 27 September - 06 October 2017 Ábrahám
More informationACTUALLY DOING IT : an Introduction to Polyhedral Computation
ACTUALLY DOING IT : an Introduction to Polyhedral Computation Jesús A. De Loera Department of Mathematics Univ. of California, Davis http://www.math.ucdavis.edu/ deloera/ 1 What is a Convex Polytope? 2
More informationLecture 5: Properties of convex sets
Lecture 5: Properties of convex sets Rajat Mittal IIT Kanpur This week we will see properties of convex sets. These properties make convex sets special and are the reason why convex optimization problems
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 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 information4 Linear Programming (LP) E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 1
4 Linear Programming (LP) E. Amaldi -- Foundations of Operations Research -- Politecnico di Milano 1 Definition: A Linear Programming (LP) problem is an optimization problem: where min f () s.t. X n the
More informationLECTURE 6: INTERIOR POINT METHOD. 1. Motivation 2. Basic concepts 3. Primal affine scaling algorithm 4. Dual affine scaling algorithm
LECTURE 6: INTERIOR POINT METHOD 1. Motivation 2. Basic concepts 3. Primal affine scaling algorithm 4. Dual affine scaling algorithm Motivation Simplex method works well in general, but suffers from exponential-time
More informationPolyhedral Computation Today s Topic: The Double Description Algorithm. Komei Fukuda Swiss Federal Institute of Technology Zurich October 29, 2010
Polyhedral Computation Today s Topic: The Double Description Algorithm Komei Fukuda Swiss Federal Institute of Technology Zurich October 29, 2010 1 Convexity Review: Farkas-Type Alternative Theorems Gale
More information6.854 Advanced Algorithms. Scribes: Jay Kumar Sundararajan. Duality
6.854 Advanced Algorithms Scribes: Jay Kumar Sundararajan Lecturer: David Karger Duality This lecture covers weak and strong duality, and also explains the rules for finding the dual of a linear program,
More informationPolar Duality and Farkas Lemma
Lecture 3 Polar Duality and Farkas Lemma October 8th, 2004 Lecturer: Kamal Jain Notes: Daniel Lowd 3.1 Polytope = bounded polyhedron Last lecture, we were attempting to prove the Minkowsky-Weyl Theorem:
More information/ Approximation Algorithms Lecturer: Michael Dinitz Topic: Linear Programming Date: 2/24/15 Scribe: Runze Tang
600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Linear Programming Date: 2/24/15 Scribe: Runze Tang 9.1 Linear Programming Suppose we are trying to approximate a minimization
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 informationLinear Programming. Linear programming provides methods for allocating limited resources among competing activities in an optimal way.
University of Southern California Viterbi School of Engineering Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research - Deterministic Models Fall
More informationLecture 12: Feasible direction methods
Lecture 12 Lecture 12: Feasible direction methods Kin Cheong Sou December 2, 2013 TMA947 Lecture 12 Lecture 12: Feasible direction methods 1 / 1 Feasible-direction methods, I Intro Consider the problem
More informationConvex Geometry arising in Optimization
Convex Geometry arising in Optimization Jesús A. De Loera University of California, Davis Berlin Mathematical School Summer 2015 WHAT IS THIS COURSE ABOUT? Combinatorial Convexity and Optimization PLAN
More informationMathematical Programming and Research Methods (Part II)
Mathematical Programming and Research Methods (Part II) 4. Convexity and Optimization Massimiliano Pontil (based on previous lecture by Andreas Argyriou) 1 Today s Plan Convex sets and functions Types
More informationCOLUMN GENERATION IN LINEAR PROGRAMMING
COLUMN GENERATION IN LINEAR PROGRAMMING EXAMPLE: THE CUTTING STOCK PROBLEM A certain material (e.g. lumber) is stocked in lengths of 9, 4, and 6 feet, with respective costs of $5, $9, and $. An order for
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 informationFACES OF CONVEX SETS
FACES OF CONVEX SETS VERA ROSHCHINA Abstract. We remind the basic definitions of faces of convex sets and their basic properties. For more details see the classic references [1, 2] and [4] for polytopes.
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 informationFinite Math Linear Programming 1 May / 7
Linear Programming Finite Math 1 May 2017 Finite Math Linear Programming 1 May 2017 1 / 7 General Description of Linear Programming Finite Math Linear Programming 1 May 2017 2 / 7 General Description of
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 informationCircuit Walks in Integral Polyhedra
Circuit Walks in Integral Polyhedra Charles Viss Steffen Borgwardt University of Colorado Denver Optimization and Discrete Geometry: Theory and Practice Tel Aviv University, April 2018 LINEAR PROGRAMMING
More informationPolyhedral Computation and their Applications. Jesús A. De Loera Univ. of California, Davis
Polyhedral Computation and their Applications Jesús A. De Loera Univ. of California, Davis 1 1 Introduction It is indeniable that convex polyhedral geometry is an important tool of modern mathematics.
More information5.3 Cutting plane methods and Gomory fractional cuts
5.3 Cutting plane methods and Gomory fractional cuts (ILP) min c T x s.t. Ax b x 0integer feasible region X Assumption: a ij, c j and b i integer. Observation: The feasible region of an ILP can be described
More informationRead: H&L chapters 1-6
Viterbi School of Engineering Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research Fall 2006 (Oct 16): Midterm Review http://www-scf.usc.edu/~ise330
More informationNotes taken by Mea Wang. February 11, 2005
CSC2411 - Linear Programming and Combinatorial Optimization Lecture 5: Smoothed Analysis, Randomized Combinatorial Algorithms, and Linear Programming Duality Notes taken by Mea Wang February 11, 2005 Summary:
More informationbe a polytope. has such a representation iff it contains the origin in its interior. For a generic, sort the inequalities so that
( Shelling (Bruggesser-Mani 1971) and Ranking Let be a polytope. has such a representation iff it contains the origin in its interior. For a generic, sort the inequalities so that. a ranking of vertices
More information