The Chvátal-Gomory Closure of a Strictly Convex Body is a Rational Polyhedron

Size: px
Start display at page:

Download "The Chvátal-Gomory Closure of a Strictly Convex Body is a Rational Polyhedron"

Transcription

1 The Chvátal-Gomory Closure of a Strictly Convex Body is a Rational Polyhedron Juan Pablo Vielma Joint work with Daniel Dadush and Santanu S. Dey July, Atlanta, GA

2 Outline Introduction Proof: Step Step Intersection with Rational Polyhedra Example of Non-Polyhedral Closure. Conclusions and Future Work /

3 Introduction CG Cuts for Rational Polyhedra /

4 Introduction CG Cuts for Rational Polyhedra /

5 Introduction CG Cuts for Rational Polyhedra /

6 Introduction CG Cuts for Rational Polyhedra /

7 Introduction CG Cuts for General Convex Sets /

8 Introduction CG Cuts for General Convex Sets /

9 Introduction CG Cuts for General Convex Sets /

10 Introduction CG Cuts for General Convex Sets /

11 Introduction CG Closure of a Convex Set CG Closure: Is CG closure a polyhedron? Finite set s.t. Yes, for rational polyhedra (Schrijver, 98) What about other convex sets? /

12 Introduction What we know for Convex Bodies There exists k s.t. (Chvátal, 97) Also for unbounded rational polyhedra ( Schrijver, 98). Result does not imply polyhedrality of /

13 Proof Proof Outline: Generation Procedure Step : Construct a finite set such that 7/

14 Proof Proof Outline: Generation Procedure Step : Construct a finite set such that Step : Construct a finite set such that CG cuts from separate all points in all of 7/

15 Proof Proof Outline: Generation Procedure Step : Construct a finite set such that Step : Construct a finite set such that CG cuts from separate all points in all of 7/

16 Proof: Step Outline of Step Step : Construct a finite set such that and (a) Separate non-integral points in. (b) Separate neighborhood of integral points in. (c) Compactness argument to construct finite. 8/

17 Proof: Step Separate non-integral points in /

18 Proof: Step Separate non-integral points in /

19 Proof: Step Separate non-integral points in /

20 Proof: Step Separate non-integral points in /

21 Proof: Step Separate non-integral points in 9/

22 Proof: Step Separate non-integral points in 9/

23 Proof: Step Separate non-integral points in 9/

24 Proof: Step Separate non-integral points in 9/

25 Proof: Step Separate neighborhood of integers /

26 Proof: Step Separate neighborhood of integers /

27 Proof: Step Separate neighborhood of integers /

28 Proof: Step Separate neighborhood of integers Similar to non-integer separation + compactness argument /

29 Proof: Step Compactness Argument /

30 Proof: Step Compactness Argument /

31 Proof: Step Compactness Argument /

32 Proof: Step Compactness Argument /

33 Proof: Step Compactness Argument /

34 Proof: Step Compactness Argument compact /

35 Proof: Step Compactness Argument /

36 Proof: Step Compactness Argument /

37 Proof: Step Compactness Argument /

38 Proof: Step Compactness Argument /

39 Proof: Step Compactness Argument /

40 Proof: Step Compactness Argument /

41 Proof: Step Compactness Argument /

42 Proof: Step Step : Separate /

43 Proof: Step Step : Separate /

44 Proof: Step Step : Separate ε > εb n +v C v V /

45 Proof: Step Step : Separate ε > εb n +v C v V /

46 Proof: Step Step : Separate ε > εb n +v C v V a ε σ C (a) σ C (a) σ v+εb n(a) = v,a+εa v,a /

47 Proof: Step Step : Separate ε > εb n +v C v V a ε σ C (a) σ C (a) σ v+εb n(a) = v,a+εa v,a S =(/ε)b Z n /

48 Strictly Convex Rational Polyhedron Example: Ellipsoid and Halfspace P polyhedron, F face of P CGC(F)=CGC(C) F (Schrijver, 98) /

49 Strictly Convex Rational Polyhedron Example: Ellipsoid and Halfspace P polyhedron, F face of P CGC(F)=CGC(C) F (Schrijver, 98) We can generalize it /

50 Strictly Convex Rational Polyhedron Example: Ellipsoid and Halfspace P polyhedron, F face of P CGC(F)=CGC(C) F (Schrijver, 98) We can generalize it. a,x σ F (a) /

51 Strictly Convex Rational Polyhedron Example: Ellipsoid and Halfspace P polyhedron, F face of P CGC(F)=CGC(C) F (Schrijver, 98) We can generalize it. a,x σ F (a) a,x σ C (a ) /

52 Example of Non-Polyhedral Closure Split Closure of an Ellipsoid Pure Integer Case: C = {x R : A(x c) } A =, c=(/,/,/) T / / Two split cuts: x x x x /

53 Conclusions and Future Work Non-Constructive because of compactness argument in step. Current work: General compact convex sets including nonrational polytopes ( Almost done). Split closure is finitely generated. Open Problems: Simpler Proof (Circle in?). R Constructive/Algorithmic proof. /

Applied Integer Programming

Applied 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 information

COMP331/557. Chapter 2: The Geometry of Linear Programming. (Bertsimas & Tsitsiklis, Chapter 2)

COMP331/557. Chapter 2: The Geometry of Linear Programming. (Bertsimas & Tsitsiklis, Chapter 2) 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 information

A Verification Based Method to Generate Cutting Planes for IPs

A Verification Based Method to Generate Cutting Planes for IPs A Verification Based Method to Generate Cutting Planes for IPs Santanu S. Dey Sebastian Pokutta Georgia Institute of Technology, USA. Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany. SIAM Conference

More information

Split-Cuts and the Stable Set Polytope of Quasi-Line Graphs

Split-Cuts and the Stable Set Polytope of Quasi-Line Graphs Split-Cuts and the Stable Set Polytope of Quasi-Line Graphs Friedrich Eisenbrand Joint work with G. Oriolo, P. Ventura and G. Stauffer Gomory cutting planes P = {x n Ax b} polyhedron, c T x δ, c n valid

More information

Integer Programming Theory

Integer 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 information

Splitting the Control Flow with Boolean Flags

Splitting the Control Flow with Boolean Flags École Normale Supérieure, Paris, France A.Simon@ens.fr July 2008 Good States are Usually Convex Declare C variable int array[12];. 0 1 2 3 4 5 6 7 8 9 10 11 i Access array[i] within bound if 0 i and i

More information

CS675: Convex and Combinatorial Optimization Spring 2018 Convex Sets. Instructor: Shaddin Dughmi

CS675: Convex and Combinatorial Optimization Spring 2018 Convex Sets. Instructor: Shaddin Dughmi CS675: Convex and Combinatorial Optimization Spring 2018 Convex Sets Instructor: Shaddin Dughmi Outline 1 Convex sets, Affine sets, and Cones 2 Examples of Convex Sets 3 Convexity-Preserving Operations

More information

Stable sets, corner polyhedra and the Chvátal closure

Stable sets, corner polyhedra and the Chvátal closure Stable sets, corner polyhedra and the Chvátal closure Manoel Campêlo Departamento de Estatística e Matemática Aplicada, Universidade Federal do Ceará, Brazil, mcampelo@lia.ufc.br. Gérard Cornuéjols Tepper

More information

Edge unfolding Cut sets Source foldouts Proof and algorithm Complexity issues Aleksandrov unfolding? Unfolding polyhedra.

Edge unfolding Cut sets Source foldouts Proof and algorithm Complexity issues Aleksandrov unfolding? Unfolding polyhedra. Unfolding polyhedra Ezra Miller University of Minnesota ezra@math.umn.edu University of Nebraska 27 April 2007 Outline 1. Edge unfolding 2. Cut sets 3. Source foldouts 4. Proof and algorithm 5. Complexity

More information

Lecture 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. 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 information

Optimality certificates for convex minimization and Helly numbers

Optimality certificates for convex minimization and Helly numbers Optimality certificates for convex minimization and Helly numbers Amitabh Basu Michele Conforti Gérard Cornuéjols Robert Weismantel Stefan Weltge October 20, 2016 Abstract We consider the problem of minimizing

More information

Integer Programming Chapter 9

Integer Programming Chapter 9 Integer Programming Chapter 9 University of Chicago Booth School of Business Kipp Martin October 25, 2017 1 / 40 Outline Key Concepts MILP Set Monoids LP set Relaxation of MILP Set Formulation Quality

More information

CS671 Parallel Programming in the Many-Core Era

CS671 Parallel Programming in the Many-Core Era 1 CS671 Parallel Programming in the Many-Core Era Polyhedral Framework for Compilation: Polyhedral Model Representation, Data Dependence Analysis, Scheduling and Data Locality Optimizations December 3,

More information

Linear programming and duality theory

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

More information

Optimality certificates for convex minimization and Helly numbers

Optimality certificates for convex minimization and Helly numbers Optimality certificates for convex minimization and Helly numbers Amitabh Basu Michele Conforti Gérard Cornuéjols Robert Weismantel Stefan Weltge May 10, 2017 Abstract We consider the problem of minimizing

More information

CS599: Convex and Combinatorial Optimization Fall 2013 Lecture 4: Convex Sets. Instructor: Shaddin Dughmi

CS599: Convex and Combinatorial Optimization Fall 2013 Lecture 4: Convex Sets. Instructor: Shaddin Dughmi CS599: Convex and Combinatorial Optimization Fall 2013 Lecture 4: Convex Sets Instructor: Shaddin Dughmi Announcements New room: KAP 158 Today: Convex Sets Mostly from Boyd and Vandenberghe. Read all of

More information

On the polyhedrality of cross and quadrilateral closures

On the polyhedrality of cross and quadrilateral closures On the polyhedrality of cross and quadrilateral closures Sanjeeb Dash IBM Research sanjeebd@us.ibm.com Oktay Günlük IBM Research gunluk@us.ibm.com December 9, 2014 Diego A. Morán R. Virginia Tech dmoran@gatech.edu

More information

Combinatorial Geometry & Topology arising in Game Theory and Optimization

Combinatorial Geometry & Topology arising in Game Theory and Optimization Combinatorial Geometry & Topology arising in Game Theory and Optimization Jesús A. De Loera University of California, Davis LAST EPISODE... We discuss the content of the course... Convex Sets A set is

More information

Lecture 4: Rational IPs, Polyhedron, Decomposition Theorem

Lecture 4: Rational IPs, Polyhedron, Decomposition Theorem IE 5: Integer Programming, Spring 29 24 Jan, 29 Lecture 4: Rational IPs, Polyhedron, Decomposition Theorem Lecturer: Karthik Chandrasekaran Scribe: Setareh Taki Disclaimer: These notes have not been subjected

More information

Convex Geometry arising in Optimization

Convex 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 information

Modeling and Analysis of Hybrid Systems

Modeling 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 information

Modeling and Analysis of Hybrid Systems

Modeling 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 information

Chapter 4 Concepts from Geometry

Chapter 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 information

Math 414 Lecture 2 Everyone have a laptop?

Math 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 information

ORIE 6300 Mathematical Programming I September 2, Lecture 3

ORIE 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 information

Combinatorial Optimization

Combinatorial Optimization Combinatorial Optimization MA4502 Michael Ritter This work is licensed under a Creative Commons Attribution- ShareAlike 4.0 International license. last updated July 14, 2015 Contents 1 Introduction and

More information

Convex Optimization Lecture 2

Convex Optimization Lecture 2 Convex Optimization Lecture 2 Today: Convex Analysis Center-of-mass Algorithm 1 Convex Analysis Convex Sets Definition: A set C R n is convex if for all x, y C and all 0 λ 1, λx + (1 λ)y C Operations that

More information

Embedding Formulations, Complexity and Representability for Unions of Convex Sets

Embedding Formulations, Complexity and Representability for Unions of Convex Sets , Complexity and Representability for Unions of Convex Sets Juan Pablo Vielma Massachusetts Institute of Technology CMO-BIRS Workshop: Modern Techniques in Discrete Optimization: Mathematics, Algorithms

More information

LP 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. 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 information

Convex Optimization - Chapter 1-2. Xiangru Lian August 28, 2015

Convex Optimization - Chapter 1-2. Xiangru Lian August 28, 2015 Convex Optimization - Chapter 1-2 Xiangru Lian August 28, 2015 1 Mathematical optimization minimize f 0 (x) s.t. f j (x) 0, j=1,,m, (1) x S x. (x 1,,x n ). optimization variable. f 0. R n R. objective

More information

Convexity: an introduction

Convexity: an introduction Convexity: an introduction Geir Dahl CMA, Dept. of Mathematics and Dept. of Informatics University of Oslo 1 / 74 1. Introduction 1. Introduction what is convexity where does it arise main concepts and

More information

maximize c, x subject to Ax b,

maximize 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 information

Intersection Cuts with Infinite Split Rank

Intersection Cuts with Infinite Split Rank Intersection Cuts with Infinite Split Rank Amitabh Basu 1,2, Gérard Cornuéjols 1,3,4 François Margot 1,5 April 2010; revised April 2011; revised October 2011 Abstract We consider mixed integer linear programs

More information

THEORY OF LINEAR AND INTEGER PROGRAMMING

THEORY OF LINEAR AND INTEGER PROGRAMMING THEORY OF LINEAR AND INTEGER PROGRAMMING ALEXANDER SCHRIJVER Centrum voor Wiskunde en Informatica, Amsterdam A Wiley-Inter science Publication JOHN WILEY & SONS^ Chichester New York Weinheim Brisbane Singapore

More information

12.1 Formulation of General Perfect Matching

12.1 Formulation of General Perfect Matching CSC5160: Combinatorial Optimization and Approximation Algorithms Topic: Perfect Matching Polytope Date: 22/02/2008 Lecturer: Lap Chi Lau Scribe: Yuk Hei Chan, Ling Ding and Xiaobing Wu In this lecture,

More information

Polar Duality and Farkas Lemma

Polar 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

CS675: 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 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 information

CS 435, 2018 Lecture 2, Date: 1 March 2018 Instructor: Nisheeth Vishnoi. Convex Programming and Efficiency

CS 435, 2018 Lecture 2, Date: 1 March 2018 Instructor: Nisheeth Vishnoi. Convex Programming and Efficiency CS 435, 2018 Lecture 2, Date: 1 March 2018 Instructor: Nisheeth Vishnoi Convex Programming and Efficiency In this lecture, we formalize convex programming problem, discuss what it means to solve it efficiently

More information

Math 5593 Linear Programming Lecture Notes

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

More information

A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed Integer Conic Quadratic Programs

A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed Integer Conic Quadratic Programs A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed Integer Conic Quadratic Programs Juan Pablo Vielma Shabbir Ahmed George L. Nemhauser H. Milton Stewart School of Industrial and Systems

More information

Submodularity Reading Group. Matroid Polytopes, Polymatroid. M. Pawan Kumar

Submodularity 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 information

DM545 Linear and Integer Programming. Lecture 2. The Simplex Method. Marco Chiarandini

DM545 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 information

5.3 Cutting plane methods and Gomory fractional cuts

5.3 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 information

Introduction to Modern Control Systems

Introduction to Modern Control Systems Introduction to Modern Control Systems Convex Optimization, Duality and Linear Matrix Inequalities Kostas Margellos University of Oxford AIMS CDT 2016-17 Introduction to Modern Control Systems November

More information

Face Width and Graph Embeddings of face-width 2 and 3

Face Width and Graph Embeddings of face-width 2 and 3 Face Width and Graph Embeddings of face-width 2 and 3 Instructor: Robin Thomas Scribe: Amanda Pascoe 3/12/07 and 3/14/07 1 Representativity Recall the following: Definition 2. Let Σ be a surface, G a graph,

More information

ACTUALLY DOING IT : an Introduction to Polyhedral Computation

ACTUALLY 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 information

Linear and Integer Programming (ADM II) Script. Rolf Möhring WS 2010/11

Linear and Integer Programming (ADM II) Script. Rolf Möhring WS 2010/11 Linear and Integer Programming (ADM II) Script Rolf Möhring WS 200/ Contents -. Algorithmic Discrete Mathematics (ADM)... 3... 4.3 Winter term 200/... 5 2. Optimization problems 2. Examples... 7 2.2 Neighborhoods

More information

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

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

More information

Week 7 Convex Hulls in 3D

Week 7 Convex Hulls in 3D 1 Week 7 Convex Hulls in 3D 2 Polyhedra A polyhedron is the natural generalization of a 2D polygon to 3D 3 Closed Polyhedral Surface A closed polyhedral surface is a finite set of interior disjoint polygons

More information

S-free Sets for Polynomial Optimization

S-free Sets for Polynomial Optimization S-free Sets for and Daniel Bienstock, Chen Chen, Gonzalo Muñoz, IEOR, Columbia University May, 2017 SIAM OPT 2017 S-free sets for PolyOpt 1 The Polyhedral Approach TighteningP with an S-free setc conv(p

More information

R n a T i x = b i} is a Hyperplane.

R n a T i x = b i} is a Hyperplane. 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.

More information

POLYHEDRAL GEOMETRY. Convex functions and sets. Mathematical Programming Niels Lauritzen Recall that a subset C R n is convex if

POLYHEDRAL 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 information

LECTURE 10 LECTURE OUTLINE

LECTURE 10 LECTURE OUTLINE We now introduce a new concept with important theoretical and algorithmic implications: polyhedral convexity, extreme points, and related issues. LECTURE 1 LECTURE OUTLINE Polar cones and polar cone theorem

More information

Lecture 2 Convex Sets

Lecture 2 Convex Sets Optimization Theory and Applications Lecture 2 Convex Sets Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Fall 2016 2016/9/29 Lecture 2: Convex Sets 1 Outline

More information

CS 473: Algorithms. Ruta Mehta. Spring University of Illinois, Urbana-Champaign. Ruta (UIUC) CS473 1 Spring / 29

CS 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 information

C&O 355 Lecture 16. N. Harvey

C&O 355 Lecture 16. N. Harvey C&O 355 Lecture 16 N. Harvey Topics Review of Fourier-Motzkin Elimination Linear Transformations of Polyhedra Convex Combinations Convex Hulls Polytopes & Convex Hulls Fourier-Motzkin Elimination Joseph

More information

arxiv: v1 [math.co] 15 Dec 2009

arxiv: v1 [math.co] 15 Dec 2009 ANOTHER PROOF OF THE FACT THAT POLYHEDRAL CONES ARE FINITELY GENERATED arxiv:092.2927v [math.co] 5 Dec 2009 VOLKER KAIBEL Abstract. In this note, we work out a simple inductive proof showing that every

More information

11 Linear Programming

11 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 information

AMS : Combinatorial Optimization Homework Problems - Week V

AMS : 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 information

In this chapter we introduce some of the basic concepts that will be useful for the study of integer programming problems.

In this chapter we introduce some of the basic concepts that will be useful for the study of integer programming problems. 2 Basics In this chapter we introduce some of the basic concepts that will be useful for the study of integer programming problems. 2.1 Notation Let A R m n be a matrix with row index set M = {1,...,m}

More information

FINITE DISJUNCTIVE PROGRAMMING CHARACTERIZATIONS FOR GENERAL MIXED-INTEGER LINEAR PROGRAMS

FINITE DISJUNCTIVE PROGRAMMING CHARACTERIZATIONS FOR GENERAL MIXED-INTEGER LINEAR PROGRAMS FINITE DISJUNCTIVE PROGRAMMING CHARACTERIZATIONS FOR GENERAL MIXED-INTEGER LINEAR PROGRAMS BINYUAN CHEN, SİMGE KÜÇÜKYAVUZ, SUVRAJEET SEN Abstract. In this paper, we give a finite disjunctive programming

More information

Lecture 2 - Introduction to Polytopes

Lecture 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 information

Lesson 17. Geometry and Algebra of Corner Points

Lesson 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 information

1. CONVEX POLYGONS. Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D.

1. CONVEX POLYGONS. Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. 1. CONVEX POLYGONS Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. Convex 6 gon Another convex 6 gon Not convex Question. Why is the third

More information

What is dimension? An investigation by Laura Escobar. Math Explorer s Club

What is dimension? An investigation by Laura Escobar. Math Explorer s Club What is dimension? An investigation by Laura Escobar Math Explorer s Club The goal of this activity is to introduce you to the notion of dimension. The movie Flatland is also a great way to learn about

More information

Hausdorff Approximation of 3D Convex Polytopes

Hausdorff Approximation of 3D Convex Polytopes Hausdorff Approximation of 3D Convex Polytopes Mario A. Lopez Department of Mathematics University of Denver Denver, CO 80208, U.S.A. mlopez@cs.du.edu Shlomo Reisner Department of Mathematics University

More information

Integer Programming: Algorithms - 2

Integer Programming: Algorithms - 2 Week 8 Integer Programming: Algorithms - 2 OPR 992 Applied Mathematical Programming OPR 992 - Applied Mathematical Programming - p. 1/13 Introduction Valid Inequalities Chvatal-Gomory Procedure Gomory

More information

Linear Programming Duality and Algorithms

Linear 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 information

Figure 2.1: An example of a convex set and a nonconvex one.

Figure 2.1: An example of a convex set and a nonconvex one. Convex Hulls 2.1 Definitions 2 Convexity is the key to understanding and simplifying geometry, and the convex hull plays a role in geometry akin to the sorted order for a collection of numbers. So what

More information

Planar Graphs. 1 Graphs and maps. 1.1 Planarity and duality

Planar Graphs. 1 Graphs and maps. 1.1 Planarity and duality Planar Graphs In the first half of this book, we consider mostly planar graphs and their geometric representations, mostly in the plane. We start with a survey of basic results on planar graphs. This chapter

More information

Linear Programming in Small Dimensions

Linear 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 information

Integer Programming Explained Through Gomory s Cutting Plane Algorithm and Column Generation

Integer Programming Explained Through Gomory s Cutting Plane Algorithm and Column Generation Integer Programming Explained Through Gomory s Cutting Plane Algorithm and Column Generation Banhirup Sengupta, Dipankar Mondal, Prajjal Kumar De, Souvik Ash Proposal Description : ILP [integer linear

More information

Open problems in convex geometry

Open problems in convex geometry Open problems in convex geometry 10 March 2017, Monash University Seminar talk Vera Roshchina, RMIT University Based on joint work with Tian Sang (RMIT University), Levent Tunçel (University of Waterloo)

More information

1. (10 points) Draw the state diagram of the DFA that recognizes the language over Σ = {0, 1}

1. (10 points) Draw the state diagram of the DFA that recognizes the language over Σ = {0, 1} CSE 5 Homework 2 Due: Monday October 6, 27 Instructions Upload a single file to Gradescope for each group. should be on each page of the submission. All group members names and PIDs Your assignments in

More information

Convex Optimization. 2. Convex Sets. Prof. Ying Cui. Department of Electrical Engineering Shanghai Jiao Tong University. SJTU Ying Cui 1 / 33

Convex Optimization. 2. Convex Sets. Prof. Ying Cui. Department of Electrical Engineering Shanghai Jiao Tong University. SJTU Ying Cui 1 / 33 Convex Optimization 2. Convex Sets Prof. Ying Cui Department of Electrical Engineering Shanghai Jiao Tong University 2018 SJTU Ying Cui 1 / 33 Outline Affine and convex sets Some important examples Operations

More information

Integer and Combinatorial Optimization

Integer and Combinatorial Optimization Integer and Combinatorial Optimization GEORGE NEMHAUSER School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, Georgia LAURENCE WOLSEY Center for Operations Research and

More information

An Eberhard-like theorem for pentagons and heptagons

An Eberhard-like theorem for pentagons and heptagons An Eberhard-like theorem for pentagons and heptagons Robert Šámal joint work with M. DeVos, A. Georgakopoulos, B. Mohar Charles University, Prague Simon Fraser University, Burnaby CanaDAM May 26, 2009

More information

Lecture 5: Properties of convex sets

Lecture 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 information

CS522: Advanced Algorithms

CS522: 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 information

cuts François Margot 1 Chicago, IL Abstract The generalized intersection cut (GIC) paradigm is a recent framework for generating

cuts François Margot 1 Chicago, IL Abstract The generalized intersection cut (GIC) paradigm is a recent framework for generating Partial hyperplane activation for generalized intersection cuts Aleksandr M. Kazachkov 1 Selvaprabu Nadarajah 2 Egon Balas 1 François Margot 1 1 Tepper School of Business, Carnegie Mellon University, Pittsburgh,

More information

CS 473: Algorithms. Ruta Mehta. Spring University of Illinois, Urbana-Champaign. Ruta (UIUC) CS473 1 Spring / 50

CS 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 information

This lecture: Convex optimization Convex sets Convex functions Convex optimization problems Why convex optimization? Why so early in the course?

This lecture: Convex optimization Convex sets Convex functions Convex optimization problems Why convex optimization? Why so early in the course? Lec4 Page 1 Lec4p1, ORF363/COS323 This lecture: Convex optimization Convex sets Convex functions Convex optimization problems Why convex optimization? Why so early in the course? Instructor: Amir Ali Ahmadi

More information

MVE165/MMG630, Applied Optimization Lecture 8 Integer linear programming algorithms. Ann-Brith Strömberg

MVE165/MMG630, Applied Optimization Lecture 8 Integer linear programming algorithms. Ann-Brith Strömberg MVE165/MMG630, Integer linear programming algorithms Ann-Brith Strömberg 2009 04 15 Methods for ILP: Overview (Ch. 14.1) Enumeration Implicit enumeration: Branch and bound Relaxations Decomposition methods:

More information

What is a cone? Anastasia Chavez. Field of Dreams Conference President s Postdoctoral Fellow NSF Postdoctoral Fellow UC Davis

What is a cone? Anastasia Chavez. Field of Dreams Conference President s Postdoctoral Fellow NSF Postdoctoral Fellow UC Davis What is a cone? Anastasia Chavez President s Postdoctoral Fellow NSF Postdoctoral Fellow UC Davis Field of Dreams Conference 2018 Roadmap for today 1 Cones 2 Vertex/Ray Description 3 Hyperplane Description

More information

2. Convex sets. x 1. x 2. affine set: contains the line through any two distinct points in the set

2. Convex sets. x 1. x 2. affine set: contains the line through any two distinct points in the set 2. Convex sets Convex Optimization Boyd & Vandenberghe affine and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes dual

More information

Stable sets, corner polyhedra and the Chvátal closure

Stable sets, corner polyhedra and the Chvátal closure Stable sets, corner polyhedra and the Chvátal closure Manoel Campêlo Departamento de Estatística e Matemática Aplicada, Universidade Federal do Ceará, Brazil, mcampelo@lia.ufc.br. Gérard Cornuéjols Tepper

More information

Polygons. An investigation by Laura Escobar. Math Explorer s Club

Polygons. An investigation by Laura Escobar. Math Explorer s Club Polygons An investigation by Laura Escobar Math Explorer s Club kl;fsdjka;fkljasd;fklj;asldkfj;askldjf;lsdjkf;klsdja;fkljas;ldfkj In this worksheet we will learn about different ways mathematicians think

More information

Convex Optimization. Convex Sets. ENSAE: Optimisation 1/24

Convex Optimization. Convex Sets. ENSAE: Optimisation 1/24 Convex Optimization Convex Sets ENSAE: Optimisation 1/24 Today affine and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes

More information

Integer Programming ISE 418. Lecture 1. Dr. Ted Ralphs

Integer Programming ISE 418. Lecture 1. Dr. Ted Ralphs Integer Programming ISE 418 Lecture 1 Dr. Ted Ralphs ISE 418 Lecture 1 1 Reading for This Lecture N&W Sections I.1.1-I.1.4 Wolsey Chapter 1 CCZ Chapter 2 ISE 418 Lecture 1 2 Mathematical Optimization Problems

More information

A Course in Convexity

A Course in Convexity A Course in Convexity Alexander Barvinok Graduate Studies in Mathematics Volume 54 American Mathematical Society Providence, Rhode Island Preface vii Chapter I. Convex Sets at Large 1 1. Convex Sets. Main

More information

EC 521 MATHEMATICAL METHODS FOR ECONOMICS. Lecture 2: Convex Sets

EC 521 MATHEMATICAL METHODS FOR ECONOMICS. Lecture 2: Convex Sets EC 51 MATHEMATICAL METHODS FOR ECONOMICS Lecture : Convex Sets Murat YILMAZ Boğaziçi University In this section, we focus on convex sets, separating hyperplane theorems and Farkas Lemma. And as an application

More information

60 2 Convex sets. {x a T x b} {x ã T x b}

60 2 Convex sets. {x a T x b} {x ã T x b} 60 2 Convex sets Exercises Definition of convexity 21 Let C R n be a convex set, with x 1,, x k C, and let θ 1,, θ k R satisfy θ i 0, θ 1 + + θ k = 1 Show that θ 1x 1 + + θ k x k C (The definition of convexity

More information

Experiments On General Disjunctions

Experiments On General Disjunctions Experiments On General Disjunctions Some Dumb Ideas We Tried That Didn t Work* and Others We Haven t Tried Yet *But that may provide some insight Ted Ralphs, Serdar Yildiz COR@L Lab, Department of Industrial

More information

Convex Hull Representation Conversion (cddlib, lrslib)

Convex Hull Representation Conversion (cddlib, lrslib) Convex Hull Representation Conversion (cddlib, lrslib) Student Seminar in Combinatorics: Mathematical Software Niklas Pfister October 31, 2014 1 Introduction In this report we try to give a short overview

More information

Pure Cutting Plane Methods for ILP: a computational perspective

Pure Cutting Plane Methods for ILP: a computational perspective Pure Cutting Plane Methods for ILP: a computational perspective Matteo Fischetti, DEI, University of Padova Rorschach test for OR disorders: can you see the tree? 1 Outline 1. Pure cutting plane methods

More information

2. Convex sets. affine and convex sets. some important examples. operations that preserve convexity. generalized inequalities

2. Convex sets. affine and convex sets. some important examples. operations that preserve convexity. generalized inequalities 2. Convex sets Convex Optimization Boyd & Vandenberghe affine and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes dual

More information

An Introduction to Computational Geometry: Arrangements and Duality

An Introduction to Computational Geometry: Arrangements and Duality An Introduction to Computational Geometry: Arrangements and Duality Joseph S. B. Mitchell Stony Brook University Some images from [O Rourke, Computational Geometry in C, 2 nd Edition, Chapter 6] Arrangement

More information

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

MA4254: Discrete Optimization. Defeng Sun. Department of Mathematics National University of Singapore Office: 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 information

Simplex Algorithm in 1 Slide

Simplex 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 information

Polyhedral Combinatorics (ADM III)

Polyhedral Combinatorics (ADM III) 1 Polyhedral Combinatorics (ADM III) Prof. Dr. Dr. h.c. mult. Martin Grötschel Sommersemester 2010, Classes: TU MA 041, Tuesdays 16:15 17:45h, first class on April 13, 2010 LV-Nr.: 3236 L 414 Diese Vorlesung

More information