Integer and Combinatorial Optimization: Introduction
|
|
- Roderick Anthony Hardy
- 5 years ago
- Views:
Transcription
1 Integer and Combinatorial Optimization: Introduction John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY USA November 2018 Mitchell Introduction 1 / 18
2 Integer and Combinatorial Optimization Outline 1 Integer and Combinatorial Optimization Complexity Solution techniques Quadratic constraints and objective functions 2 Cutting planes 3 Branch-and-bound: An example Mitchell Introduction 2 / 18
3 Integer and Combinatorial Optimization Combinatorial optimization An optimization problem is a problem of the form min x subject to f (x) x 2 S where f (x) is the objective function and S is the feasible region. A combinatorial optimization problem is one where there is only a finite number of points in S. For example: colorings of the countries of a map, so that no two adjacent coutries have the same color. Mitchell Introduction 3 / 18
4 Integer and Combinatorial Optimization Integer optimization In an integer optimization problem, the variables are constrained to take integer values. In a mixed integer optimization problem, some of the variables are required to take integer values, and the remainder can take fractional values. Mitchell Introduction 4 / 18
5 Integer and Combinatorial Optimization Integer optimization For example, a mixed integer linear optimization problem has the form min x2r n,y2r p ct x + d T y subject to Ax + By apple g x, y 0, y integer Here, A and B are given matrices, A 2 R m n, B 2 R m p. The remaining parameters c, d, g are vectors, c 2 R n, d 2 R p, g 2 R m. All vectors are understood to be column vectors in this course. Transpose is denoted by the superscript T. The dot product between two vectors c and x can be denoted c T x. In a mixed integer nonlinear optimization problem, the objective function and/or the constraints may be nonlinear functions. Mitchell Introduction 5 / 18
6 Integer and Combinatorial Optimization Complexity Outline 1 Integer and Combinatorial Optimization Complexity Solution techniques Quadratic constraints and objective functions 2 Cutting planes 3 Branch-and-bound: An example Mitchell Introduction 6 / 18
7 Integer and Combinatorial Optimization Complexity Computational complexity Linear optimization problems can be solved in polynomial time. This implies the required runtime doesn t grow too quickly as the problem size grows. By contrast, integer optimization problems are often NP-hard (to be defined later), which means that the runtime can grow very quickly in the problem size in the worst case. So mixed integer linear optimization problems are a lot harder to solve to global optimality than linear optimization problems. Mitchell Introduction 7 / 18
8 Integer and Combinatorial Optimization Solution techniques Outline 1 Integer and Combinatorial Optimization Complexity Solution techniques Quadratic constraints and objective functions 2 Cutting planes 3 Branch-and-bound: An example Mitchell Introduction 8 / 18
9 Integer and Combinatorial Optimization Solution techniques Solution techniques Our emphasis will be on methods for finding global optimal solutions to integer optimization problems. If we have a global minimizer x then there is no other feasible point x 2 S with f (x) < f (x ). In order to prove optimality, we will look at relaxations. The simplest relaxation of a mixed integer linear optimization problem is to ignore integrality, giving a linear optimization problem. The principal solution techniques we then examine are cutting planes: try to improve the relaxation branching: subdivide the feasible region. Mitchell Introduction 9 / 18
10 Integer and Combinatorial Optimization Solution techniques Solution techniques Our emphasis will be on methods for finding global optimal solutions to integer optimization problems. If we have a global minimizer x then there is no other feasible point x 2 S with f (x) < f (x ). In order to prove optimality, we will look at relaxations. The simplest relaxation of a mixed integer linear optimization problem is to ignore integrality, giving a linear optimization problem. The principal solution techniques we then examine are cutting planes: try to improve the relaxation branching: subdivide the feasible region. Mitchell Introduction 9 / 18
11 Integer and Combinatorial Optimization Quadratic constraints and objective functions Outline 1 Integer and Combinatorial Optimization Complexity Solution techniques Quadratic constraints and objective functions 2 Cutting planes 3 Branch-and-bound: An example Mitchell Introduction 10 / 18
12 Integer and Combinatorial Optimization Quadratic constraints and objective functions Quadratic constraints and objective functions f (x) = I x t I x t Qx Some combinatorial optimization problems can be formulated with quadratic constraints and/or a quadratic objective function. For other problems, it can be helpful to look at a quadratic relaxation. These problems can sometimes be attacked as semidefinite optimization problems, which have a matrix of variables that must be symmetric and positive semidefinite. A useful observation: we can replace the requirement that x is a binary variable by the requirement that x = x 2. :::::::::: -1 Mitchell Introduction 11 / 18
13 Cutting planes Outline 1 Integer and Combinatorial Optimization Complexity Solution techniques Quadratic constraints and objective functions 2 Cutting planes 3 Branch-and-bound: An example Mitchell Introduction 12 / 18
14 Cutting planes A cutting plane example Initial relaxation: Solve linear optimization relaxation min{c T x : Ax apple b, x 0} - c Solution to linear optimization relaxation Mitchell Introduction 13 / 18
15 Cutting planes A cutting plane example Initial relaxation: Solve linear optimization relaxation min{c T x : Ax apple b, x 0} Cutting plane a T 1 x = b 1 Solution to linear optimization relaxation Mitchell Introduction 13 / 18
16 Cutting planes Add a cutting plane and reoptimize Solve linear optimization relaxation min{c T x : Ax apple b, x 0, a T 1 x apple b 1} Cutting plane a T 1 x = b 1 Solution to linear optimization relaxation Mitchell Introduction 14 / 18
17 Cutting planes Add a cutting plane and reoptimize Solve linear optimization relaxation min{c T x : Ax apple b, x 0, a T 1 x apple b 1} Cutting plane a T 1 x = b 1 Solution to linear optimization relaxation Cutting plane a T 2 x = b 2 Mitchell Introduction 14 / 18
18 Cutting planes Add a cutting plane and reoptimize Solve linear optimization relaxation min{c T x : Ax apple b, x 0, a T 1 x apple b 1, a T 2 x apple b 2} Cutting plane a T 1 x = b 1 Solution to linear optimization relaxation Cutting plane a T 2 x = b 2 Solution to relaxation is integral, so it solves the integer optimization problem. Mitchell Introduction 15 / 18
19 Branch-and-bound: An example Outline 1 Integer and Combinatorial Optimization Complexity Solution techniques Quadratic constraints and objective functions 2 Cutting planes 3 Branch-and-bound: An example Mitchell Introduction 16 / 18
20 An example Branch-and-bound: An example We look at the integer optimization problem min x2r 3 5x 1 + 5x 2 13x 3 subject to x 1 + x 2 + x 3 apple 6 10x 1 8x 2 apple 15 (IOP) 6x 1 x 2 + 9x 3 apple 9 x i 0, x i integer, i = 1, 2, 3 It s easy to find a feasible solution for this problem: x =(0, 0, 0). Hence we can initialize with z u = 0. We let F = {x 2 R 3 : x 1 + x 2 + x 3 apple 6, 10x 1 8x 2 apple 15, 6x 1 x 2 + 9x 3 apple 9, x 1, x 2, x 3 0}. Mitchell Introduction 17 / 18
21 Branch-and-bound: An example The tree for the example Root node: min{c T x : x 2 F } x 0 =(1.5, 0, 2), z 0 = 18.5 Mitchell Introduction 18 / 18
22 Branch-and-bound: An example The tree for the example Root node: min{c T x : x 2 F } x 0 =(1.5, 0, 2), z 0 = 18.5 x 1 apple 1 x 1 2 x 2 =(2, 5 8, ), z 2 = Mitchell Introduction 18 / 18
23 Branch-and-bound: An example The tree for the example x 1 =(1, 0, ), z 1 = Root node: min{c T x : x 2 F } x 0 =(1.5, 0, 2), z 0 = 18.5 x 1 apple 1 x 1 2 x 2 =(2, 5 8, ), z 2 = Mitchell Introduction 18 / 18
24 Branch-and-bound: An example The tree for the example x 1 =(1, 0, ), z 1 = Root node: min{c T x : x 2 F} x 0 =(1.5, 0, 2), z 0 = 18.5 x 1 apple 1 x 1 2 x 2 =(2, 5 8, ), z 2 = x 3 3 infeasible, z 6 =+1 fathom by infeasibility Mitchell Introduction 18 / 18
25 Branch-and-bound: An example The tree for the example x 1 =(1, 0, ), z 1 = Root node: min{c T x : x 2 F} x 0 =(1.5, 0, 2), z 0 = 18.5 x 1 apple 1 x 1 2 x 5 =(2, 5 8, 2), z 5 = x 2 =(2, 5 8, ), z 2 = x 3 apple 2 x 3 3 infeasible, z 6 =+1 fathom by infeasibility Mitchell Introduction 18 / 18
26 Branch-and-bound: An example The tree for the example x 1 =(1, 0, ), z 1 = Root node: min{c T x : x 2 F } x 0 =(1.5, 0, 2), z 0 = 18.5 x 1 apple 1 *. x 1 2 x 2 =(2, 5 8, ), z 2 = x 3 apple 1 x 3 apple 2 x 3 3 x 3 =(0, 0, 1), z 3 = 13 fathom by integrality x 5 =(2, 5 8, 2), z 5 = fathom by bounds infeasible, z 6 =+1 fathom by infeasibility Mitchell Introduction 18 / 18
27 Branch-and-bound: An example The tree for the example Root node: min{c T x : x 2 F } x 1 =(1, 0, ), z 1 = x 0 =(1.5, 0, 2), z 0 = 18.5 x 1 apple 1 x 1 2 x 2 =(2, 5 8, ), z 2 = x 3 apple 1 x 3 2 x 3 apple 2 x 3 3 x 3 =(0, 0, 1), z 3 = 13 x 4 =(1, 3, 2), z 4 = 6 x 5 =(2, 5 8, 2), z 5 = infeasible, z 6 =+1 fathom by integrality fathom by bounds fathom by bounds fathom by infeasibility Mitchell Introduction 18 / 18
28 Branch-and-bound: An example The tree for the example Root node: min{c T x : x 2 F } x 1 =(1, 0, ), z 1 = x 0 =(1.5, 0, 2), z 0 = 18.5 x 1 apple 1 x 1 2 x 2 =(2, 5 8, ), z 2 = x 3 apple 1 x 3 2 x 3 apple 2 x 3 3 x 3 =(0, 0, 1), z 3 = 13 x 4 =(1, 3, 2), z 4 = 6 x 5 =(2, 5 8, 2), z 5 = infeasible, z 6 =+1 fathom by integrality fathom by bounds fathom by bounds fathom by infeasibility The optimal solution to (IOP) is x =(0, 0, 1), with value z = 13. Mitchell Introduction 18 / 18
Math Models of OR: Branch-and-Bound
Math Models of OR: Branch-and-Bound John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA November 2018 Mitchell Branch-and-Bound 1 / 15 Branch-and-Bound Outline 1 Branch-and-Bound
More informationLecture 23 Branch-and-Bound Algorithm. November 3, 2009
Branch-and-Bound Algorithm November 3, 2009 Outline Lecture 23 Modeling aspect: Either-Or requirement Special ILPs: Totally unimodular matrices Branch-and-Bound Algorithm Underlying idea Terminology Formal
More informationMath Models of OR: Sensitivity Analysis
Math Models of OR: Sensitivity Analysis John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 8 USA October 8 Mitchell Sensitivity Analysis / 9 Optimal tableau and pivot matrix Outline Optimal
More informationLinear integer programming and its application
Linear integer programming and its application Presented by Dr. Sasthi C. Ghosh Associate Professor Advanced Computing & Microelectronics Unit Indian Statistical Institute Kolkata, India Outline Introduction
More informationThe Ellipsoid Algorithm
The Ellipsoid Algorithm John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA 9 February 2018 Mitchell The Ellipsoid Algorithm 1 / 28 Introduction Outline 1 Introduction 2 Assumptions
More informationMath Models of OR: Extreme Points and Basic Feasible Solutions
Math Models of OR: Extreme Points and Basic Feasible Solutions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 1180 USA September 018 Mitchell Extreme Points and Basic Feasible Solutions
More informationStructured Problems and Algorithms
Integer and quadratic optimization problems Dept. of Engg. and Comp. Sci., Univ. of Cal., Davis Aug. 13, 2010 Table of contents Outline 1 2 3 Benefits of Structured Problems Optimization problems may become
More informationLP Relaxations of Mixed Integer Programs
LP Relaxations of Mixed Integer Programs John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA February 2015 Mitchell LP Relaxations 1 / 29 LP Relaxations LP relaxations We want
More informationTravelling Salesman Problem
Travelling Salesman Problem Fabio Furini November 10th, 2014 Travelling Salesman Problem 1 Outline 1 Traveling Salesman Problem Separation Travelling Salesman Problem 2 (Asymmetric) Traveling Salesman
More informationNetwork Flows. 6. Lagrangian Relaxation. Programming. Fall 2010 Instructor: Dr. Masoud Yaghini
In the name of God Network Flows 6. Lagrangian Relaxation 6.3 Lagrangian Relaxation and Integer Programming Fall 2010 Instructor: Dr. Masoud Yaghini Integer Programming Outline Branch-and-Bound Technique
More informationPart 4. Decomposition Algorithms
In the name of God Part 4. 4.4. Column Generation for the Constrained Shortest Path Problem Spring 2010 Instructor: Dr. Masoud Yaghini Constrained Shortest Path Problem Constrained Shortest Path Problem
More informationChapter 6 Constraint Satisfaction Problems
Chapter 6 Constraint Satisfaction Problems CS5811 - Artificial Intelligence Nilufer Onder Department of Computer Science Michigan Technological University Outline CSP problem definition Backtracking search
More informationInteger Programming. Wolfram Wiesemann. December 6, 2007
Integer Programming Wolfram Wiesemann December 6, 2007 Contents of this Lecture Revision: Mixed Integer Programming Problems Branch & Bound Algorithms: The Big Picture Solving MIP s: Complete Enumeration
More informationSection Notes 8. Integer Programming II. Applied Math 121. Week of April 5, expand your knowledge of big M s and logical constraints.
Section Notes 8 Integer Programming II Applied Math 121 Week of April 5, 2010 Goals for the week understand IP relaxations be able to determine the relative strength of formulations understand the branch
More informationLearning to Branch. Ellen Vitercik. Joint work with Nina Balcan, Travis Dick, and Tuomas Sandholm. Published in ICML 2018
Learning to Branch Ellen Vitercik Joint work with Nina Balcan, Travis Dick, and Tuomas Sandholm Published in ICML 2018 1 Integer Programs (IPs) a maximize subject to c x Ax b x {0,1} n 2 Facility location
More informationAM 121: Intro to Optimization! Models and Methods! Fall 2018!
AM 121: Intro to Optimization Models and Methods Fall 2018 Lecture 13: Branch and Bound (I) Yiling Chen SEAS Example: max 5x 1 + 8x 2 s.t. x 1 + x 2 6 5x 1 + 9x 2 45 x 1, x 2 0, integer 1 x 2 6 5 x 1 +x
More informationGeometric Steiner Trees
Geometric Steiner Trees From the book: Optimal Interconnection Trees in the Plane By Marcus Brazil and Martin Zachariasen Part 3: Computational Complexity and the Steiner Tree Problem Marcus Brazil 2015
More informationMath Models of OR: Traveling Salesman Problem
Math Models of OR: Traveling Salesman Problem John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA November 2018 Mitchell Traveling Salesman Problem 1 / 19 Outline 1 Examples 2
More informationMath Models of OR: Some Definitions
Math Models of OR: Some Definitions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Some Definitions 1 / 20 Active constraints Outline 1 Active constraints
More informationIntroduction to integer programming II
Introduction to integer programming II Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects of Optimization
More information23. Cutting planes and branch & bound
CS/ECE/ISyE 524 Introduction to Optimization Spring 207 8 23. Cutting planes and branch & bound ˆ Algorithms for solving MIPs ˆ Cutting plane methods ˆ Branch and bound methods Laurent Lessard (www.laurentlessard.com)
More informationMath Models of OR: Handling Upper Bounds in Simplex
Math Models of OR: Handling Upper Bounds in Simplex John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 280 USA September 208 Mitchell Handling Upper Bounds in Simplex / 8 Introduction Outline
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 02: Optimization (Convex and Otherwise) What is Optimization? An Optimization Problem has 3 parts. x F f(x) :
More information1 The linear algebra of linear programs (March 15 and 22, 2015)
1 The linear algebra of linear programs (March 15 and 22, 2015) Many optimization problems can be formulated as linear programs. The main features of a linear program are the following: Variables are real
More informationModule 04 Optimization Problems KKT Conditions & Solvers
Module 04 Optimization Problems KKT Conditions & Solvers Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September
More informationSemidefinite Programming Basics and Applications
Semidefinite Programming Basics and Applications Ray Pörn, principal lecturer Åbo Akademi University Novia University of Applied Sciences Content What is semidefinite programming (SDP)? How to represent
More informationSection #2: Linear and Integer Programming
Section #2: Linear and Integer Programming Prof. Dr. Sven Seuken 8.3.2012 (with most slides borrowed from David Parkes) Housekeeping Game Theory homework submitted? HW-00 and HW-01 returned Feedback on
More informationColumn Generation. i = 1,, 255;
Column Generation The idea of the column generation can be motivated by the trim-loss problem: We receive an order to cut 50 pieces of.5-meter (pipe) segments, 250 pieces of 2-meter segments, and 200 pieces
More informationInteger Programming Part II
Be the first in your neighborhood to master this delightful little algorithm. Integer Programming Part II The Branch and Bound Algorithm learn about fathoming, bounding, branching, pruning, and much more!
More informationWeek Cuts, Branch & Bound, and Lagrangean Relaxation
Week 11 1 Integer Linear Programming This week we will discuss solution methods for solving integer linear programming problems. I will skip the part on complexity theory, Section 11.8, although this is
More informationBasic notions of Mixed Integer Non-Linear Programming
Basic notions of Mixed Integer Non-Linear Programming Claudia D Ambrosio CNRS & LIX, École Polytechnique 5th Porto Meeting on Mathematics for Industry, April 10, 2014 C. D Ambrosio (CNRS) April 10, 2014
More informationto work with) can be solved by solving their LP relaxations with the Simplex method I Cutting plane algorithms, e.g., Gomory s fractional cutting
Summary so far z =max{c T x : Ax apple b, x 2 Z n +} I Modeling with IP (and MIP, and BIP) problems I Formulation for a discrete set that is a feasible region of an IP I Alternative formulations for the
More information1 Strict local optimality in unconstrained optimization
ORF 53 Lecture 14 Spring 016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Thursday, April 14, 016 When in doubt on the accuracy of these notes, please cross check with the instructor s
More informationAppendix A Taylor Approximations and Definite Matrices
Appendix A Taylor Approximations and Definite Matrices Taylor approximations provide an easy way to approximate a function as a polynomial, using the derivatives of the function. We know, from elementary
More informationThe final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts.
Math 141 Review for Final The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Part 1 (no calculator) graphing (polynomial, rational, linear, exponential, and logarithmic
More information3.3 Easy ILP problems and totally unimodular matrices
3.3 Easy ILP problems and totally unimodular matrices Consider a generic ILP problem expressed in standard form where A Z m n with n m, and b Z m. min{c t x : Ax = b, x Z n +} (1) P(b) = {x R n : Ax =
More informationDual bounds: can t get any better than...
Bounds, relaxations and duality Given an optimization problem z max{c(x) x 2 }, how does one find z, or prove that a feasible solution x? is optimal or close to optimal? I Search for a lower and upper
More information4y Springer NONLINEAR INTEGER PROGRAMMING
NONLINEAR INTEGER PROGRAMMING DUAN LI Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin, N. T. Hong Kong XIAOLING SUN Department of Mathematics Shanghai
More informationAPPLIED MECHANISM DESIGN FOR SOCIAL GOOD
APPLIED MECHANISM DESIGN FOR SOCIAL GOOD JOHN P DICKERSON Lecture #4 09/08/2016 CMSC828M Tuesdays & Thursdays 12:30pm 1:45pm PRESENTATION LIST IS ONLINE! SCRIBE LIST COMING SOON 2 THIS CLASS: (COMBINATORIAL)
More informationMIT Algebraic techniques and semidefinite optimization February 14, Lecture 3
MI 6.97 Algebraic techniques and semidefinite optimization February 4, 6 Lecture 3 Lecturer: Pablo A. Parrilo Scribe: Pablo A. Parrilo In this lecture, we will discuss one of the most important applications
More informationDecision Procedures An Algorithmic Point of View
An Algorithmic Point of View ILP References: Integer Programming / Laurence Wolsey Deciding ILPs with Branch & Bound Intro. To mathematical programming / Hillier, Lieberman Daniel Kroening and Ofer Strichman
More information16.1 L.P. Duality Applied to the Minimax Theorem
CS787: Advanced Algorithms Scribe: David Malec and Xiaoyong Chai Lecturer: Shuchi Chawla Topic: Minimax Theorem and Semi-Definite Programming Date: October 22 2007 In this lecture, we first conclude our
More informationLinear Programming: Simplex
Linear Programming: Simplex Stephen J. Wright 1 2 Computer Sciences Department, University of Wisconsin-Madison. IMA, August 2016 Stephen Wright (UW-Madison) Linear Programming: Simplex IMA, August 2016
More informationConic optimization under combinatorial sparsity constraints
Conic optimization under combinatorial sparsity constraints Christoph Buchheim and Emiliano Traversi Abstract We present a heuristic approach for conic optimization problems containing sparsity constraints.
More informationOn the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems
MATHEMATICS OF OPERATIONS RESEARCH Vol. 35, No., May 010, pp. 84 305 issn 0364-765X eissn 156-5471 10 350 084 informs doi 10.187/moor.1090.0440 010 INFORMS On the Power of Robust Solutions in Two-Stage
More informationComputational and Statistical Tradeoffs via Convex Relaxation
Computational and Statistical Tradeoffs via Convex Relaxation Venkat Chandrasekaran Caltech Joint work with Michael Jordan Time-constrained Inference o Require decision after a fixed (usually small) amount
More informationSeparation Techniques for Constrained Nonlinear 0 1 Programming
Separation Techniques for Constrained Nonlinear 0 1 Programming Christoph Buchheim Computer Science Department, University of Cologne and DEIS, University of Bologna MIP 2008, Columbia University, New
More informationwhere X is the feasible region, i.e., the set of the feasible solutions.
3.5 Branch and Bound Consider a generic Discrete Optimization problem (P) z = max{c(x) : x X }, where X is the feasible region, i.e., the set of the feasible solutions. Branch and Bound is a general semi-enumerative
More informationBounds on the Traveling Salesman Problem
Bounds on the Traveling Salesman Problem Sean Zachary Roberson Texas A&M University MATH 613, Graph Theory A common routing problem is as follows: given a collection of stops (for example, towns, stations,
More informationNonconvex Quadratic Programming: Return of the Boolean Quadric Polytope
Nonconvex Quadratic Programming: Return of the Boolean Quadric Polytope Kurt M. Anstreicher Dept. of Management Sciences University of Iowa Seminar, Chinese University of Hong Kong, October 2009 We consider
More informationIntroduction to Mathematical Programming IE406. Lecture 21. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 21 Dr. Ted Ralphs IE406 Lecture 21 1 Reading for This Lecture Bertsimas Sections 10.2, 10.3, 11.1, 11.2 IE406 Lecture 21 2 Branch and Bound Branch
More informationAcyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs
Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs Raphael Louca & Eilyan Bitar School of Electrical and Computer Engineering American Control Conference (ACC) Chicago,
More informationChapter 3 Deterministic planning
Chapter 3 Deterministic planning In this chapter we describe a number of algorithms for solving the historically most important and most basic type of planning problem. Two rather strong simplifying assumptions
More informationIntractable Problems Part Two
Intractable Problems Part Two Announcements Problem Set Five graded; will be returned at the end of lecture. Extra office hours today after lecture from 4PM 6PM in Clark S250. Reminder: Final project goes
More informationDiscrete (and Continuous) Optimization WI4 131
Discrete (and Continuous) Optimization WI4 131 Kees Roos Technische Universiteit Delft Faculteit Electrotechniek, Wiskunde en Informatica Afdeling Informatie, Systemen en Algoritmiek e-mail: C.Roos@ewi.tudelft.nl
More informationCS675: Convex and Combinatorial Optimization Fall 2016 Combinatorial Problems as Linear and Convex Programs. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Fall 2016 Combinatorial Problems as Linear and Convex Programs Instructor: Shaddin Dughmi Outline 1 Introduction 2 Shortest Path 3 Algorithms for Single-Source
More informationOperations Research Lecture 6: Integer Programming
Operations Research Lecture 6: Integer Programming Notes taken by Kaiquan Xu@Business School, Nanjing University May 12th 2016 1 Integer programming (IP) formulations The integer programming (IP) is the
More informationApproximation Algorithms
Approximation Algorithms Chapter 26 Semidefinite Programming Zacharias Pitouras 1 Introduction LP place a good lower bound on OPT for NP-hard problems Are there other ways of doing this? Vector programs
More informationIP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, You wish to solve the IP below with a cutting plane technique.
IP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, 31 14. You wish to solve the IP below with a cutting plane technique. Maximize 4x 1 + 2x 2 + x 3 subject to 14x 1 + 10x 2 + 11x 3 32 10x 1 +
More information16.410/413 Principles of Autonomy and Decision Making
6.4/43 Principles of Autonomy and Decision Making Lecture 8: (Mixed-Integer) Linear Programming for Vehicle Routing and Motion Planning Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute
More informationData Mining and Analysis: Fundamental Concepts and Algorithms
Data Mining and Analysis: Fundamental Concepts and Algorithms dataminingbook.info Mohammed J. Zaki 1 Wagner Meira Jr. 2 1 Department of Computer Science Rensselaer Polytechnic Institute, Troy, NY, USA
More informationPOLYNOMIAL OPTIMIZATION WITH SUMS-OF-SQUARES INTERPOLANTS
POLYNOMIAL OPTIMIZATION WITH SUMS-OF-SQUARES INTERPOLANTS Sercan Yıldız syildiz@samsi.info in collaboration with Dávid Papp (NCSU) OPT Transition Workshop May 02, 2017 OUTLINE Polynomial optimization and
More information15.083J/6.859J Integer Optimization. Lecture 2: Efficient Algorithms and Computational Complexity
15.083J/6.859J Integer Optimization Lecture 2: Efficient Algorithms and Computational Complexity 1 Outline Efficient algorithms Slide 1 Complexity The classes P and N P The classes N P-complete and N P-hard
More informationRelations between Semidefinite, Copositive, Semi-infinite and Integer Programming
Relations between Semidefinite, Copositive, Semi-infinite and Integer Programming Author: Faizan Ahmed Supervisor: Dr. Georg Still Master Thesis University of Twente the Netherlands May 2010 Relations
More informationCS675: Convex and Combinatorial Optimization Fall 2014 Combinatorial Problems as Linear Programs. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Fall 2014 Combinatorial Problems as Linear Programs Instructor: Shaddin Dughmi Outline 1 Introduction 2 Shortest Path 3 Algorithms for Single-Source Shortest
More informationSection Notes 9. IP: Cutting Planes. Applied Math 121. Week of April 12, 2010
Section Notes 9 IP: Cutting Planes Applied Math 121 Week of April 12, 2010 Goals for the week understand what a strong formulations is. be familiar with the cutting planes algorithm and the types of cuts
More informationFrom structures to heuristics to global solvers
From structures to heuristics to global solvers Timo Berthold Zuse Institute Berlin DFG Research Center MATHEON Mathematics for key technologies OR2013, 04/Sep/13, Rotterdam Outline From structures to
More informationMath Models of OR: The Revised Simplex Method
Math Models of OR: The Revised Simplex Method John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell The Revised Simplex Method 1 / 25 Motivation Outline 1
More informationMulti-objective branch-and-cut algorithm and multi-modal traveling salesman problem
Multi-objective branch-and-cut algorithm and multi-modal traveling salesman problem Nicolas Jozefowiez 1, Gilbert Laporte 2, Frédéric Semet 3 1. LAAS-CNRS, INSA, Université de Toulouse, Toulouse, France,
More informationConstraint Satisfaction Problems
Constraint Satisfaction Problems Chapter 5 AIMA2e Slides, Stuart Russell and Peter Norvig, Completed by Kazim Fouladi, Fall 2008 Chapter 5 1 Outline CSP examples Backtracking search for CSPs Problem structure
More information1 Column Generation and the Cutting Stock Problem
1 Column Generation and the Cutting Stock Problem In the linear programming approach to the traveling salesman problem we used the cutting plane approach. The cutting plane approach is appropriate when
More informationNP-Completeness I. Lecture Overview Introduction: Reduction and Expressiveness
Lecture 19 NP-Completeness I 19.1 Overview In the past few lectures we have looked at increasingly more expressive problems that we were able to solve using efficient algorithms. In this lecture we introduce
More informationBilevel Integer Optimization: Theory and Algorithms
: Theory and Algorithms Ted Ralphs 1 Joint work with Sahar Tahernajad 1, Scott DeNegre 3, Menal Güzelsoy 2, Anahita Hassanzadeh 4 1 COR@L Lab, Department of Industrial and Systems Engineering, Lehigh University
More informationPrincipal Components Analysis (PCA)
Principal Components Analysis (PCA) Principal Components Analysis (PCA) a technique for finding patterns in data of high dimension Outline:. Eigenvectors and eigenvalues. PCA: a) Getting the data b) Centering
More informationLimitations of Algorithm Power
Limitations of Algorithm Power Objectives We now move into the third and final major theme for this course. 1. Tools for analyzing algorithms. 2. Design strategies for designing algorithms. 3. Identifying
More informationMultivalued Decision Diagrams. Postoptimality Analysis Using. J. N. Hooker. Tarik Hadzic. Cork Constraint Computation Centre
Postoptimality Analysis Using Multivalued Decision Diagrams Tarik Hadzic Cork Constraint Computation Centre J. N. Hooker Carnegie Mellon University London School of Economics June 2008 Postoptimality Analysis
More informationOn the Computational Hardness of Graph Coloring
On the Computational Hardness of Graph Coloring Steven Rutherford June 3, 2011 Contents 1 Introduction 2 2 Turing Machine 2 3 Complexity Classes 3 4 Polynomial Time (P) 4 4.1 COLORED-GRAPH...........................
More informationBranch-and-Bound. Leo Liberti. LIX, École Polytechnique, France. INF , Lecture p. 1
Branch-and-Bound Leo Liberti LIX, École Polytechnique, France INF431 2011, Lecture p. 1 Reminders INF431 2011, Lecture p. 2 Problems Decision problem: a question admitting a YES/NO answer Example HAMILTONIAN
More informationComplexity and Simplicity of Optimization Problems
Complexity and Simplicity of Optimization Problems Yurii Nesterov, CORE/INMA (UCL) February 17 - March 23, 2012 (ULg) Yu. Nesterov Algorithmic Challenges in Optimization 1/27 Developments in Computer Sciences
More informationThe Class NP. NP is the problems that can be solved in polynomial time by a nondeterministic machine.
The Class NP NP is the problems that can be solved in polynomial time by a nondeterministic machine. NP The time taken by nondeterministic TM is the length of the longest branch. The collection of all
More informationNP-problems continued
NP-problems continued Page 1 Since SAT and INDEPENDENT SET can be reduced to each other we might think that there would be some similarities between the two problems. In fact, there is one such similarity.
More informationInteger Quadratic Programming is in NP
Alberto Del Pia 1 Santanu S. Dey 2 Marco Molinaro 2 1 IBM T. J. Watson Research Center, Yorktown Heights. 2 School of Industrial and Systems Engineering, Atlanta Outline 1 4 Program: Definition Definition
More informationPART 4 INTEGER PROGRAMMING
PART 4 INTEGER PROGRAMMING 102 Read Chapters 11 and 12 in textbook 103 A capital budgeting problem We want to invest $19 000 Four investment opportunities which cannot be split (take it or leave it) 1.
More informationReconnect 04 Introduction to Integer Programming
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, Reconnect 04 Introduction to Integer Programming Cynthia Phillips, Sandia National Laboratories Integer programming
More information4-1 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall, ECC
4-1 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall, ECC 2003 2003.09.02.02 4. The Algebraic-Geometric Dictionary Equality constraints Ideals and Varieties Feasibility problems and duality The
More informationCHAPTER 3: INTEGER PROGRAMMING
CHAPTER 3: INTEGER PROGRAMMING Overview To this point, we have considered optimization problems with continuous design variables. That is, the design variables can take any value within a continuous feasible
More informationCopositive Programming and Combinatorial Optimization
Copositive Programming and Combinatorial Optimization Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria joint work with I.M. Bomze (Wien) and F. Jarre (Düsseldorf) IMA
More informationOnline generation via offline selection - Low dimensional linear cuts from QP SDP relaxation -
Online generation via offline selection - Low dimensional linear cuts from QP SDP relaxation - Radu Baltean-Lugojan Ruth Misener Computational Optimisation Group Department of Computing Pierre Bonami Andrea
More informationMDD-based Postoptimality Analysis for Mixed-integer Programs
MDD-based Postoptimality Analysis for Mixed-integer Programs John Hooker, Ryo Kimura Carnegie Mellon University Thiago Serra Mitsubishi Electric Research Laboratories Symposium on Decision Diagrams for
More informationStochastic Integer Programming
IE 495 Lecture 20 Stochastic Integer Programming Prof. Jeff Linderoth April 14, 2003 April 14, 2002 Stochastic Programming Lecture 20 Slide 1 Outline Stochastic Integer Programming Integer LShaped Method
More informationLecture 6: Conic Optimization September 8
IE 598: Big Data Optimization Fall 2016 Lecture 6: Conic Optimization September 8 Lecturer: Niao He Scriber: Juan Xu Overview In this lecture, we finish up our previous discussion on optimality conditions
More informationA new family of facet defining inequalities for the maximum edge-weighted clique problem
A new family of facet defining inequalities for the maximum edge-weighted clique problem Franklin Djeumou Fomeni June 2016 Abstract This paper considers a family of cutting planes, recently developed for
More informationInteger Programming and Branch and Bound
Courtesy of Sommer Gentry. Used with permission. Integer Programming and Branch and Bound Sommer Gentry November 4 th, 003 Adapted from slides by Eric Feron and Brian Williams, 6.40, 00. Integer Programming
More informationELE539A: Optimization of Communication Systems Lecture 16: Pareto Optimization and Nonconvex Optimization
ELE539A: Optimization of Communication Systems Lecture 16: Pareto Optimization and Nonconvex Optimization Professor M. Chiang Electrical Engineering Department, Princeton University March 16, 2007 Lecture
More informationLecture 2. MATH3220 Operations Research and Logistics Jan. 8, Pan Li The Chinese University of Hong Kong. Integer Programming Formulations
Lecture 2 MATH3220 Operations Research and Logistics Jan. 8, 2015 Pan Li The Chinese University of Hong Kong 2.1 Agenda 1 2 3 2.2 : a linear program plus the additional constraints that some or all of
More informationMATHEMATICS. Units Topics Marks I Relations and Functions 10
MATHEMATICS Course Structure Units Topics Marks I Relations and Functions 10 II Algebra 13 III Calculus 44 IV Vectors and 3-D Geometry 17 V Linear Programming 6 VI Probability 10 Total 100 Course Syllabus
More informationInteger Linear Programs
Lecture 2: Review, Linear Programming Relaxations Today we will talk about expressing combinatorial problems as mathematical programs, specifically Integer Linear Programs (ILPs). We then see what happens
More informationA A x i x j i j (i, j) (j, i) Let. Compute the value of for and
7.2 - Quadratic Forms quadratic form on is a function defined on whose value at a vector in can be computed by an expression of the form, where is an symmetric matrix. The matrix R n Q R n x R n Q(x) =
More information4/12/2011. Chapter 8. NP and Computational Intractability. Directed Hamiltonian Cycle. Traveling Salesman Problem. Directed Hamiltonian Cycle
Directed Hamiltonian Cycle Chapter 8 NP and Computational Intractability Claim. G has a Hamiltonian cycle iff G' does. Pf. Suppose G has a directed Hamiltonian cycle Γ. Then G' has an undirected Hamiltonian
More informationDevelopment of an algorithm for solving mixed integer and nonconvex problems arising in electrical supply networks
Development of an algorithm for solving mixed integer and nonconvex problems arising in electrical supply networks E. Wanufelle 1 S. Leyffer 2 A. Sartenaer 1 Ph. Toint 1 1 FUNDP, University of Namur 2
More information