Discrete Optimization


 Percival Bryant
 3 years ago
 Views:
Transcription
1 Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and by Johan Högdahl and Victoria Svedberg Seminar 2,
2 Todays presentation Chapter 3 Transforms using 01 variables 3.1 Transform logical expressions 3.2 Transform Nonbinary to 01 variable 3.3 Transform Piecewise Linear Functions 3.4 Transform 01 Polynomial Functions 3.5 Transform Functions with Products of 01 and Cont. Variables 3.6 Transform Nonsimultaneous Constraints Chapter 4 Better formulation by preprocessing 4.1 Better Formulation 4.2 Automatic Problem Preprocessing 4.3 Tightening bounds on Variables
3 3.1 Transform logical expressions It may be natural to formulate expressions as logical expressions Example: If we select project A then we will also select project B The logical relations is Conjuntion (A and B) Disjunction (A or B) Simple implication (if A then B) Double implication (A if and only if B) Negation (not A)
4 3.1 Transform logical expressions Consider the project planning problem. Decision variables: Let y j = 1 if we select project j, else 0. Formulated as logical statements A: project A is selected (y A = 1) or not selected (y A = 0) B: project B is selected (y B = 1) or not selected (y B = 0)
5 3.1 Transform logical expressions Conjunction (A and B) A and B => both A and B are selected Transforms into: y A = 1 and y B = 1 Alternative formulation: y A + y B = 2 Disjunction (A or B) A or B => either A or B are selected or both are selected Transforms into: y A + y B 1
6 3.1 Transform logical expressions Simple implication (if A then B) If A then B => if A are selected then B are selected, else B are either selected or not selected Transforms into: y A y B Double implication (A if and only if B) A if and only if B <=> (if A then B) and (if B then A) which is expressed as y A y B and y B y A Transforms into: y A = y B
7 3.1 Transform logical expressions Negation (not A) Not A reverses the statement A, that is not(y A = 1) => y A = 0. Relation between either/or and if/then statements If A then B <=> not A or B True since when A is true; not A is false so B must be true if the statement shall be true. In the other case when A is false the statement is true if independent of B. Multiple boolean operations on variables
8 3.2 Transform nonbinary to 01 variable Nonbinary variables General integer variables y {0,1,2,...} Discrete variables that takes on nonconsecutive integer values. For example y {2,5,9,21} Other cases are easily transformed into one of the to categories above
9 3.2 Transform nonbinary to 01 variable Transform integer variables Any finite upper bounded integer variable can be expressed by a set of 01 variables. Example: x 20 can be expressed as Where y j {0,1} for each j = 0,...,4 Generally we can express an integer x u as Where y j {0,1} for each j = 0,...,k
10 3.2 Transform nonbinary to 01 variable How many binary variables do we need? The sequence of coefficients is given recursively by (1) With k coefficients u is bounded by Which implies that Taking log 2 of (2) gives (2) We need binary variables to represent a integer x u.
11 3.2 Transform nonbinary to 01 variable Comments If b z u and b > 0 then we just et z' = z + b so we have 0 z' u + b. The number of variables grows logarithmic. Transforming general integer variables is useful when There is a small number of variables and each having a low upper bound The proposed 01 algorithm is much more efficient than the existing general integer algorithm
12 3.2 Transform nonbinary to 01 variable Transforming discrete variables If a variable is only allowed to take on one value in a list of integer numbers then we can replace that variable with a set of 01 variables. Example If z {1,5,7,9,23} then we may introduce y i = 1 if the i:th element of the list is chosen, else 0. That is z = y 1 + 5y 2 + 7y 3 + 9y y 5 y 1 + y 2 + y 3 + y 4 + y 5 = 1 (this is also called a multiple choice constraint) y 1 {0,1} for each i = 1,...,5
13 Transform piecewise linear functions f (x) = 10x if 0 x 100 (1) f (x) = x if 100 x 300 (2) f (x) = x if 300 x 500 (3)
14 Transform piecewise linear functions Every point x in the linesegment between the consecutive points a i and a i+1 can be described as: x = λ k a k + (1 λ k )a k+1 where 0 λ k 1 (4)
15 Transform piecewise linear functions f (x) is also a linesegment between f (a i ) and f (a i+1 ) it can therefore be described similarly: f (x) = λ k f (a k ) + (1 λ k )f (a k+1 ) where 0 λ k 1 (5)
16 General model: Transform piecewise linear functions x = λ 1a 1 + λ 2a λ r+1a r+1 (6) f (x) = λ 1f (a 1) + λ 2f (a 2) + + λ r+1f (a r+1) (7) λ 1 y 1 (8) λ 2 y 1 + y 2 (9) λ 3 y 2 + y 3 (10). (11) λ r y r 1 + y r (12) λ r+1 y r (13) r y k = 1 (14) k=1 r+1 λ k = 1 (15) k=1 y k 0 k (16) y k {0, 1}, 0 λ k 1 (17)
17 Transform concave piecewise linear functions Due to properties of a concave function, the model can be improved the following way: Each linesegment may be expressed as: The intercepts are formulated as: f (x) = b i + s i x (18) t 0 = 0, at a 0 = 0 (19) t i = t i 1 + s i 1 a i s i a i (20) The model can now be expressed as: (t i y i + s i x i ) (21) constraints i x = x i (22) i a i y i x a i+1 y i i i y i = 1 (23) y i {0, 1} i
18 Transform 01 polynomial functions Quadratic binary function f (y 1, y 2,..., y n) = j y 2 j + i k y i y k (24) f (y 1, y 2,..., y n) = j y j + i k y jk (25) 2y jk y j + y k y jk + 1 j k (26)
19 Transform 01 polynomial functions Binary function of general degree f (y 1, y 2,..., y n) = y j (27) j {1,...,n} f (y 1, y 2,..., y n) = y Q (28) Q y Q y j y Q + ( Q 1) j Q (29)
20 Transform functions with products of binary and continuous variables Bundle pricing problem Pricing of individual components and bundled components to maximize profit. If there are n components, there are also 2 n 1 bundling possibilities. General problem max s.t. ( ni j y ) ijx j i j y ij = 1 i (30) j (r ij x j )y ij r ij x j But this is nonlinear!
21 Transform functions with products of binary and continuous variables Bundle pricing problem Pricing of individual components and bundled components to maximize profit. If there are n components, there are also 2 n 1 bundling possibilities. General linear problem Replace y ij x j by z i j and add constraints: max s.t. ( i ni j z ) ij j y ij = 1 i j (r ijy ij z ij ) r ij x j z ij x j (31) z ij r ij y ij z ij x j (1 y ij )M j
22 3.6 Transform nonsimultaneous constraints In order for a given problem formulation to be classified as a MIP each constraint must be satisfied simultaneous, according to the assumptions that we saw last week. But it may happen that we end up with nonsimultaneous constraints when we model a problem.
23 3.6 Transform nonsimultaneous constraints Either/or constraints A decision variable may be defined in disjunctive regions, for example out side the interval [a,b]. That is, x a or x b. Such constraints is transformed into x  a My x + b M(1  y) y {0,1}
24 3.6 Transform nonsimultaneous constraints p out m constraints must hold (a generalization of the previous) If p out of m constraints must hold and we can choose any combination then we can transform that in the following way. Introduce y i = 1 if constraint i must hold, else 0, and write the constraints as f i (x)  b i My i y 1 + y y m = m  p y i {0,1}
25 3.6 Transform nonsimultaneous constraints Disjunctive constraint sets (also a generalization) Suppose that either must one subset of constraints hold or else must another subset hold, but not both. That is Either subset 1: { a i^t*x  b i 0, i = 1...,m 1 } Or subset 2: { c i^t*x d i 0, i = 1...,m 2 } We transform it into simultaneous constraints with the 01 variable y by writing the constraints as a i^t*x  b i My for each i = 1...,m 1 c i^t*x d i M(1  y) for each i = 1...,m 2 y {0,1}
26 3.6 Transform nonsimultaneous constraints Negation of a constraint The negation of a constraint f(x)  b 0 is f(x)  b > 0 <=> f(x) + b < 0 If/then constraints Since 3.2 we know that (if A then B) <=> (not A or B) Let A be f 1 (x)  b 1 0 and B be f 2 (x)  b 2 0 then Not A is f 1 (x) + b 1 < 0 So, if f 1 (x)  b 1 0 then f 2 (x)  b 2 0 is equivalent to f 1 (x) + b 1 < My f 2 (x)  b 2 M(1  y) y {0,1}
27 4 Better formulation by preprocessing For every IPproblem there exists many, possibly infinite, alternative formulations and some of these are better then others. Better formulation = easier problem to solve. Some definitions are needed.
28 4.1 Better formulation First an example Consider the following three pure IP constraints IP1: 2y 1 + 2y 2 3 y 1, y 2 integer IP2: 3y 1 + 2y 2 3 y 1, y 2 integer IP3: y 1 + y 2 1 y 1, y 2 integer All three contain the same set of feasible points S = {(0,0),(0,1),(1,0)} The LP relaxations contains the same feasible solutions and are called alternative formulations of the set S
29 4.1 Better formulation Polyhedron Formulation S y is the set of feasible integer solutions. S xy is the set of feasible mixed integer solutions.
30 4.1 Better formulation Comments Any feasible region of a linear program is a polyhedron A polyhedron P is a formulation of S if it contains the exactly same set of feasible solutions as S Example Consider the constraint set
31 4.1 Better formulation Better formulation Ideal formulation
32 4.1 Better formulation Comments Which points do we mean with the extreme points? From definition 8.4: extreme points are those points on a convex set that cannot be represented as a strict (0 < t < 1) convex combination of two points. That means that the ideal formulation is the convex hull of S y. When P is a ideal formulation of S y then the optimal LP solution is the same as the optimal IP solution. Ideal formulation means that the integer program is easy.
33 Automatic problem preprocessing Tightening bounds on variables Fixing variables Eliminating redundant constraints Identifying feasibility Tightening constraints Decomposing the problem into independent subproblems Scaling the coefficient matrix
34 Tightening bounds on continuous variables min s.t. j c jx j j a ijx j b j i l j x j u j j (32) Isolate the variable one wants to investigate (x k ) and separate positive and negative a j. a ik x ik + a ij x j + a ij x j b i i (33) j k:a ij >0 j k:a ij <0 If a ik > 0, then an upper bound is defined to û k = 1 ( b i a ij l j a ik j k:a ij >0 j k:a ij <0 a ij u j ) (34) If a ik < 0, then a lower bound is defined to ˆlk = 1 ( b i a ij l j a ik j k:a ij >0 j k:a ij <0 a ij u j ) (35)
35 Tightening bounds on integer variables If x Z: if a ik > 0, then x k û k if a ik < 0, then x k ˆl k If x {0, 1}: Same method, but one knows that 0 x i 1 i.
36 Variable fixing, redundant constraints and infeasibility max s.t. j c jx j j a ijx j b j i l j x j u j j (36) If a ij > 0 i and c j < 0, fix x j at l j. If a ij < 0 i and c j > 0, fix x j at u j. If l k = u k when applying the bound tightening routine previously described x k can be fixed to l k.
37 Variable fixing, redundant constraints and infeasibility U i = a ij u j + a ij l j (37) j:a ij >0 j:a ij <0 L i = a ij l j + a ij u j (38) j:a ij >0 j:a ij <0 Check L i b i U i If b i U i then the ith constraint is redundant and can be removed. If b i L i then the ith constraint can not be satisfied and no feasible solution exists. If b i = L i then all x j with a ij > 0 can be fixed at x j = l j and all x j with a ij < 0 can be fixed at x j = u j.
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationApproximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai
Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization
More informationSome Optimization Fundamentals
ISyE 3133B Engineering Optimization Some Optimization Fundamentals Shabbir Ahmed Email: sahmed@isye.gatech.edu Homepage: www.isye.gatech.edu/~sahmed Basic Building Blocks min or max s.t. objective as
More informationLecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method
Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming
More informationOptimization Modeling for Mining Engineers
Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2
More informationIntegrating Benders decomposition within Constraint Programming
Integrating Benders decomposition within Constraint Programming Hadrien Cambazard, Narendra Jussien email: {hcambaza,jussien}@emn.fr École des Mines de Nantes, LINA CNRS FRE 2729 4 rue Alfred Kastler BP
More informationLecture 11: 01 Quadratic Program and Lower Bounds
Lecture :  Quadratic Program and Lower Bounds (3 units) Outline Problem formulations Reformulation: Linearization & continuous relaxation Branch & Bound Method framework Simple bounds, LP bound and semidefinite
More informationINTEGER PROGRAMMING. Integer Programming. Prototype example. BIP model. BIP models
Integer Programming INTEGER PROGRAMMING In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is
More informationWeek 5 Integral Polyhedra
Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory
More informationDantzigWolfe bound and DantzigWolfe cookbook
DantzigWolfe bound and DantzigWolfe cookbook thst@man.dtu.dk DTUManagement Technical University of Denmark 1 Outline LP strength of the DantzigWolfe The exercise from last week... The DantzigWolfe
More informationScheduling Home Health Care with Separating Benders Cuts in Decision Diagrams
Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams André Ciré University of Toronto John Hooker Carnegie Mellon University INFORMS 2014 Home Health Care Home health care delivery
More informationNonlinear Optimization: Algorithms 3: Interiorpoint methods
Nonlinear Optimization: Algorithms 3: Interiorpoint methods INSEAD, Spring 2006 JeanPhilippe Vert Ecole des Mines de Paris JeanPhilippe.Vert@mines.org Nonlinear optimization c 2006 JeanPhilippe Vert,
More informationProximal mapping via network optimization
L. Vandenberghe EE236C (Spring 234) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:
More informationA Constraint Programming based Column Generation Approach to Nurse Rostering Problems
Abstract A Constraint Programming based Column Generation Approach to Nurse Rostering Problems Fang He and Rong Qu The Automated Scheduling, Optimisation and Planning (ASAP) Group School of Computer Science,
More informationTHE SCHEDULING OF MAINTENANCE SERVICE
THE SCHEDULING OF MAINTENANCE SERVICE Shoshana Anily Celia A. Glass Refael Hassin Abstract We study a discrete problem of scheduling activities of several types under the constraint that at most a single
More informationScheduling and (Integer) Linear Programming
Scheduling and (Integer) Linear Programming Christian Artigues LAAS  CNRS & Université de Toulouse, France artigues@laas.fr Master Class CPAIOR 2012  Nantes Christian Artigues Scheduling and (Integer)
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More informationDuality in General Programs. Ryan Tibshirani Convex Optimization 10725/36725
Duality in General Programs Ryan Tibshirani Convex Optimization 10725/36725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T
More informationChapter 13: Binary and MixedInteger Programming
Chapter 3: Binary and MixedInteger Programming The general branch and bound approach described in the previous chapter can be customized for special situations. This chapter addresses two special situations:
More information. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2
4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the
More informationCHAPTER 9. Integer Programming
CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral
More informationDefinition of a Linear Program
Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1
More informationA Branch and Bound Algorithm for Solving the Binary Bilevel Linear Programming Problem
A Branch and Bound Algorithm for Solving the Binary Bilevel Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,
More informationIEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2
IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3
More informationDiscuss the size of the instance for the minimum spanning tree problem.
3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can
More informationSolving Integer Programming with BranchandBound Technique
Solving Integer Programming with BranchandBound Technique This is the divide and conquer method. We divide a large problem into a few smaller ones. (This is the branch part.) The conquering part is done
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationWhat is Linear Programming?
Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to
More informationMinimizing costs for transport buyers using integer programming and column generation. Eser Esirgen
MASTER STHESIS Minimizing costs for transport buyers using integer programming and column generation Eser Esirgen DepartmentofMathematicalSciences CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG
More informationTwoStage Stochastic Linear Programs
TwoStage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 TwoStage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic
More informationSolutions to Homework 6
Solutions to Homework 6 Debasish Das EECS Department, Northwestern University ddas@northwestern.edu 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More informationRecovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branchandbound approach
MASTER S THESIS Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branchandbound approach PAULINE ALDENVIK MIRJAM SCHIERSCHER Department of Mathematical
More informationScheduling Algorithm with Optimization of Employee Satisfaction
Washington University in St. Louis Scheduling Algorithm with Optimization of Employee Satisfaction by Philip I. Thomas Senior Design Project http : //students.cec.wustl.edu/ pit1/ Advised By Associate
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More informationClassification  Examples
Lecture 2 Scheduling 1 Classification  Examples 1 r j C max given: n jobs with processing times p 1,...,p n and release dates r 1,...,r n jobs have to be scheduled without preemption on one machine taking
More informationScheduling of Mixed BatchContinuous Production Lines
Université Catholique de Louvain Faculté des Sciences Appliquées Scheduling of Mixed BatchContinuous Production Lines Thèse présentée en vue de l obtention du grade de Docteur en Sciences Appliquées par
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationBinary Search. Search for x in a sorted array A.
Divide and Conquer A general paradigm for algorithm design; inspired by emperors and colonizers. Threestep process: 1. Divide the problem into smaller problems. 2. Conquer by solving these problems. 3.
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationOptimal shift scheduling with a global service level constraint
Optimal shift scheduling with a global service level constraint Ger Koole & Erik van der Sluis Vrije Universiteit Division of Mathematics and Computer Science De Boelelaan 1081a, 1081 HV Amsterdam The
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationLargest FixedAspect, AxisAligned Rectangle
Largest FixedAspect, AxisAligned Rectangle David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created: February 21, 2004 Last Modified: February
More information3 Does the Simplex Algorithm Work?
Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. !approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of
More informationLocating and sizing bankbranches by opening, closing or maintaining facilities
Locating and sizing bankbranches by opening, closing or maintaining facilities Marta S. Rodrigues Monteiro 1,2 and Dalila B. M. M. Fontes 2 1 DMCT  Universidade do Minho Campus de Azurém, 4800 Guimarães,
More informationInverse Optimization by James Orlin
Inverse Optimization by James Orlin based on research that is joint with Ravi Ahuja Jeopardy 000  the Math Programming Edition The category is linear objective functions The answer: When you maximize
More informationTutorial: Operations Research in Constraint Programming
Tutorial: Operations Research in Constraint Programming John Hooker Carnegie Mellon University May 2009 Revised June 2009 May 2009 Slide 1 Motivation Benders decomposition allows us to apply CP and OR
More informationA MODEL TO SOLVE EN ROUTE AIR TRAFFIC FLOW MANAGEMENT PROBLEM:
A MODEL TO SOLVE EN ROUTE AIR TRAFFIC FLOW MANAGEMENT PROBLEM: A TEMPORAL AND SPATIAL CASE V. Tosic, O. Babic, M. Cangalovic and Dj. Hohlacov Faculty of Transport and Traffic Engineering, University of
More informationIntroduction to Linear Programming (LP) Mathematical Programming (MP) Concept
Introduction to Linear Programming (LP) Mathematical Programming Concept LP Concept Standard Form Assumptions Consequences of Assumptions Solution Approach Solution Methods Typical Formulations Massachusetts
More informationLecture 3: Linear methods for classification
Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,
More information2007/26. A tighter continuous time formulation for the cyclic scheduling of a mixed plant
CORE DISCUSSION PAPER 2007/26 A tighter continuous time formulation for the cyclic scheduling of a mixed plant Yves Pochet 1, François Warichet 2 March 2007 Abstract In this paper, based on the cyclic
More informationSingle machine parallel batch scheduling with unbounded capacity
Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University
More informationTwo objective functions for a real life Split Delivery Vehicle Routing Problem
International Conference on Industrial Engineering and Systems Management IESM 2011 May 25  May 27 METZ  FRANCE Two objective functions for a real life Split Delivery Vehicle Routing Problem Marc Uldry
More informationMinimize subject to. x S R
Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such
More informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationMaxMin Representation of Piecewise Linear Functions
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 297302. MaxMin Representation of Piecewise Linear Functions Sergei Ovchinnikov Mathematics Department,
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationTransportation Polytopes: a Twenty year Update
Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,
More informationLecture 10 Scheduling 1
Lecture 10 Scheduling 1 Transportation Models 1 large variety of models due to the many modes of transportation roads railroad shipping airlines as a consequence different type of equipment and resources
More informationLecture 7: Approximation via Randomized Rounding
Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining
More informationSome representability and duality results for convex mixedinteger programs.
Some representability and duality results for convex mixedinteger programs. Santanu S. Dey Joint work with Diego Morán and Juan Pablo Vielma December 17, 2012. Introduction About Motivation Mixed integer
More informationFinal Report. to the. Center for Multimodal Solutions for Congestion Mitigation (CMS) CMS Project Number: 2010018
Final Report to the Center for Multimodal Solutions for Congestion Mitigation (CMS) CMS Project Number: 2010018 CMS Project Title: Impacts of Efficient Transportation Capacity Utilization via MultiProduct
More informationLinear Programming Sensitivity Analysis
Linear Programming Sensitivity Analysis Massachusetts Institute of Technology LP Sensitivity Analysis Slide 1 of 22 Sensitivity Analysis Rationale Shadow Prices Definition Use Sign Range of Validity Opportunity
More informationChapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
More informationClassification  Examples 1 1 r j C max given: n jobs with processing times p 1,..., p n and release dates
Lecture 2 Scheduling 1 Classification  Examples 11 r j C max given: n jobs with processing times p 1,..., p n and release dates r 1,..., r n jobs have to be scheduled without preemption on one machine
More informationScheduling a sequence of tasks with general completion costs
Scheduling a sequence of tasks with general completion costs Francis Sourd CNRSLIP6 4, place Jussieu 75252 Paris Cedex 05, France Francis.Sourd@lip6.fr Abstract Scheduling a sequence of tasks in the acceptation
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationconstraint. Let us penalize ourselves for making the constraint too big. We end up with a
Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the
More informationEfficiency of algorithms. Algorithms. Efficiency of algorithms. Binary search and linear search. Best, worst and average case.
Algorithms Efficiency of algorithms Computational resources: time and space Best, worst and average case performance How to compare algorithms: machineindependent measure of efficiency Growth rate Complexity
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora
princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of
More informationConvex Programming Tools for Disjunctive Programs
Convex Programming Tools for Disjunctive Programs João Soares, Departamento de Matemática, Universidade de Coimbra, Portugal Abstract A Disjunctive Program (DP) is a mathematical program whose feasible
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More informationOn Quantum Hamming Bound
On Quantum Hamming Bound Salah A. Aly Department of Computer Science, Texas A&M University, College Station, TX 778433112, USA Email: salah@cs.tamu.edu We prove quantum Hamming bound for stabilizer codes
More informationNotes from Week 1: Algorithms for sequential prediction
CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 2226 Jan 2007 1 Introduction In this course we will be looking
More informationNumerisches Rechnen. (für Informatiker) M. Grepl J. Berger & J.T. Frings. Institut für Geometrie und Praktische Mathematik RWTH Aachen
(für Informatiker) M. Grepl J. Berger & J.T. Frings Institut für Geometrie und Praktische Mathematik RWTH Aachen Wintersemester 2010/11 Problem Statement Unconstrained Optimality Conditions Constrained
More informationModels in Transportation. Tim Nieberg
Models in Transportation Tim Nieberg Transportation Models large variety of models due to the many modes of transportation roads railroad shipping airlines as a consequence different type of equipment
More informationNonlinear Algebraic Equations Example
Nonlinear Algebraic Equations Example Continuous Stirred Tank Reactor (CSTR). Look for steady state concentrations & temperature. s r (in) p,i (in) i In: N spieces with concentrations c, heat capacities
More informationErdős on polynomials
Erdős on polynomials Vilmos Totik University of Szeged and University of South Florida totik@mail.usf.edu Vilmos Totik (SZTE and USF) Polynomials 1 / * Erdős on polynomials Vilmos Totik (SZTE and USF)
More informationThe Multiplicative Weights Update method
Chapter 2 The Multiplicative Weights Update method The Multiplicative Weights method is a simple idea which has been repeatedly discovered in fields as diverse as Machine Learning, Optimization, and Game
More informationLecture 13 Linear quadratic Lyapunov theory
EE363 Winter 289 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discretetime
More informationLecture Notes on Linear Search
Lecture Notes on Linear Search 15122: Principles of Imperative Computation Frank Pfenning Lecture 5 January 29, 2013 1 Introduction One of the fundamental and recurring problems in computer science is
More informationInternational Doctoral School Algorithmic Decision Theory: MCDA and MOO
International Doctoral School Algorithmic Decision Theory: MCDA and MOO Lecture 2: Multiobjective Linear Programming Department of Engineering Science, The University of Auckland, New Zealand Laboratoire
More informationIntroduction to Online Learning Theory
Introduction to Online Learning Theory Wojciech Kot lowski Institute of Computing Science, Poznań University of Technology IDSS, 04.06.2013 1 / 53 Outline 1 Example: Online (Stochastic) Gradient Descent
More informationSolutions Of Some NonLinear Programming Problems BIJAN KUMAR PATEL. Master of Science in Mathematics. Prof. ANIL KUMAR
Solutions Of Some NonLinear Programming Problems A PROJECT REPORT submitted by BIJAN KUMAR PATEL for the partial fulfilment for the award of the degree of Master of Science in Mathematics under the supervision
More informationMinimizing the Number of Machines in a UnitTime Scheduling Problem
Minimizing the Number of Machines in a UnitTime Scheduling Problem Svetlana A. Kravchenko 1 United Institute of Informatics Problems, Surganova St. 6, 220012 Minsk, Belarus kravch@newman.basnet.by Frank
More informationSeveral Views of Support Vector Machines
Several Views of Support Vector Machines Ryan M. Rifkin Honda Research Institute USA, Inc. Human Intention Understanding Group 2007 Tikhonov Regularization We are considering algorithms of the form min
More informationLAGRANGIAN RELAXATION TECHNIQUES FOR LARGE SCALE OPTIMIZATION
LAGRANGIAN RELAXATION TECHNIQUES FOR LARGE SCALE OPTIMIZATION Kartik Sivaramakrishnan Department of Mathematics NC State University kksivara@ncsu.edu http://www4.ncsu.edu/ kksivara SIAM/MGSA Brown Bag
More informationSYSTEMS OF EQUATIONS
SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which
More informationEquilibrium computation: Part 1
Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium
More informationDistributed and Scalable QoS Optimization for Dynamic Web Service Composition
Distributed and Scalable QoS Optimization for Dynamic Web Service Composition Mohammad Alrifai L3S Research Center Leibniz University of Hannover, Germany alrifai@l3s.de Supervised by: Prof. Dr. tech.
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More informationCan linear programs solve NPhard problems?
Can linear programs solve NPhard problems? p. 1/9 Can linear programs solve NPhard problems? Ronald de Wolf Linear programs Can linear programs solve NPhard problems? p. 2/9 Can linear programs solve
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationLecture 2: August 29. Linear Programming (part I)
10725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.
More informationKey words. Mixedinteger programming, mixing sets, convex hull descriptions, lotsizing.
MIXING SETS LINKED BY BIDIRECTED PATHS MARCO DI SUMMA AND LAURENCE A. WOLSEY Abstract. Recently there has been considerable research on simple mixedinteger sets, called mixing sets, and closely related
More informationInteger Programming: Algorithms  3
Week 9 Integer Programming: Algorithms  3 OPR 992 Applied Mathematical Programming OPR 992  Applied Mathematical Programming  p. 1/12 DantzigWolfe Reformulation Example Strength of the Linear Programming
More information