# Lecture 11: 0-1 Quadratic Program and Lower Bounds

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 relaxation Variable fixation / 25

2 Problem formulation Standard form with - variables: (-QP) min f (x) = x {,} n 2 x T Qx + c T x where Q is an n n symmetric matrix and c R n. Homogenous form: Binary variables: (-QP h ) (BQP) min x T Qx. x {,} n min x T Qx + c T x. x {,} n Transformation: x i = 2 (y i + ). Homogenous form with binary variables: (BQP h ) min x T Qx, x {,} n (BQP) ( (BQP h ) with ) Q := 2 ct, x := (±, x c Q T ) {, } n+. 2 / 25

3 Max-Cut problem Consider a graph G = (E, V ) with vertex set V = {,..., n} and edge set E = {ij i < j n}. For every edge ij E, there is an associated weight w ij. Cut: For a given set S V, a cut δ(s) is the set of all edges with one endpoint in S and the other in V \ S, and the weight of cut δ(s) is ij δ(s) w ij. Max-Cut: find a cut δ(s) with maximum weight. Binary quadratic problem: (Max-Cut) max 2 i<j n s.t. x {, } n. w ij ( x i x j ) x {, } n S = {i V x i = } and V \ S = {i V x i = }. 3 / 25

4 Linearization Method - Quadratic problem: (P) n min Q(x) = c i x i + x {,} n i= i<j n q ij x i x j. For x i, x j {, }, y ij = x i x j iff y ij = max{x i + x j, }, y ij {, }, or y ij = min{x i, x j }, y ij {, }. 4 / 25

5 (P) is equivalent to the following - linear integer program: n min c i x i + q ij y ij + q ij y ij x,y i= (i,j) I + (i,j) I s.t. y ij x i, y ij x j, (i, j) I (q ij < ) y ij x i + x j, (i, j) I + (q ij ), x i {, }, i =,..., n, y ij {, }, i < j n. A polynomially solvable case: q ij : n c i x i + q ij y ij min x,y i= i<j n s.t. y ij x i, i < j n y ij x j, i < j n x i, x j, y ij {, }, i < j n. The constraint matrix is totally unimodular! Can we use transformation: z i = x i for q ij >? 5 / 25

6 Continuous relaxation Consider the continuous relaxation of (P): ( P) n min Q(x) = c i x i + x [,] n i= i<j n q ij x i x j. Then, at least one of the optimal solutions of ( P) is located at an extreme point of [, ] n. Therefore v(p) = v( P). Unfortunately, the objective function of ( P) is nonconvex and nonconcave. Define n Q p (x) = c i x i + x T Qx px T x + pe T x. i= Q p (x) = Q(x) for x {, } n. For large p, Q p (x) is a concave function. Thus, (P) is equivalent to the concave minimization problem: (P c ) min x [,] n Q p(x) 6 / 25

7 Branch and Bound Framework Computing lower bound; Branching on x i = or x i = ; Fixing variable by certain optimality condition. 7 / 25

8 Basic lower bounding methods Simple lower bounds Continuous relaxation LP relaxation Lagrangian relaxation & SDP relaxation 8 / 25

9 Simple bounds Lower bound. Let Q(x) = n c i x i + 2 x T Qx = i= n c i x i + 2 x T Qx. i= An obvious lower bound of Q(x) over {, } n is: LB s = 2 n n min(q ij, ) + min(c i + 2 q ii, ). i= j i i= Lower bound 2. An improved simple lower bound is derived by noting that: since x, if Qx a, then 2 x T Qx 2 at x. Let Q i denote the ith row of Q. Then a i = min Q i x = min(q ij, ). x {,} n j i 9 / 25

10 So = min x {,} n 2 x T Qx + c T x min x {,} n( 2 a + c)t x n min{c i + 2 q ii + min(q ij, ), } 2 i= = LB 2 s. j i It is easy to show that LB 2 s is better than LB s, i.e., LB 2 s LB s. / 25

11 Continuous relaxation Since Q(x) = 2 x T (Q + diag(u))x + (c 2 u)t x takes the same value on {, } n as Q(x), it is natural to compute a lower bound via solving the continuous relaxation: ( P) β(u) = min x [,] n 2 x T (Q + diag(u))x + (c 2 u)t x. Some observations: If ui s are large enough, then Q = Q + diag(u) will be diagonally dominant (thus positive definite). u = λ min e is an obvious choice to make Q positive semidefinite (but not necessarily the best one). The optimal solution to ( P) tends to ( 2, 2,..., 2 )T as u i s are increased. / 25

12 Consider a small example of (P) where ( ) ( 3 Q =, c = 3 ). For this example, we have x = (, ) T with Q(x ) =. The two simple bounds for this problem are: LB s = 3 and LB 2 s =. The eigenvalues of Q is ( 2, 4). 2 / 25

13 Figure: The figure of Q(x) over [, ] 2, which is nonconvex 3 / 25

14 Figure: u = λ min e, x u = (.864, ) T, β(u) = / 25

15 Figure: u = 2e, x u = (.577,.5538) T, β(u) = / 25

16 Figure: u = e, x u = (.53,.5) T, β(u) = / 25

17 Another way of choosing u is to find a u such that β(u ) = max{β(u) (Q diag(u)), u R n }. The above problem is equivalent to a semidefinite quadratic program which can be solved efficiently (polynomially). 7 / 25

18 LP relaxation The continuous relaxation of the - linearized problem is a linear program (y ij = x i x j ): min x,y i<j n q ij y ij + n (c i + 2 q ii)x i i= s.t. y ij x i, y ij x j, i < j n, q ij <, x i + x j y ij, i < j n, q ij >, x i, i =,..., n, y ij, i < j n. 8 / 25

19 SDP relaxation Consider Q(x) = 2 x T Qx + c T. The Lagrangian dual is: v(d) = max λ R n d(λ) = max λ R n x R n{ 2 x T [Q + 2diag(λ)]x + c T x e T λ} = max τ (λ,τ) R n+ s.t. 2 x T [Q + 2diag(λ)]x + c T x e T λ τ x R n. 9 / 25

20 2 x T [Q + 2diag(λ)]x + c T x e T λ τ, x R n ( (x T Q + 2diag(λ) c, t) c T 2τ 2e T λ (t, x) R n+, ) ( x t ) ( Q + 2diag(λ) c c T 2τ 2e T λ ), 2 / 25

21 Variable fixation Let x denote the optimal solution of (P). Let l i = c i + 2 q ii + j i min(, q ij ), u i = c i + 2 q ii + j i max(, q ij ). (i) If l i > then x i = ; (ii) If u i <, then x i =. 2 / 25

22 Fixing variable by outerbox Equivalent perturbed problem: (P µ ) min x {,} n f µ(x) = 2 x T (Q + µi )x + (c 2 µe)t x, where µ, I is the n n identity matrix and e = (,..., ) T. The shape of contour changes when µ changes. The properties of (P µ ), e.g., the conditional number, change when µ changes. 22 / 25

23 (,) (,) (,) (,) x 2 µ= µ=5 µ=3 2 µ= x (µ) 3 4 µ= x 23 / 25

24 Ellipsoid contour: Let x {, } n. E µ = {x f µ (x) = f (x ). Let the minimum box that contains ellipsoid E µ be B = [a, b], where a = x s, b = x + s. Then Let x be an optimal solution to (P). Then (i) If b i < for some i, then x i = ; (ii) If a i > for some i, then x i =. 24 / 25

25 .5 (,) (,).5 (,) (,).5 x (µ) Eµ(ṽ) Bµ(ṽ) / 25

### Lecture 5 Principal Minors and the Hessian

Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and

### Proximal mapping via network optimization

L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

### Discrete Optimization

Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

### An Introduction on SemiDefinite Program

An Introduction on SemiDefinite Program from the viewpoint of computation Hayato Waki Institute of Mathematics for Industry, Kyushu University 2015-10-08 Combinatorial Optimization at Work, Berlin, 2015

### Lecture 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

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### 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

### Nonlinear Programming Methods.S2 Quadratic Programming

Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective

### CHAPTER 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

### Some Optimization Fundamentals

ISyE 3133B Engineering Optimization Some Optimization Fundamentals Shabbir Ahmed E-mail: sahmed@isye.gatech.edu Homepage: www.isye.gatech.edu/~sahmed Basic Building Blocks min or max s.t. objective as

### An Overview Of Software For Convex Optimization. Brian Borchers Department of Mathematics New Mexico Tech Socorro, NM 87801 borchers@nmt.

An Overview Of Software For Convex Optimization Brian Borchers Department of Mathematics New Mexico Tech Socorro, NM 87801 borchers@nmt.edu In fact, the great watershed in optimization isn t between linearity

### Support Vector Machines Explained

March 1, 2009 Support Vector Machines Explained Tristan Fletcher www.cs.ucl.ac.uk/staff/t.fletcher/ Introduction This document has been written in an attempt to make the Support Vector Machines (SVM),

### Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725

Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 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

### 5.1 Bipartite Matching

CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

### Lecture 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

### Numerical Methods I Eigenvalue Problems

Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 September 30th, 2010 A. Donev (Courant Institute)

### 2.3 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

### Lecture 13 Linear quadratic Lyapunov theory

EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time

### LAGRANGIAN 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

### Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.

### On SDP- and CP-relaxations and on connections between SDP-, CP and SIP

On SDP- and CP-relaations and on connections between SDP-, CP and SIP Georg Still and Faizan Ahmed University of Twente p 1/12 1. IP and SDP-, CP-relaations Integer program: IP) : min T Q s.t. a T j =

### Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach

MASTER S THESIS Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach PAULINE ALDENVIK MIRJAM SCHIERSCHER Department of Mathematical

### Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a non-zero column vector V such that AV = λv then λ is called an eigenvalue of A and V is

### Solving polynomial least squares problems via semidefinite programming relaxations

Solving polynomial least squares problems via semidefinite programming relaxations Sunyoung Kim and Masakazu Kojima August 2007, revised in November, 2007 Abstract. A polynomial optimization problem whose

### Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

### MAP-Inference for Highly-Connected Graphs with DC-Programming

MAP-Inference for Highly-Connected Graphs with DC-Programming Jörg Kappes and Christoph Schnörr Image and Pattern Analysis Group, Heidelberg Collaboratory for Image Processing, University of Heidelberg,

### Two-Stage Stochastic Linear Programs

Two-Stage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 Two-Stage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic

### 11. APPROXIMATION ALGORITHMS

11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005

### Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### Week 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

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Nonlinear Optimization: Algorithms 3: Interior-point methods

Nonlinear Optimization: Algorithms 3: Interior-point methods INSEAD, Spring 2006 Jean-Philippe Vert Ecole des Mines de Paris Jean-Philippe.Vert@mines.org Nonlinear optimization c 2006 Jean-Philippe Vert,

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

### Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

### Lecture Topic: Low-Rank Approximations

Lecture Topic: Low-Rank Approximations Low-Rank Approximations We have seen principal component analysis. The extraction of the first principle eigenvalue could be seen as an approximation of the original

### Absolute Value Programming

Computational Optimization and Aplications,, 1 11 (2006) c 2006 Springer Verlag, Boston. Manufactured in The Netherlands. Absolute Value Programming O. L. MANGASARIAN olvi@cs.wisc.edu Computer Sciences

### A QCQP Approach to Triangulation. Chris Aholt, Sameer Agarwal, and Rekha Thomas University of Washington 2 Google, Inc.

A QCQP Approach to Triangulation 1 Chris Aholt, Sameer Agarwal, and Rekha Thomas 1 University of Washington 2 Google, Inc. 2 1 The Triangulation Problem X Given: -n camera matrices P i R 3 4 -n noisy observations

### Using the Theory of Reals in. Analyzing Continuous and Hybrid Systems

Using the Theory of Reals in Analyzing Continuous and Hybrid Systems Ashish Tiwari Computer Science Laboratory (CSL) SRI International (SRI) Menlo Park, CA 94025 Email: ashish.tiwari@sri.com Ashish Tiwari

### 1 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

### Massive Data Classification via Unconstrained Support Vector Machines

Massive Data Classification via Unconstrained Support Vector Machines Olvi L. Mangasarian and Michael E. Thompson Computer Sciences Department University of Wisconsin 1210 West Dayton Street Madison, WI

### Derivative Free Optimization

Department of Mathematics Derivative Free Optimization M.J.D. Powell LiTH-MAT-R--2014/02--SE Department of Mathematics Linköping University S-581 83 Linköping, Sweden. Three lectures 1 on Derivative Free

### Inverse 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

### Minimize 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

### Explicit inverses of some tridiagonal matrices

Linear Algebra and its Applications 325 (2001) 7 21 wwwelseviercom/locate/laa Explicit inverses of some tridiagonal matrices CM da Fonseca, J Petronilho Depto de Matematica, Faculdade de Ciencias e Technologia,

### Solving Integer Programming with Branch-and-Bound Technique

Solving Integer Programming with Branch-and-Bound 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

### Branch and Cut for TSP

Branch and Cut for TSP jla,jc@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark 1 Branch-and-Cut for TSP Branch-and-Cut is a general technique applicable e.g. to solve symmetric

### MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

### (67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7

(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition

### Randomization Approaches for Network Revenue Management with Customer Choice Behavior

Randomization Approaches for Network Revenue Management with Customer Choice Behavior Sumit Kunnumkal Indian School of Business, Gachibowli, Hyderabad, 500032, India sumit kunnumkal@isb.edu March 9, 2011

### A Lagrangian-DNN Relaxation: a Fast Method for Computing Tight Lower Bounds for a Class of Quadratic Optimization Problems

A Lagrangian-DNN Relaxation: a Fast Method for Computing Tight Lower Bounds for a Class of Quadratic Optimization Problems Sunyoung Kim, Masakazu Kojima and Kim-Chuan Toh October 2013 Abstract. We propose

### 8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

### Algorithm Design and Analysis

Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;

Quadratic Functions, Optimization, and Quadratic Forms Robert M. Freund February, 2004 2004 Massachusetts Institute of echnology. 1 2 1 Quadratic Optimization A quadratic optimization problem is an optimization

### The tree-number and determinant expansions (Biggs 6-7)

The tree-number and determinant expansions (Biggs 6-7) André Schumacher March 20, 2006 Overview Biggs 6-7 [1] The tree-number κ(γ) κ(γ) and the Laplacian matrix The σ function Elementary (sub)graphs Coefficients

### MOSEK modeling manual

MOSEK modeling manual August 12, 2014 Contents 1 Introduction 1 2 Linear optimization 3 2.1 Introduction....................................... 3 2.1.1 Basic notions.................................. 3

### NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems

NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems Amir Ali Ahmadi, Alex Olshevsky, Pablo A. Parrilo, and John N. Tsitsiklis Abstract We show that unless P=NP, there exists no

### On Minimal Valid Inequalities for Mixed Integer Conic Programs

On Minimal Valid Inequalities for Mixed Integer Conic Programs Fatma Kılınç Karzan June 27, 2013 Abstract We study mixed integer conic sets involving a general regular (closed, convex, full dimensional,

### Split Nonthreshold Laplacian Integral Graphs

Split Nonthreshold Laplacian Integral Graphs Stephen Kirkland University of Regina, Canada kirkland@math.uregina.ca Maria Aguieiras Alvarez de Freitas Federal University of Rio de Janeiro, Brazil maguieiras@im.ufrj.br

### Gossip Algorithms. Devavrat Shah MIT

Gossip Algorithms Devavrat Shah MIT Motivation Ad-hoc networks Not deliberately designed with an infrastructure Some examples Sensor networks formed by randomly deployed sensors in a geographic area for

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

### Load-Balanced Virtual Backbone Construction for Wireless Sensor Networks

Load-Balanced Virtual Backbone Construction for Wireless Sensor Networks Jing (Selena) He, Shouling Ji, Yi Pan, Zhipeng Cai Department of Computer Science, Georgia State University, Atlanta, GA, USA, {jhe9,

### 4.6 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

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### SF2940: Probability theory Lecture 8: Multivariate Normal Distribution

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,

### 15-750 Graduate Algorithms Spring 2010

15-750 Graduate Algorithms Spring 2010 1 Cheap Cut (25 pts.) Suppose we are given a directed network G = (V, E) with a root node r and a set of terminal nodes T V. We would like to remove the fewest number

### Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### Optimization 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

### Applied 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

### Approximation 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

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### A Metaheuristic Optimization Algorithm for Binary Quadratic Problems

OSE SEMINAR 22 A Metaheuristic Optimization Algorithm for Binary Quadratic Problems Otto Nissfolk CENTER OF EXCELLENCE IN OPTIMIZATION AND SYSTEMS ENGINEERING ÅBO AKADEMI UNIVERSITY ÅBO, NOVEMBER 29 th

### Dantzig-Wolfe bound and Dantzig-Wolfe cookbook

Dantzig-Wolfe bound and Dantzig-Wolfe cookbook thst@man.dtu.dk DTU-Management Technical University of Denmark 1 Outline LP strength of the Dantzig-Wolfe The exercise from last week... The Dantzig-Wolfe

### Compressing Forwarding Tables for Datacenter Scalability

TECHNICAL REPORT TR12-03, TECHNION, ISRAEL 1 Compressing Forwarding Tables for Datacenter Scalability Ori Rottenstreich, Marat Radan, Yuval Cassuto, Isaac Keslassy, Carmi Arad, Tal Mizrahi, Yoram Revah

### Maximum Margin Clustering

Maximum Margin Clustering Linli Xu James Neufeld Bryce Larson Dale Schuurmans University of Waterloo University of Alberta Abstract We propose a new method for clustering based on finding maximum margin

### Zeros of Polynomial Functions

Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + 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

### Transportation 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,

### Math 2280 - Assignment 6

Math 2280 - Assignment 6 Dylan Zwick Spring 2014 Section 3.8-1, 3, 5, 8, 13 Section 4.1-1, 2, 13, 15, 22 Section 4.2-1, 10, 19, 28 1 Section 3.8 - Endpoint Problems and Eigenvalues 3.8.1 For the eigenvalue

### LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005

LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005 DAVID L. BERNICK dbernick@soe.ucsc.edu 1. Overview Typical Linear Programming problems Standard form and converting

### Lecture 1: Schur s Unitary Triangularization Theorem

Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections

### Some representability and duality results for convex mixed-integer programs.

Some representability and duality results for convex mixed-integer programs. Santanu S. Dey Joint work with Diego Morán and Juan Pablo Vielma December 17, 2012. Introduction About Motivation Mixed integer

### CONSTRAINED NONLINEAR PROGRAMMING

149 CONSTRAINED NONLINEAR PROGRAMMING We now turn to methods for general constrained nonlinear programming. These may be broadly classified into two categories: 1. TRANSFORMATION METHODS: In this approach

### Advanced Lecture on Mathematical Science and Information Science I. Optimization in Finance

Advanced Lecture on Mathematical Science and Information Science I Optimization in Finance Reha H. Tütüncü Visiting Associate Professor Dept. of Mathematical and Computing Sciences Tokyo Institute of Technology

### SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation

SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline

### Scheduling Parallel Jobs with Linear Speedup

Scheduling Parallel Jobs with Linear Speedup Alexander Grigoriev and Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands. Email: {a.grigoriev,m.uetz}@ke.unimaas.nl

### On the Unique Games Conjecture

On the Unique Games Conjecture Antonios Angelakis National Technical University of Athens June 16, 2015 Antonios Angelakis (NTUA) Theory of Computation June 16, 2015 1 / 20 Overview 1 Introduction 2 Preliminary

### 24. 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 NP-complete. Then one can conclude according to the present state of science that no

### Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

### Minimum Makespan Scheduling

Minimum Makespan Scheduling Minimum makespan scheduling: Definition and variants Factor 2 algorithm for identical machines PTAS for identical machines Factor 2 algorithm for unrelated machines Martin Zachariasen,

### Facility Location: Discrete Models and Local Search Methods

Facility Location: Discrete Models and Local Search Methods Yury KOCHETOV Sobolev Institute of Mathematics, Novosibirsk, Russia Abstract. Discrete location theory is one of the most dynamic areas of operations

### Scheduling Parallel Machine Scheduling. Tim Nieberg

Scheduling Parallel Machine Scheduling Tim Nieberg Problem P C max : m machines n jobs with processing times p 1,..., p n Problem P C max : m machines n jobs with processing times p 1,..., p { n 1 if job

### Conic optimization: examples and software

Conic optimization: examples and software Etienne de Klerk Tilburg University, The Netherlands Etienne de Klerk (Tilburg University) Conic optimization: examples and software 1 / 16 Outline Conic optimization

### Lecture 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

### NP-Hardness Results Related to PPAD

NP-Hardness Results Related to PPAD Chuangyin Dang Dept. of Manufacturing Engineering & Engineering Management City University of Hong Kong Kowloon, Hong Kong SAR, China E-Mail: mecdang@cityu.edu.hk Yinyu

### Statistical Machine Learning

Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

### IEOR 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