Lecture 11: 0-1 Quadratic Program and Lower Bounds
|
|
- Lily Cathleen Parks
- 8 years ago
- Views:
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
More informationProximal 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:
More informationDiscrete 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
More informationAn 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
More informationDATA 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
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 information5 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 informationNonlinear 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
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 informationAn 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
More informationSupport 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),
More informationDuality 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
More information5.1 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
More informationNumerical 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)
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 informationLecture 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
More informationLinear 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.
More informationOn 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 =
More informationRecovery 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
More informationSolving 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
More informationTwo-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
More informationMAP-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,
More informationPermutation 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
More information11. 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
More informationUsing 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
More informationZeros 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
More information! 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
More informationNonlinear 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,
More informationDefinition 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
More informationLecture 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
More informationA 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
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 informationMassive 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
More informationDerivative 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
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 NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one
More informationBranch 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
More informationRandomization 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
More information8.1 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
More informationMATH 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
More information(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
More informationA 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
More informationAlgorithm 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;
More informationNP-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
More information! 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
More informationSplit 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
More informationLoad-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,
More informationOn 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,
More informationApproximation 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
More informationGossip 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
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 informationPermutation 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
More informationSF2940: 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,
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 informationA 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
More informationMA107 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
More informationDantzig-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
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 informationCompressing 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
More informationMaximum 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
More informationZeros 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
More informationAdvanced 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
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 informationMath 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
More informationLECTURE: 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
More informationSome 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
More informationLecture 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
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 NP-complete. Then one can conclude according to the present state of science that no
More informationCONSTRAINED 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
More informationConic 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
More informationFacility 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
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 informationNP-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
More informationStatistical 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
More informationOn 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
More informationIntroduction 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
More informationHow the European day-ahead electricity market works
How the European day-ahead electricity market works ELEC0018-1 - Marché de l'énergie - Pr. D. Ernst! Bertrand Cornélusse, Ph.D.! bertrand.cornelusse@ulg.ac.be! October 2014! 1 Starting question How is
More informationEconomics 326: Duality and the Slutsky Decomposition. Ethan Kaplan
Economics 326: Duality and the Slutsky Decomposition Ethan Kaplan September 19, 2011 Outline 1. Convexity and Declining MRS 2. Duality and Hicksian Demand 3. Slutsky Decomposition 4. Net and Gross Substitutes
More informationCyber-Security Analysis of State Estimators in Power Systems
Cyber-Security Analysis of State Estimators in Electric Power Systems André Teixeira 1, Saurabh Amin 2, Henrik Sandberg 1, Karl H. Johansson 1, and Shankar Sastry 2 ACCESS Linnaeus Centre, KTH-Royal Institute
More informationOptimization Methods in Finance
Optimization Methods in Finance Gerard Cornuejols Reha Tütüncü Carnegie Mellon University, Pittsburgh, PA 15213 USA January 2006 2 Foreword Optimization models play an increasingly important role in financial
More informationSECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA
SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the
More informationOn the effect of forwarding table size on SDN network utilization
IBM Haifa Research Lab On the effect of forwarding table size on SDN network utilization Rami Cohen IBM Haifa Research Lab Liane Lewin Eytan Yahoo Research, Haifa Seffi Naor CS Technion, Israel Danny Raz
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 informationA characterization of trace zero symmetric nonnegative 5x5 matrices
A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the
More informationWarshall s Algorithm: Transitive Closure
CS 0 Theory of Algorithms / CS 68 Algorithms in Bioinformaticsi Dynamic Programming Part II. Warshall s Algorithm: Transitive Closure Computes the transitive closure of a relation (Alternatively: all paths
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
More informationEfficient and Robust Allocation Algorithms in Clouds under Memory Constraints
Efficient and Robust Allocation Algorithms in Clouds under Memory Constraints Olivier Beaumont,, Paul Renaud-Goud Inria & University of Bordeaux Bordeaux, France 9th Scheduling for Large Scale Systems
More informationMinimizing the Number of Machines in a Unit-Time Scheduling Problem
Minimizing the Number of Machines in a Unit-Time Scheduling Problem Svetlana A. Kravchenko 1 United Institute of Informatics Problems, Surganova St. 6, 220012 Minsk, Belarus kravch@newman.bas-net.by Frank
More informationSolutions Of Some Non-Linear Programming Problems BIJAN KUMAR PATEL. Master of Science in Mathematics. Prof. ANIL KUMAR
Solutions Of Some Non-Linear 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 informationLogistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression
Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationSF2940: Probability theory Lecture 8: Multivariate Normal Distribution
SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2014 Timo Koski () Mathematisk statistik 24.09.2014 1 / 75 Learning outcomes Random vectors, mean vector, covariance
More informationSummer course on Convex Optimization. Fifth Lecture Interior-Point Methods (1) Michel Baes, K.U.Leuven Bharath Rangarajan, U.
Summer course on Convex Optimization Fifth Lecture Interior-Point Methods (1) Michel Baes, K.U.Leuven Bharath Rangarajan, U.Minnesota Interior-Point Methods: the rebirth of an old idea Suppose that f is
More informationScheduling 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
More informationOctober 3rd, 2012. Linear Algebra & Properties of the Covariance Matrix
Linear Algebra & Properties of the Covariance Matrix October 3rd, 2012 Estimation of r and C Let rn 1, rn, t..., rn T be the historical return rates on the n th asset. rn 1 rṇ 2 r n =. r T n n = 1, 2,...,
More informationHet inplannen van besteld ambulancevervoer (Engelse titel: Scheduling elected ambulance transportation)
Technische Universiteit Delft Faculteit Elektrotechniek, Wiskunde en Informatica Delft Institute of Applied Mathematics Het inplannen van besteld ambulancevervoer (Engelse titel: Scheduling elected ambulance
More informationIntroduction to Support Vector Machines. Colin Campbell, Bristol University
Introduction to Support Vector Machines Colin Campbell, Bristol University 1 Outline of talk. Part 1. An Introduction to SVMs 1.1. SVMs for binary classification. 1.2. Soft margins and multi-class classification.
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More information