Proximal mapping via network optimization

Save this PDF as:

Size: px
Start display at page:

Transcription

1 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

2 Introduction this lecture: network flow algorithm for evaluating prox-operator of h(x) = = n i= n i,j= i j= A ij x i x j A ij max{,x i x j } coefficients A ij = A ji are nonnegative associated undirected graph has n nodes, edges (i,j) when A ij > applications in image processing and machine learning Proximal mapping via network optimization 2

3 Outline minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping

4 Minimum cut problem find subset I {,2,...,n} that minimizes C(I) = i I,j IA ij + i I b i A S n with A ij = A ji ; no sign restrictions on b R n cost can be expressed as C(I) = x T A( x)+b T x x is the incidence vector of I: x k = if k I, x k = if k I graph interpretation optimal two-way partition of n nodes of undirected graph first term in C(I) is cost of the cut (edges removed by partitioning) Proximal mapping via network optimization 3

5 Discrete optimization formulations binary quadratic maximization minimize x T Ax+(A+b) T x subject to x {,} n cost function is equal to C(I) if x is incidence vector of I binary piecewise-linear minimization minimize i>j subject to x {,} n A ij x i x j +b T x cost function is equal to C(I) if x is incidence vector of I Proximal mapping via network optimization 4

6 Convex relaxation relaxation: replace x {,} n with x (componentwise) minimize i>j subject to x A ij x i x j +b T x we will use LP duality to show that the relaxation is exact relaxation has an optimal solution x {,} n if x {,} n is optimal for the relaxation, then rounding x as ˆx i = if x i > /2, ˆx i = if x i /2 gives an integer optimal solution ˆx {,} n Proximal mapping via network optimization 5

7 Linear program formulation relaxed problem as LP: introduce matrix variable Y R n n minimize tr(ay)+b T x subject to Y x T x T Y x (componentwise) inequalities on Y are equivalent to Y ij max{,x i x j }, i,j =,...,n at optimum, Y ij = max{,x i x j } because A ij ; therefore tr(ay)+b T x = i>j A ij x i x j +b T x Proximal mapping via network optimization 6

8 Dual problem dual linear program (variables U R n n, v,w R n ) maximize T w subject to U A (U U T ) v +w +b = v, w complementary slackness conditions: if x, Y, U, v, w are optimal (Y x T +x T ) U = Y (A U) = x v = ( x) w = denotes Hadamard (componentwise) matrix product Proximal mapping via network optimization 7

9 Exactness of relaxation let x be optimal for the relaxation; round x to ˆx {,} n as ˆx i = if x i > /2, ˆx i = if x i /2 by complementary slackness, if ˆx i = and ˆx j =, then x i > x j ; hence U ij = A ij, U ji =, v i =, w j = implies ˆx T A( ˆx)+b Tˆx is equal to the lower bound from relaxation ˆx T A( ˆx)+b Tˆx = ˆx T (U U T )( ˆx) ˆx T v ( ˆx) T w+b Tˆx = ˆx T ((U U T ) v +w +b) ˆx T (U U T )ˆx T w = T w Proximal mapping via network optimization 8

10 Network flow interpretation of dual LP change of variables (with b +,k = max{b k,}, b,k = max{ b k,}) Z = U U T, z s = b w, z t = b + v reformulated dual problem maximize T z s T b subject to Z z s +z t = A Z A, z s b, z t b + Z ij = Z ji is flow from node i to node j z s is vector of flows from an added source node to nodes,...,n z t is vector of flows from nodes,...,n to an added sink node maximize flow T z s from source to sink, subject to capacity constraints exactness of relaxation is known as the max-flow min-cut theorem Proximal mapping via network optimization 9

11 Outline minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping

12 Parametric min-cut problem min-cut problem: take b = α c with α a scalar parameter minimize n i,j= subject to y {,} n A ij max{,y i y j }+(α c) T y equivalent LP and its dual min. tr(ay)+(α c) T y s.t. Y y T y T Y y max. T w s.t. U A (U U T ) v +w+α = c v, w primal variables y R n, Y R n n ; dual variables U R n n, v,w R n Proximal mapping via network optimization

13 p (α) = inf y {,} n Optimal value function n i,j= A ij max{,y i y j }+(α c) T y immediate properties p (α) is piecewise-linear and concave p (α) = nα T c for α c, with optimal solution y = p (α) = for α c, with optimal solution y = less obvious: p (α) hat at most n+ linear segments Proximal mapping via network optimization

14 Monotonicity parametric min-cut problem minimize f α (y) = y T A( y)+(α c) T y subject to y {,} n monotonicity of solutions if y β is optimal for α = β and y γ is optimal for α = γ > β then y γ y β y γ is zero in all positions where y β is zero implies that optimal value function p (α) has at most n+ segments Proximal mapping via network optimization 2

15 proof of monotonicity property denote component-wise maximum and minimum of y β and y γ as y min = min{y β,y γ }, y max = max{y β,y γ } (submodularity) it is readily verified that y T maxa( y max )+y T mina( y min ) y T βa( y β )+y T γa( y γ ) from submodularity and optimality of y β, y γ : f β (y β )+f β (y γ ) = f β (y β )+f γ (y γ ) (γ β) T y γ f β (y max )+f γ (y min ) (γ β) T y γ = f β (y max )+f β (y min ) (γ β) T (y γ y min ) f β (y β )+f β (y γ ) (γ β) T (y γ y min ) therefore γ > β implies y γ = y min Proximal mapping via network optimization 3

16 Example A = c = α p (α) Proximal mapping via network optimization 4

17 Outline minimum cut and maximum flow problems parametric minimum cut problem proximal mapping via parametric flow maximization

18 Proximal mapping piecewise-linear convex function h(x) = i>j A ij x i x j = n i,j= A ij max{,x i x j } proximal mapping: x = prox h (c) is the solution of minimize h(x)+ 2 x c 2 2 equivalent to a quadratic program efficiently computed from solution of parametric minimum cut problem Proximal mapping via network optimization 5

19 Quadratic program formulation minimize tr(ay)+ 2 x c 2 2 subject to Y x T x T Y at optimum, x = prox h (c) and Y ij = max{,y i y j } optimality conditions: there exists a U R n n that satisfies U A, x+(u U T ) = c and the complementary slackness conditions (Y x T +x T ) U =, Y (A U) = in particular, if x i > x j, then U ij = A ij and U ji = Proximal mapping via network optimization 6

20 Relation with parametric min-cut problem parametric min-cut problem minimize f α (y) = n i,j= subject to y {,} n A ij max{,y i y j }+(α c) T y parametric min-cut solution from proximal mapping if x = prox h (c) then a solution of the parametric min-cut problem is y i = if x i α, y i = if x i < α proximal mapping from parametric min-cut solution x i = sup{α parametric min-cut problem has a solution with y i = } (follows from monotonicity of min-cut solution) Proximal mapping via network optimization 7

21 proof: let x = prox h (c) and U the optimal multiplier (page 6); define w = (x α) +, v = (α x) + U, v, w are dual feasible for parametric LP on page ; therefore f α (ŷ) T (x α) + ŷ {,} n equality holds for y defined on p. 7: f α (y) = y T (U U T )( y)+(α c) T y = y T ((U U T )+x c) y T (U U T )y (x α) T y = T (x α) + first line follows from U ij = A ij, U ji = if y i =, y j = (see p. 6) last line follows from definition of U (p. 6) and construction of y Proximal mapping via network optimization 8

22 Example α p (α) prox h (c) = (,,,3,2) prox h (c) k is value of α at breakpoint where y k switches from to Proximal mapping via network optimization 9

23 Summary proximal mapping of h(x) = i>j A ij x i x j can be computed by solving a parametric min-cut/max-flow problem very efficiently solved by algorithms from network optimization complexity O(mnlog(n 2 /m)) for general graphs (m is # edges) faster algorithms for graphs with special structure Proximal mapping via network optimization 2

24 References D. Goldfarb and W. Yin, Parametric maximum flow algorithms for fast total variation minimization, SIAM Journal on Scientic Computing (29) A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows, International Journal of Computer Vision (29) J. Mairal, R. Jenatton, G. Obozinski, F. Bach, Network flow algorithms for structured sparsity, arxiv.org/abs/8.529 (2) Proximal mapping via network optimization 2

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

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

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

Definition 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

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

Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

10. Proximal point method

L. Vandenberghe EE236C Spring 2013-14) 10. Proximal point method proximal point method augmented Lagrangian method Moreau-Yosida smoothing 10-1 Proximal point method a conceptual algorithm for minimizing

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

Lecture 11: 0-1 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

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

LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method

LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right

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

What 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

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

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

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;

Lecture 4: BK inequality 27th August and 6th September, 2007

CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

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

! 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

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.

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

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

1 Solving LPs: The Simplex Algorithm of George Dantzig

Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

Discuss 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

Optimization in R n Introduction

Optimization in R n Introduction Rudi Pendavingh Eindhoven Technical University Optimization in R n, lecture Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4 Some optimization problems designing

Linear Programming I

Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

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

26 Linear Programming

The greatest flood has the soonest ebb; the sorest tempest the most sudden calm; the hottest love the coldest end; and from the deepest desire oftentimes ensues the deadliest hate. Th extremes of glory

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

Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

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

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

Introduction 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.

Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

Chapter 13: Binary and Mixed-Integer Programming

Chapter 3: Binary and Mixed-Integer Programming The general branch and bound approach described in the previous chapter can be customized for special situations. This chapter addresses two special situations:

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 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

Optimal 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

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

3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

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

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

Introduction to Convex Optimization for Machine Learning

Introduction to Convex Optimization for Machine Learning John Duchi University of California, Berkeley Practical Machine Learning, Fall 2009 Duchi (UC Berkeley) Convex Optimization for Machine Learning

Bargaining Solutions in a Social Network

Bargaining Solutions in a Social Network Tanmoy Chakraborty and Michael Kearns Department of Computer and Information Science University of Pennsylvania Abstract. We study the concept of bargaining solutions,

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

! 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

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

Linear Programming in Matrix Form

Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,

International 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

princeton 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

Support Vector Machines

Support Vector Machines Charlie Frogner 1 MIT 2011 1 Slides mostly stolen from Ryan Rifkin (Google). Plan Regularization derivation of SVMs. Analyzing the SVM problem: optimization, duality. Geometric

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),

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

A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem

A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,

Linear 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

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

The Equivalence of Linear Programs and Zero-Sum Games

The Equivalence of Linear Programs and Zero-Sum Games Ilan Adler, IEOR Dep, UC Berkeley adler@ieor.berkeley.edu Abstract In 1951, Dantzig showed the equivalence of linear programming problems and two-person

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

Scheduling Parallel Jobs with Monotone Speedup 1

Scheduling Parallel Jobs with Monotone Speedup 1 Alexander Grigoriev, Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands, {a.grigoriev@ke.unimaas.nl,

Lecture 5: Conic Optimization: Overview

EE 227A: Conve Optimization and Applications January 31, 2012 Lecture 5: Conic Optimization: Overview Lecturer: Laurent El Ghaoui Reading assignment: Chapter 4 of BV. Sections 3.1-3.6 of WTB. 5.1 Linear

1. spectral. Either global (e.g., Cheeger inequality,) or local.

CS369M: Algorithms for Modern Massive Data Set Analysis Lecture 12-11/04/2009 Introduction to Graph Partitioning Lecturer: Michael Mahoney Scribes: Noah Youngs and Weidong Shao *Unedited Notes 1 Graph

Several 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

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

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

Optimal energy trade-off schedules Neal Barcelo, Daniel G. Cole, Dimitrios Letsios, Michael Nugent, Kirk R. Pruhs To cite this version: Neal Barcelo, Daniel G. Cole, Dimitrios Letsios, Michael Nugent,

Date: April 12, 2001. Contents

2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........

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

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

Numerisches 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

Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

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,

Minimizing 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

A Note on the Bertsimas & Sim Algorithm for Robust Combinatorial Optimization Problems

myjournal manuscript No. (will be inserted by the editor) A Note on the Bertsimas & Sim Algorithm for Robust Combinatorial Optimization Problems Eduardo Álvarez-Miranda Ivana Ljubić Paolo Toth Received:

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan

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

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

Duality in Linear Programming

Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow

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,

Max Flow. Lecture 4. Optimization on graphs. C25 Optimization Hilary 2013 A. Zisserman. Max-flow & min-cut. The augmented path algorithm

Lecture 4 C5 Optimization Hilary 03 A. Zisserman Optimization on graphs Max-flow & min-cut The augmented path algorithm Optimization for binary image graphs Applications Max Flow Given: a weighted directed

Approximation Algorithms. Scheduling. Approximation algorithms. Scheduling jobs on a single machine

Approximation algorithms Approximation Algorithms Fast. Cheap. Reliable. Choose two. NP-hard problems: choose 2 of optimal polynomial time all instances Approximation algorithms. Trade-off between time

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

C&O 370 Deterministic OR Models Winter 2011

C&O 370 Deterministic OR Models Winter 2011 Assignment 1 Due date: Friday Jan. 21, 2011 Assignments are due at the start of class on the due date. Write your name and ID# clearly, and underline your last

1 Portfolio mean and variance

Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a one-period investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring

The Impact of Linear Optimization on Promotion Planning

The Impact of Linear Optimization on Promotion Planning Maxime C. Cohen Operations Research Center, MIT, Cambridge, MA 02139, maxcohen@mit.edu Ngai-Hang Zachary Leung Operations Research Center, MIT, Cambridge,

A Robust Formulation of the Uncertain Set Covering Problem

A Robust Formulation of the Uncertain Set Covering Problem Dirk Degel Pascal Lutter Chair of Management, especially Operations Research Ruhr-University Bochum Universitaetsstrasse 150, 44801 Bochum, Germany

Linear programming and reductions

Chapter 7 Linear programming and reductions Many of the problems for which we want algorithms are optimization tasks: the shortest path, the cheapest spanning tree, the longest increasing subsequence,

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,

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

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

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

Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.

Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian

A Network Flow Approach in Cloud Computing

1 A Network Flow Approach in Cloud Computing Soheil Feizi, Amy Zhang, Muriel Médard RLE at MIT Abstract In this paper, by using network flow principles, we propose algorithms to address various challenges

Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology

Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology Yair Weiss School of Computer Science and Engineering The Hebrew University of Jerusalem www.cs.huji.ac.il/ yweiss

Cyber-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

Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints

Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and

Lecture 2: August 29. Linear Programming (part I)

10-725: 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.

Linear 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

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