# Conic optimization: examples and software

Save this PDF as:

Size: px
Start display at page:

## Transcription

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

2 Outline Conic optimization Second order cone optimization example: robust linear programming; Semidefinite programming examples: Lyapunov stability and data fitting; Software. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 2 / 16

3 Cones Convex cones The set K R n is a convex cone if it is a convex set and for all x K and λ > 0 one has λx K. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 3 / 16

4 Convex cones Conic optimization problem Data: A convex cone K R n ; A linear operator A : R n R m ; Vectors c R n and b R m, and an inner product, on R n. Conic optimization problem inf { c, x : Ax = b}. x K Etienne de Klerk (Tilburg University) Conic optimization: examples and software 4 / 16

5 Choices for K Convex cones We consider the conic optimization problem for three choices of the cone K (or Cartesian products of cones of this type): Linear Programming (LP): K is the nonnegative orthant in R n : R n + := {x R n : x i 0 (i = 1,..., n)}, Second order cone programming (SOCP): K is the second order (Lorentz) cone: {[ ] } x : x R n, t R, t x. t Semidefinite programming (SDP): K is the cone of symmetric positive semidefinite matrices. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 5 / 16

6 Robust LP Examples for the second order cone We consider an LP problem with uncertain data. Robust LP Problem min c T x subject to ai T x b i (i = 1,..., m) a i E i (i = 1,..., m), where the E i are given ellipsoids: E i = {ā i + P i u : u 1}, with P i symmetric positive semidefinite. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 6 / 16

7 Examples for the second order cone Robust LP: SOCP formulation We had Notice that E i := {ā i + P i u : u 1}. a T i x b i a i E i ā T i x + P i x b i Robust LP Problem: SOCP reformulation subject to min c T x ā T i x + P i x b i (i = 1,..., m). Note that this is indeed an SOCP problem. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 7 / 16

8 References and info Examples for the second order cone The solutions of at least 13 of the 90 Netlib LP problems are meaningless if there is 0.01% uncertainty in the data entries! Solving the robust LP instead overcomes this difficulty. Ben-Tal, A., Nemirovski, A. Robust solutions of Linear Programming problems contaminated with uncertain data. Math. Progr. 88 (2000), More SOCP examples in the online paper: M. Lobo, L. Vandenberghe, S. Boyd, H. Lebret, Applications of second-order cone programming. Linear Algebra and its Applications, Applications include robust least squares problems, portfolio selection, filter design... Etienne de Klerk (Tilburg University) Conic optimization: examples and software 8 / 16

9 Examples for the positive semidefinite cone Example: sum of squares polynomials Example Is p(x) := 2x x 3 1 x 2 x 2 1 x x 4 2 a sum of squared polynomials? YES, because p(x) = x 2 1 x 2 2 x 1 x 2 T x 2 1 x 2 2 x 1 x 2. The 3 3 matrix (say M) is positive semidefinite and: and consequently p(x) = 1 2 M = L T L, L = 1 2 [ ], ( 2x 2 1 3x2 2 ) 2 1 ( ) + x 1 x 2 + x x 1 x 2. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 9 / 16

10 Examples for the positive semidefinite cone Discussion: sum of squares polynomials The example illustrates the fact that deciding if a polynomial is a sum of squares is equivalent to an SDP problem; This has application in polynomial optimization problems, data fitting using nonnegative or monotone polynomials,... and finding polynomial Lyapunov functions to prove stability of dynamical systems. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 10 / 16

11 Examples for the positive semidefinite cone Example: Lyapunov stability Definition The origin is asymptotically stable for a dynamical system ẋ(t) = f (x(t)), x(0) = x 0 if lim t x(t) = 0 whenever x 0 is sufficiently close to 0. A sufficient condition for stability is a nonnegative Lyapunov function V : R n R such that V (0) = 0 and V (x) T f (x) < 0 if x 0. Example (Parrilo): ẋ 1 (t) = x 2 (t) x 2 1 (t) 1 2 x 3 1 (t) ẋ 2 (t) = 3x 1 (t) x 2 (t). Using SDP, one may find a degree 4 polynomial V to prove stability, where both V (x) and V (x) T f (x) are sums of squares. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 11 / 16

12 Examples for the positive semidefinite cone Example: Lyapunov stability (ctd.) contours of V (x); trajectories. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 12 / 16

13 Examples for the positive semidefinite cone Example: Nonnegative data fitting Etienne de Klerk (Tilburg University) Conic optimization: examples and software 13 / 16

14 References and info Examples for the positive semidefinite cone Lyapunov stability example from: P. Parrilo. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD thesis, Caltech, May Data fitting example from: Siem, A.Y.D., Klerk, E. de, and Hertog, D. den (2008). Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions. Structural and Multidisciplinary Optimization, 35(4), Other SDP applications include free material optimization, sensor network localization, low rank matrix completion,... Etienne de Klerk (Tilburg University) Conic optimization: examples and software 14 / 16

15 Examples for the positive semidefinite cone Free material optimization: wing design of the Airbus A380 Further reading M. Kočvara, M. Stingl and J. Zowe. Free material optimization: recent progress. Optimization, 57(1), , Etienne de Klerk (Tilburg University) Conic optimization: examples and software 15 / 16

16 Examples for the positive semidefinite cone Software Software that implements interior point methods for conic programming: Commercial LP solvers: CPLEX, MOSEK, XPRESS-MP,... SOCP solvers: MOSEK, LOQO, SeDuMi SDP solvers: SDPT3, SeDuMi, CSDP, SDPA... Sizes of problems that can be solved in reasonable time (sparse data in the LP/SOCP case): LP SOCP SDP n m Etienne de Klerk (Tilburg University) Conic optimization: examples and software 16 / 16

### 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

### 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

### Advances in Convex Optimization: Interior-point Methods, Cone Programming, and Applications

Advances in Convex Optimization: Interior-point Methods, Cone Programming, and Applications Stephen Boyd Electrical Engineering Department Stanford University (joint work with Lieven Vandenberghe, UCLA)

### Lecture 7: Finding Lyapunov Functions 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1

### 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

### Semidefinite characterisation of invariant measures for one-dimensional discrete dynamical systems

Semidefinite characterisation of invariant measures for one-dimensional discrete dynamical systems Didier Henrion 1,2,3 April 24, 2012 Abstract Using recent results on measure theory and algebraic geometry,

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### 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

### An Accelerated First-Order Method for Solving SOS Relaxations of Unconstrained Polynomial Optimization Problems

An Accelerated First-Order Method for Solving SOS Relaxations of Unconstrained Polynomial Optimization Problems Dimitris Bertsimas, Robert M. Freund, and Xu Andy Sun December 2011 Abstract Our interest

### Convex Optimization. Lieven Vandenberghe University of California, Los Angeles

Convex Optimization Lieven Vandenberghe University of California, Los Angeles Tutorial lectures, Machine Learning Summer School University of Cambridge, September 3-4, 2009 Sources: Boyd & Vandenberghe,

### 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

### Completely Positive Cone and its Dual

On the Computational Complexity of Membership Problems for the Completely Positive Cone and its Dual Peter J.C. Dickinson Luuk Gijben July 3, 2012 Abstract Copositive programming has become a useful tool

### 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

### Optimisation et simulation numérique.

Optimisation et simulation numérique. Lecture 1 A. d Aspremont. M2 MathSV: Optimisation et simulation numérique. 1/106 Today Convex optimization: introduction Course organization and other gory details...

### Summer 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

### The Price of Robustness

The Price of Robustness Dimitris Bertsimas Melvyn Sim August 001; revised August, 00 Abstract A robust approach to solving linear optimization problems with uncertain data has been proposed in the early

### [1] F. Jarre and J. Stoer, Optimierung, Lehrbuch, Springer Verlag 2003.

References Lehrbuch: [1] F. Jarre and J. Stoer, Optimierung, Lehrbuch, Springer Verlag 2003. Referierte Zeitschriftenbeiträge: [2] F. Jarre, On the Convergence of the Method of Analytic Centers when applied

### 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

### Optimization 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

### Robust solutions of Linear Programming problems contaminated with uncertain data 1

Robust solutions of Linear Programming problems contaminated with uncertain data 1 Aharon Ben-Tal and Arkadi Nemirovski morbt@ie.technion.ac.il nemirovs@ie.technion.ac.il Faculty of Industrial Engineering

### 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

### 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,

### Similar matrices and Jordan form

Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive

### Theory and applications of Robust Optimization

Theory and applications of Robust Optimization Dimitris Bertsimas, David B. Brown, Constantine Caramanis July 6, 2007 Abstract In this paper we survey the primary research, both theoretical and applied,

### 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

### Nonlinear Systems and Control Lecture # 15 Positive Real Transfer Functions & Connection with Lyapunov Stability. p. 1/?

Nonlinear Systems and Control Lecture # 15 Positive Real Transfer Functions & Connection with Lyapunov Stability p. 1/? p. 2/? Definition: A p p proper rational transfer function matrix G(s) is positive

### Model Predictive Control Lecture 5

Model Predictive Control Lecture 5 Klaus Trangbæk ktr@es.aau.dk Automation & Control Aalborg University Denmark. http://www.es.aau.dk/staff/ktr/mpckursus/mpckursus.html mpc5 p. 1 Exercise from last time

### 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

### Real-Time Embedded Convex Optimization

Real-Time Embedded Convex Optimization Stephen Boyd joint work with Michael Grant, Jacob Mattingley, Yang Wang Electrical Engineering Department, Stanford University Spencer Schantz Lecture, Lehigh University,

### Distributionally Robust Optimization with ROME (part 2)

Distributionally Robust Optimization with ROME (part 2) Joel Goh Melvyn Sim Department of Decision Sciences NUS Business School, Singapore 18 Jun 2009 NUS Business School Guest Lecture J. Goh, M. Sim (NUS)

### Additional Exercises for Convex Optimization

Additional Exercises for Convex Optimization Stephen Boyd Lieven Vandenberghe February 11, 2016 This is a collection of additional exercises, meant to supplement those found in the book Convex Optimization,

### 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

### Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management

Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management Shu-Shang Zhu Department of Management Science, School of Management, Fudan University, Shanghai 200433, China, sszhu@fudan.edu.cn

### 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

### 9. Numerical linear algebra background

Convex Optimization Boyd & Vandenberghe 9. Numerical linear algebra background matrix structure and algorithm complexity solving linear equations with factored matrices LU, Cholesky, LDL T factorization

### 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

### 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

### Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach

Outline Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach The University of New South Wales SPOM 2013 Joint work with V. Jeyakumar, B.S. Mordukhovich and

### 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

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### 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

### A Robust Optimization Approach to Supply Chain Management

A Robust Optimization Approach to Supply Chain Management Dimitris Bertsimas and Aurélie Thiele Massachusetts Institute of Technology, Cambridge MA 0139, dbertsim@mit.edu, aurelie@mit.edu Abstract. We

### Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization

Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization Thesis by Pablo A. Parrilo In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

### Las Vegas and Monte Carlo Randomized Algorithms for Systems and Control

Las Vegas and Monte Carlo Randomized Algorithms for Systems and Control Roberto Tempo IEIIT-CNR Politecnico di Torino roberto.tempo@polito.it BasarFest,, Urbana RT 2006 1 CSL UIUC Six months at CSL in

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### Graph Implementations for Nonsmooth Convex Programs

Graph Implementations for Nonsmooth Convex Programs Michael C. Grant I and Stephen P. Boyd 2 1 Stanford University mcgrant9 ord. edu 2 Stanford University boydostanf ord. edu Summary. We describe graph

### 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

### Robust Optimization with Application in Asset Management

Technische Universität München Zentrum Mathematik HVB-Stiftungsinstitut für Finanzmathematik Robust Optimization with Application in Asset Management Katrin Schöttle Vollständiger Abdruck der von der Fakultät

### Robust Adaptive Resource Allocation in Container Terminals

Robust Adaptive Resource Allocation in Container Terminals Marco Laumanns IBM Research - Zurich, 8803 Rueschlikon, Switzerland, mlm@zurich.ibm.com Rico Zenklusen, Kaspar Schüpbach Institute for Operations

### Pre-Calculus Semester 1 Course Syllabus

Pre-Calculus Semester 1 Course Syllabus The Plano ISD eschool Mission is to create a borderless classroom based on a positive student-teacher relationship that fosters independent, innovative critical

### 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

### Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

### 4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

### 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

### 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

### Optimal monitoring in large networks by Successive c-optimal Designs

Optimal monitoring in large networks by Successive c-optimal Designs Guillaume Sagnol, Stéphane Gaubert INRIA Saclay & CMAP, Ecole Polytechnique Email: {guillaume.sagnol,stephane.gaubert}@inria.fr Mustapha

### Math Common Core Sampler Test

High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests

### Miguel Sousa Lobo Curriculum Vitae January 2012

Miguel Sousa Lobo miguel.lobo@insead.edu www.sousalobo.com Education 1995-2000 PhD, Stanford University, Information Systems Laboratory, Electrical Engineering, with minor in Engineering-Economic Systems

### 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.

### Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10

Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction

### 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

### 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

### Recovering Risk-Neutral Probability Density Functions from Options Prices using Cubic Splines

Recovering Risk-Neutral Probability Density Functions from Options Prices using Cubic Splines Ana Margarida Monteiro Reha H. Tütüncü Luís N. Vicente July 2, 24 Abstract We present a new approach to estimate

### Mathematics Courses. (All Math courses not used to fulfill core requirements count as academic electives.)

(All Math courses not used to fulfill core requirements count as academic electives.) Course Number Course Name Grade Level Course Description Prerequisites Who Signs for Course 27.04810 GSE Foundations

### A Log-Robust Optimization Approach to Portfolio Management

A Log-Robust Optimization Approach to Portfolio Management Dr. Aurélie Thiele Lehigh University Joint work with Ban Kawas Research partially supported by the National Science Foundation Grant CMMI-0757983

### Numerical Methods for Pricing Exotic Options

Numerical Methods for Pricing Exotic Options Dimitra Bampou Supervisor: Dr. Daniel Kuhn Second Marker: Professor Berç Rustem 18 June 2008 2 Numerical Methods for Pricing Exotic Options 0BAbstract 3 Abstract

### A FIRST COURSE IN OPTIMIZATION THEORY

A FIRST COURSE IN OPTIMIZATION THEORY RANGARAJAN K. SUNDARAM New York University CAMBRIDGE UNIVERSITY PRESS Contents Preface Acknowledgements page xiii xvii 1 Mathematical Preliminaries 1 1.1 Notation

### Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

### Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours

MAT 051 Pre-Algebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT

### Linear-Quadratic Optimal Controller 10.3 Optimal Linear Control Systems

Linear-Quadratic Optimal Controller 10.3 Optimal Linear Control Systems In Chapters 8 and 9 of this book we have designed dynamic controllers such that the closed-loop systems display the desired transient

### Perron vector Optimization applied to search engines

Perron vector Optimization applied to search engines Olivier Fercoq INRIA Saclay and CMAP Ecole Polytechnique May 18, 2011 Web page ranking The core of search engines Semantic rankings (keywords) Hyperlink

### Linear Algebra Review Part 2: Ax=b

Linear Algebra Review Part 2: Ax=b Edwin Olson University of Michigan The Three-Day Plan Geometry of Linear Algebra Vectors, matrices, basic operations, lines, planes, homogeneous coordinates, transformations

### Applications to Data Smoothing and Image Processing I

Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is

### Recovering Risk-Neutral Probability Density Functions from Options Prices using Cubic Splines and Ensuring Nonnegativity

Recovering Risk-Neutral Probability Density Functions from Options Prices using Cubic Splines and Ensuring Nonnegativity Ana Margarida Monteiro Reha H. Tütüncü Luís N. Vicente Abstract We present a new

### A. Factoring out the Greatest Common Factor.

DETAILED SOLUTIONS AND CONCEPTS - FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

### Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

### Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.

An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points

### Using the Singular Value Decomposition

Using the Singular Value Decomposition Emmett J. Ientilucci Chester F. Carlson Center for Imaging Science Rochester Institute of Technology emmett@cis.rit.edu May 9, 003 Abstract This report introduces

### Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve

### Copositive Cones. Qingxia Kong. Department of Decision Sciences, NUS. Email: qingxia@nus.edu.sg. Chung-Yee Lee

Scheduling Arrivals to a Stochastic Service Delivery System using Copositive Cones Qingxia Kong Department of Decision Sciences, NUS. Email: qingxia@nus.edu.sg Chung-Yee Lee Department of Industrial Engineering

### Bindel, 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

### New and Forthcoming Developments in the AMPL Modeling Language & System

New and Forthcoming Developments in the AMPL Modeling Language & System Robert Fourer*, David M. Gay** AMPL Optimization LLC www.ampl.com 773-336-AMPL * Industrial Eng & Management Sciences, Northwestern

### Robust Approximation to Multi-Period Inventory Management

Robust Approximation to Multi-Period Inventory Management Chuen-Teck See Melvyn Sim April 2007, Revised Jan 2008 Abstract We propose a robust optimization approach to address a multi-period, inventory

### Mathematical Background

Appendix A Mathematical Background A.1 Joint, Marginal and Conditional Probability Let the n (discrete or continuous) random variables y 1,..., y n have a joint joint probability probability p(y 1,...,

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### Robust International Portfolio Management

Computational Optimization Methods in Statistics, Econometrics and Finance - Marie Curie Research and Training Network funded by the EU Commission through MRTN-CT-2006-034270 - COMISEF WORKING PAPERS SERIES

### CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

### 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)

Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible

### LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

### COURSE SYLLABUS Pre-Calculus A/B Last Modified: April 2015

COURSE SYLLABUS Pre-Calculus A/B Last Modified: April 2015 Course Description: In this year-long Pre-Calculus course, students will cover topics over a two semester period (as designated by A and B sections).

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### Modeling, Optimization and Computation for Software Verification

odeling, Optimization and Computation for Software Verification ardavij Roozbehani 1,EricFeron 2, and Alexandre egrestki 3 1 mardavij@mit.edu, 2 feron@mit.edu, 3 ameg@mit.edu Laboratory for Information

### On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems

Dynamics at the Horsetooth Volume 2, 2010. On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems Eric Hanson Department of Mathematics Colorado State University

### Quadratic Functions, Optimization, and Quadratic Forms

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

### 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

### Degree Reduction of Interval SB Curves

International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:13 No:04 1 Degree Reduction of Interval SB Curves O. Ismail, Senior Member, IEEE Abstract Ball basis was introduced

### 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

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are