Tutorial on Convex Optimization for Engineers Part I


 Colleen Stanley
 2 years ago
 Views:
Transcription
1 Tutorial on Convex Optimization for Engineers Part I M.Sc. Jens Steinwandt Communications Research Laboratory Ilmenau University of Technology PO Box D Ilmenau, Germany January 2014
2 Course mechanics strongly based on the advanced course Convex Optimization I by Prof. Stephen Boyd at Stanford University, CA info, slides, video lectures, textbook, software on web page: mandatory assignment (30 % of the final grade) Prerequisites working knowledge of linear algebra
3 Course objectives to provide you with overview of convex optimization working knowledge on convex optimization in particular, to provide you with skills about how to recognize convex problems model nonconvex problems as convex get a feel for easy and difficult problems efficiently solve the problems using optimization tools
4 Outline 1. Introduction 2. Convex sets 3. Convex functions 4. Convex optimization problems 5. Lagrangian duality theory 6. Disciplined convex programming and CVX
5 1. Introduction
6 Mathematical optimization (mathematical) optimization problem minimize f 0 (x) subject to f i (x) b i, i = 1,...,m x = (x 1,...,x n ): optimization variables f 0 : R n R: objective function f i : R n R, i = 1,...,m: constraint functions optimal solution x has smallest value of f 0 among all vectors that satisfy the constraints Introduction 1 2
7 Solving optimization problems general optimization problem very difficult to solve methods involve some compromise, e.g., very long computation time, or not always finding the solution exceptions: certain problem classes can be solved efficiently and reliably leastsquares problems linear programming problems convex optimization problems Introduction 1 4
8 Leastsquares minimize Ax b 2 2 solving leastsquares problems analytical solution: x = (A T A) 1 A T b reliable and efficient algorithms and software computation time proportional to n 2 k (A R k n ); less if structured a mature technology using leastsquares leastsquares problems are easy to recognize a few standard techniques increase flexibility (e.g., including weights, adding regularization terms) Introduction 1 5
9 Linear programming solving linear programs no analytical formula for solution minimize c T x subject to a T i x b i, i = 1,...,m reliable and efficient algorithms and software computation time proportional to n 2 m if m n; less with structure a mature technology using linear programming not as easy to recognize as leastsquares problems a few standard tricks used to convert problems into linear programs (e.g., problems involving l 1  or l norms, piecewiselinear functions) Introduction 1 6
10 Convex optimization problem minimize f 0 (x) subject to f i (x) b i, i = 1,...,m objective and constraint functions are convex: f i (αx+βy) αf i (x)+βf i (y) if α+β = 1, α 0, β 0 includes leastsquares problems and linear programs as special cases Introduction 1 7
11 Why is convex optimization so essential? convex optimization formulation always achieves global minimum, no local traps certificate for infeasibility can be solved by polynomial time complexity algorithms (e.g., interior point methods) highly efficient software available (e.g., SeDuMi, CVX) the dividing line between easy and difficult problems (compare with solving linear equations) = whenever possible, always go for convex formulation Introduction
12 Brief history of convex optimization theory (convex analysis): ca algorithms 1947: simplex algorithm for linear programming (Dantzig) 1960s: early interiorpoint methods (Fiacco & McCormick, Dikin,... ) 1970s: ellipsoid method and other subgradient methods 1980s: polynomialtime interiorpoint methods for linear programming (Karmarkar 1984) late 1980s now: polynomialtime interiorpoint methods for nonlinear convex optimization (Nesterov & Nemirovski 1994) applications before 1990: mostly in operations research; few in engineering since 1990: many new applications in engineering (control, signal processing, communications, circuit design,... ); new problem classes (semidefinite and secondorder cone programming, robust optimization) Introduction 1 15
13 2. Convex sets
14 Subspaces a space S R n is a subspace if x 1, x 2 S, θ 1, θ 2 R = θ 1x 1 + θ 2x 2 S geometrically: plane through x 1, x 2 S and the origin representation: range(a) = {Aw w R q } (A = [a 1,... a q]) = {w 1a 1 + w qa q w i R} = span {a 1, a 2,..., a q} null(a) = {x Bx = 0} (B = [b 1,... b p] T ) = { x b T 1 x = 0,..., b T p x = 0 } Convex sets
15 Affine set line through x 1, x 2 : all points x = θx 1 +(1 θ)x 2 (θ R) θ = 1.2 x 1 θ = 1 θ = 0.6 x 2 θ = 0 θ = 0.2 affine set: contains the line through any two distinct points in the set example: solution set of linear equations {x Ax = b} (conversely, every affine set can be expressed as solution set of system of linear equations) Convex sets 2 2
16 Convex set line segment between x 1 and x 2 : all points with 0 θ 1 x = θx 1 +(1 θ)x 2 convex set: contains line segment between any two points in the set x 1,x 2 C, 0 θ 1 = θx 1 +(1 θ)x 2 C examples (one convex, two nonconvex sets) Convex sets 2 3
17 Convex combination and convex hull convex combination of x 1,..., x k : any point x of the form with θ 1 + +θ k = 1, θ i 0 x = θ 1 x 1 +θ 2 x 2 + +θ k x k convex hull convs: set of all convex combinations of points in S Convex sets 2 4
18 Convex cone conic (nonnegative) combination of x 1 and x 2 : any point of the form with θ 1 0, θ 2 0 x = θ 1 x 1 +θ 2 x 2 x 1 0 x 2 convex cone: set that contains all conic combinations of points in the set Convex sets 2 5
19 Hyperplanes and halfspaces hyperplane: set of the form {x a T x = b} (a 0) a x 0 x a T x = b halfspace: set of the form {x a T x b} (a 0) a x 0 a T x b a T x b a is the normal vector hyperplanes are affine and convex; halfspaces are convex Convex sets 2 6
20 Euclidean balls and ellipsoids (Euclidean) ball with center x c and radius r: B(x c,r) = {x x x c 2 r} = {x c +ru u 2 1} ellipsoid: set of the form {x (x x c ) T P 1 (x x c ) 1} with P S n ++ (i.e., P symmetric positive definite) x c other representation: {x c +Au u 2 1} with A square and nonsingular Convex sets 2 7
21 Norm balls and norm cones norm: a function that satisfies x 0; x = 0 if and only if x = 0 tx = t x for t R x+y x + y notation: is general (unspecified) norm; symb is particular norm norm ball with center x c and radius r: {x x x c r} 1 norm cone: {(x,t) x t} Euclidean norm cone is called secondorder cone norm balls and cones are convex t x x Convex sets 2 8
22 Polyhedra solution set of finitely many linear inequalities and equalities Ax b, Cx = d (A R m n, C R p n, is componentwise inequality) a 1 a2 a 5 P a 3 a 4 polyhedron is intersection of finite number of halfspaces and hyperplanes Convex sets 2 9
23 Positive semidefinite cone notation: S n is set of symmetric n n matrices S n + = {X S n X 0}: positive semidefinite n n matrices X S n + z T Xz 0 for all z S n + is a convex cone S n ++ = {X S n X 0}: positive definite n n matrices 1 example: [ x y y z ] S 2 + z y 1 0 x Convex sets 2 10
24 Operations that preserve convexity practical methods for establishing convexity of a set C 1. apply definition x 1,x 2 C, 0 θ 1 = θx 1 +(1 θ)x 2 C 2. show that C is obtained from simple convex sets (hyperplanes, halfspaces, norm balls,... ) by operations that preserve convexity intersection affine functions perspective function linearfractional functions Convex sets 2 11
25 Examples of convex sets linear subspace: S = {x Ax = 0} is a convex cone affine subspace: S = {x Ax = b} is a convex set polyhedral set: S = {x Ax b} is a convex set PSD matrix cone: S = {A A is symmetric, A 0} is convex second order cone: S = {(t, x) t x } is convex intersection intersection of linear subspaces affine subspaces convex cones convex sets is also a linear subspace affine subspace convex cone convex set example: a polyhedron is intersection of a finite number of halfspaces Convex sets
26 3. Convex functions
27 Definition f : R n R is convex if domf is a convex set and f(θx+(1 θ)y) θf(x)+(1 θ)f(y) for all x,y domf, 0 θ 1 (x,f(x)) (y,f(y)) f is concave if f is convex f is strictly convex if domf is convex and f(θx+(1 θ)y) < θf(x)+(1 θ)f(y) for x,y domf, x y, 0 < θ < 1 Convex functions 3 2
28 Examples on R convex: affine: ax+b on R, for any a,b R exponential: e ax, for any a R powers: x α on R ++, for α 1 or α 0 powers of absolute value: x p on R, for p 1 negative entropy: xlogx on R ++ concave: affine: ax+b on R, for any a,b R powers: x α on R ++, for 0 α 1 logarithm: logx on R ++ Convex functions 3 3
29 Examples on R n and R m n affine functions are convex and concave; all norms are convex examples on R n affine function f(x) = a T x+b norms: x p = ( n i=1 x i p ) 1/p for p 1; x = max k x k examples on R m n (m n matrices) affine function f(x) = tr(a T X)+b = m i=1 n A ij X ij +b j=1 spectral (maximum singular value) norm f(x) = X 2 = σ max (X) = (λ max (X T X)) 1/2 Convex functions 3 4
30 Firstorder condition f is differentiable if domf is open and the gradient f(x) = exists at each x domf ( f(x), f(x),..., f(x) ) x 1 x 2 x n 1storder condition: differentiable f with convex domain is convex iff f(y) f(x)+ f(x) T (y x) for all x,y domf f(y) f(x) + f(x) T (y x) (x,f(x)) firstorder approximation of f is global underestimator Convex functions 3 7
31 Secondorder conditions f is twice differentiable if domf is open and the Hessian 2 f(x) S n, exists at each x domf 2 f(x) ij = 2 f(x) x i x j, i,j = 1,...,n, 2ndorder conditions: for twice differentiable f with convex domain f is convex if and only if 2 f(x) 0 for all x domf if 2 f(x) 0 for all x domf, then f is strictly convex Convex functions 3 8
32 Examples quadratic function: f(x) = (1/2)x T Px+q T x+r (with P S n ) f(x) = Px+q, 2 f(x) = P convex if P 0 leastsquares objective: f(x) = Ax b 2 2 f(x) = 2A T (Ax b), 2 f(x) = 2A T A convex (for any A) quadraticoverlinear: f(x,y) = x 2 /y 2 2 f(x,y) = 2 y 3 [ y x ][ y x ] T 0 f(x,y) convex for y > 0 1 y 0 2 x 0 Convex functions 3 9
33 Epigraph and sublevel set αsublevel set of f : R n R: C α = {x domf f(x) α} sublevel sets of convex functions are convex (converse is false) epigraph of f : R n R: epif = {(x,t) R n+1 x domf, f(x) t} epif f f is convex if and only if epif is a convex set Convex functions 3 11
34 Properties of convex functions convexity over all lines: f (x) is convex f (x 0 + th) is convex in t for all x 0, h positive multiple: f (x) is convex = αf (x) is convex for all α 0 sum of convex functions: f 1(x), f 2(x) convex = f 1(x) + f 2(x) is convex pointwise maximum: f 1(x), f 2(x) convex = max{f 1(x), f 2(x)} is convex affine transformation of domain: f (x) is convex = f (Ax + b) is convex Convex functions
35 Operations that preserve convexity practical methods for establishing convexity of a function 1. verify definition (often simplified by restricting to a line) 2. for twice differentiable functions, show 2 f(x) 0 3. show that f is obtained from simple convex functions by operations that preserve convexity nonnegative weighted sum composition with affine function pointwise maximum and supremum composition minimization perspective Convex functions 3 13
36 Quasiconvex functions f : R n R is quasiconvex if domf is convex and the sublevel sets are convex for all α S α = {x domf f(x) α} β α a b c f is quasiconcave if f is quasiconvex f is quasilinear if it is quasiconvex and quasiconcave Convex functions 3 23
3. Convex functions. basic properties and examples. operations that preserve convexity. the conjugate function. quasiconvex functions
3. Convex functions Convex Optimization Boyd & Vandenberghe basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions logconcave and logconvex functions
More informationOptimisation 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...
More informationSummer course on Convex Optimization. Fifth Lecture InteriorPoint Methods (1) Michel Baes, K.U.Leuven Bharath Rangarajan, U.
Summer course on Convex Optimization Fifth Lecture InteriorPoint Methods (1) Michel Baes, K.U.Leuven Bharath Rangarajan, U.Minnesota InteriorPoint Methods: the rebirth of an old idea Suppose that f is
More informationConvex 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 34, 2009 Sources: Boyd & Vandenberghe,
More informationMOSEK 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
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 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 3.13.6 of WTB. 5.1 Linear
More informationIntroduction 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
More informationDuality 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
More informationWhat is Linear Programming?
Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to
More information3. Proximal gradient method
Algorithms for largescale convex optimization DTU 2010 3. Proximal gradient method introduction proximal mapping proximal gradient method convergence analysis accelerated proximal gradient method forwardbackward
More information3. 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
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 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 informationLinear Algebra Review Part 2: Ax=b
Linear Algebra Review Part 2: Ax=b Edwin Olson University of Michigan The ThreeDay Plan Geometry of Linear Algebra Vectors, matrices, basic operations, lines, planes, homogeneous coordinates, transformations
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
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 informationMathematical 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,...,
More informationLecture 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 informationAn Introduction on SemiDefinite Program
An Introduction on SemiDefinite Program from the viewpoint of computation Hayato Waki Institute of Mathematics for Industry, Kyushu University 20151008 Combinatorial Optimization at Work, Berlin, 2015
More informationThinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks
Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationA NEW LOOK AT CONVEX ANALYSIS AND OPTIMIZATION
1 A NEW LOOK AT CONVEX ANALYSIS AND OPTIMIZATION Dimitri Bertsekas M.I.T. FEBRUARY 2003 2 OUTLINE Convexity issues in optimization Historical remarks Our treatment of the subject Three unifying lines of
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 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 information(Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties
Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More information1 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.
More informationQuadratic 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
More informationIntroduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test
Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are
More informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More informationCan linear programs solve NPhard problems?
Can linear programs solve NPhard problems? p. 1/9 Can linear programs solve NPhard problems? Ronald de Wolf Linear programs Can linear programs solve NPhard problems? p. 2/9 Can linear programs solve
More informationAffine Transformations. University of Texas at Austin CS384G  Computer Graphics Fall 2010 Don Fussell
Affine Transformations University of Texas at Austin CS384G  Computer Graphics Fall 2010 Don Fussell Logistics Required reading: Watt, Section 1.1. Further reading: Foley, et al, Chapter 5.15.5. David
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 informationLinear 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
More informationLecture 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
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationWASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,
More informationActually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York.
1: 1. Compute a random 4dimensional polytope P as the convex hull of 10 random points using rand sphere(4,10). Run VISUAL to see a Schlegel diagram. How many 3dimensional polytopes do you see? How many
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationINTERIOR POINT POLYNOMIAL TIME METHODS IN CONVEX PROGRAMMING
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF INDUSTRIAL AND SYSTEMS ENGINEERING LECTURE NOTES INTERIOR POINT POLYNOMIAL TIME METHODS IN CONVEX PROGRAMMING ISYE 8813 Arkadi Nemirovski On sabbatical leave from
More informationAdvances in Convex Optimization: Interiorpoint Methods, Cone Programming, and Applications
Advances in Convex Optimization: Interiorpoint Methods, Cone Programming, and Applications Stephen Boyd Electrical Engineering Department Stanford University (joint work with Lieven Vandenberghe, UCLA)
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 3 Linear Least Squares Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign Copyright c 2002. Reproduction
More informationMATHEMATICS (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)
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationLarson, R. and Boswell, L. (2016). Big Ideas Math, Algebra 2. Erie, PA: Big Ideas Learning, LLC. ISBN
ALG B Algebra II, Second Semester #PR0, BK04 (v.4.0) To the Student: After your registration is complete and your proctor has been approved, you may take the Credit by Examination for ALG B. WHAT TO
More informationLAGRANGIAN RELAXATION TECHNIQUES FOR LARGE SCALE OPTIMIZATION
LAGRANGIAN RELAXATION TECHNIQUES FOR LARGE SCALE OPTIMIZATION Kartik Sivaramakrishnan Department of Mathematics NC State University kksivara@ncsu.edu http://www4.ncsu.edu/ kksivara SIAM/MGSA Brown Bag
More informationSequence of Mathematics Courses
Sequence of ematics Courses Where do I begin? Associates Degree and Nontransferable Courses (For math course below prealgebra, see the Learning Skills section of the catalog) MATH M09 PREALGEBRA 3 UNITS
More informationDefinition of a Linear Program
Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1
More 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 informationMathematical 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
More informationConvex Optimization SVM s and Kernel Machines
Convex Optimization SVM s and Kernel Machines S.V.N. Vishy Vishwanathan vishy@axiom.anu.edu.au National ICT of Australia and Australian National University Thanks to Alex Smola and Stéphane Canu S.V.N.
More informationNotes on Symmetric Matrices
CPSC 536N: Randomized Algorithms 201112 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.
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 information. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2
4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the
More 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 informationNPhardness of Deciding Convexity of Quartic Polynomials and Related Problems
NPhardness 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 informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More information4. 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
More informationMATH 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
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More informationMathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 PreAlgebra 4 Hours
MAT 051 PreAlgebra 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
More informationMyMathLab ecourse for Developmental Mathematics
MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and
More informationDevelopmental Math Course Outcomes and Objectives
Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/PreAlgebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and
More informationMATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSAMAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationSolutions Of Some NonLinear Programming Problems BIJAN KUMAR PATEL. Master of Science in Mathematics. Prof. ANIL KUMAR
Solutions Of Some NonLinear Programming Problems A PROJECT REPORT submitted by BIJAN KUMAR PATEL for the partial fulfilment for the award of the degree of Master of Science in Mathematics under the supervision
More informationNPhardness of Deciding Convexity of Quartic Polynomials and Related Problems
NPhardness 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 informationNonlinear Optimization: Algorithms 3: Interiorpoint methods
Nonlinear Optimization: Algorithms 3: Interiorpoint methods INSEAD, Spring 2006 JeanPhilippe Vert Ecole des Mines de Paris JeanPhilippe.Vert@mines.org Nonlinear optimization c 2006 JeanPhilippe Vert,
More informationOverview of Math Standards
Algebra 2 Welcome to math curriculum design maps for Manhattan Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse
More information10. Proximal point method
L. Vandenberghe EE236C Spring 201314) 10. Proximal point method proximal point method augmented Lagrangian method MoreauYosida smoothing 101 Proximal point method a conceptual algorithm for minimizing
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationCompletely 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
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationDuality in General Programs. Ryan Tibshirani Convex Optimization 10725/36725
Duality in General Programs Ryan Tibshirani Convex Optimization 10725/36725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T
More informationAlgebra II. Weeks 13 TEKS
Algebra II Pacing Guide Weeks 13: Equations and Inequalities: Solve Linear Equations, Solve Linear Inequalities, Solve Absolute Value Equations and Inequalities. Weeks 46: Linear Equations and Functions:
More informationPrentice 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
More informationA 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
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
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 informationPrerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.
Course Syllabus Math 1314 College Algebra Revision Date: 82115 Catalog Description: Indepth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems
More informationPCHS ALGEBRA PLACEMENT TEST
MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More informationWe shall turn our attention to solving linear systems of equations. Ax = b
59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationMath 1050 Khan Academy Extra Credit Algebra Assignment
Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina  Beaufort Lisa S. Yocco, Georgia Southern University
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationInterior Point Methods and Linear Programming
Interior Point Methods and Linear Programming Robert Robere University of Toronto December 13, 2012 Abstract The linear programming problem is usually solved through the use of one of two algorithms: either
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,
More information