3. Convex functions. basic properties and examples. operations that preserve convexity. the conjugate function. quasiconvex functions


 Rodney Hardy
 1 years ago
 Views:
Transcription
1 3. Convex functions Convex Optimization Boyd & Vandenberghe basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions logconcave and logconvex functions convexity with respect to generalized inequalities 3 1
2 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
3 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
4 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
5 Restriction of a convex function to a line f : R n R is convex if and only if the function g : R R, g(t) = f(x+tv), domg = {t x+tv domf} is convex (in t) for any x domf, v R n can check convexity of f by checking convexity of functions of one variable example. f : S n R with f(x) = logdetx, domf = S n ++ g(t) = logdet(x +tv) = logdetx +logdet(i +tx 1/2 VX 1/2 ) n = logdetx + log(1+tλ i ) where λ i are the eigenvalues of X 1/2 VX 1/2 g is concave in t (for any choice of X 0, V); hence f is concave i=1 Convex functions 3 5
6 Extendedvalue extension extendedvalue extension f of f is f(x) = f(x), x domf, f(x) =, x domf often simplifies notation; for example, the condition 0 θ 1 = f(θx+(1 θ)y) θ f(x)+(1 θ) f(y) (as an inequality in R { }), means the same as the two conditions domf is convex for x,y domf, 0 θ 1 = f(θx+(1 θ)y) θf(x)+(1 θ)f(y) Convex functions 3 6
7 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
8 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
9 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
10 logsumexp: f(x) = log n k=1 expx k is convex 2 f(x) = 1 1 T z diag(z) 1 (1 T z) 2zzT (z k = expx k ) to show 2 f(x) 0, we must verify that v T 2 f(x)v 0 for all v: v T 2 f(x)v = ( k z kv 2 k )( k z k) ( k v kz k ) 2 ( k z k) 2 0 since ( k v kz k ) 2 ( k z kv 2 k )( k z k) (from CauchySchwarz inequality) geometric mean: f(x) = ( n k=1 x k) 1/n on R n ++ is concave (similar proof as for logsumexp) Convex functions 3 10
11 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
12 Jensen s inequality basic inequality: if f is convex, then for 0 θ 1, f(θx+(1 θ)y) θf(x)+(1 θ)f(y) extension: if f is convex, then f(ez) Ef(z) for any random variable z basic inequality is special case with discrete distribution prob(z = x) = θ, prob(z = y) = 1 θ Convex functions 3 12
13 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
14 Positive weighted sum & composition with affine function nonnegative multiple: αf is convex if f is convex, α 0 sum: f 1 +f 2 convex if f 1,f 2 convex (extends to infinite sums, integrals) composition with affine function: f(ax+b) is convex if f is convex examples log barrier for linear inequalities f(x) = m log(b i a T i x), i=1 domf = {x a T i x < b i,i = 1,...,m} (any) norm of affine function: f(x) = Ax+b Convex functions 3 14
15 Pointwise maximum if f 1,..., f m are convex, then f(x) = max{f 1 (x),...,f m (x)} is convex examples piecewiselinear function: f(x) = max i=1,...,m (a T i x+b i) is convex sum of r largest components of x R n : f(x) = x [1] +x [2] + +x [r] is convex (x [i] is ith largest component of x) proof: f(x) = max{x i1 +x i2 + +x ir 1 i 1 < i 2 < < i r n} Convex functions 3 15
16 Pointwise supremum if f(x,y) is convex in x for each y A, then is convex examples g(x) = supf(x,y) y A support function of a set C: S C (x) = sup y C y T x is convex distance to farthest point in a set C: f(x) = sup x y y C maximum eigenvalue of symmetric matrix: for X S n, λ max (X) = sup y T Xy y 2 =1 Convex functions 3 16
17 Composition with scalar functions composition of g : R n R and h : R R: f(x) = h(g(x)) f is convex if g convex, h convex, h nondecreasing g concave, h convex, h nonincreasing proof (for n = 1, differentiable g,h) f (x) = h (g(x))g (x) 2 +h (g(x))g (x) note: monotonicity must hold for extendedvalue extension h examples expg(x) is convex if g is convex 1/g(x) is convex if g is concave and positive Convex functions 3 17
18 Vector composition composition of g : R n R k and h : R k R: f(x) = h(g(x)) = h(g 1 (x),g 2 (x),...,g k (x)) f is convex if g i convex, h convex, h nondecreasing in each argument g i concave, h convex, h nonincreasing in each argument proof (for n = 1, differentiable g,h) f (x) = g (x) T 2 h(g(x))g (x)+ h(g(x)) T g (x) examples m i=1 logg i(x) is concave if g i are concave and positive log m i=1 expg i(x) is convex if g i are convex Convex functions 3 18
19 Minimization if f(x,y) is convex in (x,y) and C is a convex set, then is convex examples g(x) = inf y C f(x,y) f(x,y) = x T Ax+2x T By +y T Cy with [ A B B T C ] 0, C 0 minimizing over y gives g(x) = inf y f(x,y) = x T (A BC 1 B T )x g is convex, hence Schur complement A BC 1 B T 0 distance to a set: dist(x,s) = inf y S x y is convex if S is convex Convex functions 3 19
20 Perspective the perspective of a function f : R n R is the function g : R n R R, g(x,t) = tf(x/t), domg = {(x,t) x/t domf, t > 0} g is convex if f is convex examples f(x) = x T x is convex; hence g(x,t) = x T x/t is convex for t > 0 negative logarithm f(x) = logx is convex; hence relative entropy g(x,t) = tlogt tlogx is convex on R 2 ++ if f is convex, then g(x) = (c T x+d)f ( (Ax+b)/(c T x+d) ) is convex on {x c T x+d > 0, (Ax+b)/(c T x+d) domf} Convex functions 3 20
21 the conjugate of a function f is The conjugate function f (y) = sup (y T x f(x)) x dom f f(x) xy x (0, f (y)) f is convex (even if f is not) will be useful in chapter 5 Convex functions 3 21
22 examples negative logarithm f(x) = logx f (y) = sup(xy +logx) = x>0 { 1 log( y) y < 0 otherwise strictly convex quadratic f(x) = (1/2)x T Qx with Q S n ++ f (y) = sup(y T x (1/2)x T Qx) x = 1 2 yt Q 1 y Convex functions 3 22
23 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
24 x is quasiconvex on R Examples ceil(x) = inf{z Z z x} is quasilinear logx is quasilinear on R ++ f(x 1,x 2 ) = x 1 x 2 is quasiconcave on R 2 ++ linearfractional function is quasilinear distance ratio f(x) = at x+b c T x+d, domf = {x ct x+d > 0} f(x) = x a 2 x b 2, domf = {x x a 2 x b 2 } is quasiconvex Convex functions 3 24
25 internal rate of return cash flow x = (x 0,...,x n ); x i is payment in period i (to us if x i > 0) we assume x 0 < 0 and x 0 +x 1 + +x n > 0 present value of cash flow x, for interest rate r: PV(x,r) = n (1+r) i x i i=0 internal rate of return is smallest interest rate for which PV(x,r) = 0: IRR(x) = inf{r 0 PV(x,r) = 0} IRR is quasiconcave: superlevel set is intersection of open halfspaces IRR(x) R n (1+r) i x i > 0 for 0 r < R i=0 Convex functions 3 25
26 Properties modified Jensen inequality: for quasiconvex f 0 θ 1 = f(θx+(1 θ)y) max{f(x),f(y)} firstorder condition: differentiable f with cvx domain is quasiconvex iff f(y) f(x) = f(x) T (y x) 0 x f(x) sums of quasiconvex functions are not necessarily quasiconvex Convex functions 3 26
27 Logconcave and logconvex functions a positive function f is logconcave if logf is concave: f(θx+(1 θ)y) f(x) θ f(y) 1 θ for 0 θ 1 f is logconvex if logf is convex powers: x a on R ++ is logconvex for a 0, logconcave for a 0 many common probability densities are logconcave, e.g., normal: f(x) = 1 (2π)n detσ e 1 2 (x x)t Σ 1 (x x) cumulative Gaussian distribution function Φ is logconcave Φ(x) = 1 2π x e u2 /2 du Convex functions 3 27
28 Properties of logconcave functions twice differentiable f with convex domain is logconcave if and only if f(x) 2 f(x) f(x) f(x) T for all x domf product of logconcave functions is logconcave sum of logconcave functions is not always logconcave integration: if f : R n R m R is logconcave, then g(x) = f(x,y) dy is logconcave (not easy to show) Convex functions 3 28
29 consequences of integration property convolution f g of logconcave functions f, g is logconcave (f g)(x) = f(x y)g(y)dy if C R n convex and y is a random variable with logconcave pdf then is logconcave f(x) = prob(x+y C) proof: write f(x) as integral of product of logconcave functions p is pdf of y f(x) = g(x+y)p(y)dy, g(u) = { 1 u C 0 u C, Convex functions 3 29
30 example: yield function Y(x) = prob(x+w S) x R n : nominal parameter values for product w R n : random variations of parameters in manufactured product S: set of acceptable values if S is convex and w has a logconcave pdf, then Y is logconcave yield regions {x Y(x) α} are convex Convex functions 3 30
31 Convexity with respect to generalized inequalities f : R n R m is Kconvex if domf is convex and for x, y domf, 0 θ 1 f(θx+(1 θ)y) K θf(x)+(1 θ)f(y) example f : S m S m, f(x) = X 2 is S m +convex proof: for fixed z R m, z T X 2 z = Xz 2 2 is convex in X, i.e., z T (θx +(1 θ)y) 2 z θz T X 2 z +(1 θ)z T Y 2 z for X,Y S m, 0 θ 1 therefore (θx +(1 θ)y) 2 θx 2 +(1 θ)y 2 Convex functions 3 31
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...
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 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 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 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 informationAdditional 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,
More informationPattern Analysis. Logistic Regression. 12. Mai 2009. Joachim Hornegger. Chair of Pattern Recognition Erlangen University
Pattern Analysis Logistic Regression 12. Mai 2009 Joachim Hornegger Chair of Pattern Recognition Erlangen University Pattern Analysis 2 / 43 1 Logistic Regression Posteriors and the Logistic Function Decision
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 informationLecture 4: Equality Constrained Optimization. Tianxi Wang
Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x
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 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 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 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 informationSF2940: Probability theory Lecture 8: Multivariate Normal Distribution
SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,
More informationInner 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
More informationRecitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere
Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x
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 informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationSMOOTHING APPROXIMATIONS FOR TWO CLASSES OF CONVEX EIGENVALUE OPTIMIZATION PROBLEMS YU QI. (B.Sc.(Hons.), BUAA)
SMOOTHING APPROXIMATIONS FOR TWO CLASSES OF CONVEX EIGENVALUE OPTIMIZATION PROBLEMS YU QI (B.Sc.(Hons.), BUAA) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL
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 informationLecture 13 Linear quadratic Lyapunov theory
EE363 Winter 289 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discretetime
More informationOctober 3rd, 2012. Linear Algebra & Properties of the Covariance Matrix
Linear Algebra & Properties of the Covariance Matrix October 3rd, 2012 Estimation of r and C Let rn 1, rn, t..., rn T be the historical return rates on the n th asset. rn 1 rṇ 2 r n =. r T n n = 1, 2,...,
More 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 informationMath 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.
More informationChapter 3 Joint Distributions
Chapter 3 Joint Distributions 3.6 Functions of Jointly Distributed Random Variables Discrete Random Variables: Let f(x, y) denote the joint pdf of random variables X and Y with A denoting the twodimensional
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
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 informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationNOV  30211/II. 1. Let f(z) = sin z, z C. Then f(z) : 3. Let the sequence {a n } be given. (A) is bounded in the complex plane
Mathematical Sciences Paper II Time Allowed : 75 Minutes] [Maximum Marks : 100 Note : This Paper contains Fifty (50) multiple choice questions. Each question carries Two () marks. Attempt All questions.
More informationCost Minimization and the Cost Function
Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This is
More informationModern 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
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationEE364a Homework 3 solutions
EE364a, Winter 200708 Prof. S. Boyd EE364a Homework 3 solutions 3.42 Approximation width. Let f 0,...,f n : R R be given continuous functions. We consider the problem of approximating f 0 as a linear
More informationTim Kerins. Leaving Certificate Honours Maths  Algebra. Tim Kerins. the date
Leaving Certificate Honours Maths  Algebra the date Chapter 1 Algebra This is an important portion of the course. As well as generally accounting for 2 3 questions in examination it is the basis for many
More information1 Homogenous and Homothetic Functions
1 Homogenous and Homothetic Functions Reading: [Simon], Chapter 20, p. 483504. 1.1 Homogenous Functions Definition 1 A real valued function f(x 1,..., x n ) is homogenous of degree k if for all t > 0
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationSF2940: Probability theory Lecture 8: Multivariate Normal Distribution
SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2014 Timo Koski () Mathematisk statistik 24.09.2014 1 / 75 Learning outcomes Random vectors, mean vector, covariance
More 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 informationProximal mapping via network optimization
L. Vandenberghe EE236C (Spring 234) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationMATH PROBLEMS, WITH SOLUTIONS
MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More information6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms
AAU  Business Mathematics I Lecture #6, March 16, 2009 6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms 6.1 Rational Inequalities: x + 1 x 3 > 1, x + 1 x 2 3x + 5
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 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 informationAdaptive Online Gradient Descent
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
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
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 informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationsekolahsultanalamshahkoleksisoalansi jilpelajaranmalaysiasekolahsultanala mshahkoleksisoalansijilpelajaranmala ysiasekolahsultanalamshahkoleksisoal
sekolahsultanalamshahkoleksisoalansi jilpelajaranmalaysiasekolahsultanala mshahkoleksisoalansijilpelajaranmala ysiasekolahsultanalamshahkoleksisoal KOLEKSI SOALAN SPM ansijilpelajaranmalaysiasekolahsultan
More informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
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 informationLABEL PROPAGATION ON GRAPHS. SEMISUPERVISED LEARNING. Changsheng Liu 10302014
LABEL PROPAGATION ON GRAPHS. SEMISUPERVISED LEARNING Changsheng Liu 10302014 Agenda Semi Supervised Learning Topics in Semi Supervised Learning Label Propagation Local and global consistency Graph
More informationExercises with solutions (1)
Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal
More informationTwoStage Stochastic Linear Programs
TwoStage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 TwoStage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationError 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
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationIntroduction to Probability
Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence
More informationThe Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Linesearch Method
The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Linesearch Method Robert M. Freund February, 004 004 Massachusetts Institute of Technology. 1 1 The Algorithm The problem
More informationEigenvalues and eigenvectors of a matrix
Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a nonzero column vector V such that AV = λv then λ is called an eigenvalue of A and V is
More information3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field
3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important
More informationMaximum Entropy. Information Theory 2013 Lecture 9 Chapter 12. Tohid Ardeshiri. May 22, 2013
Maximum Entropy Information Theory 2013 Lecture 9 Chapter 12 Tohid Ardeshiri May 22, 2013 Why Maximum Entropy distribution? max f (x) h(f ) subject to E r(x) = α Temperature of a gas corresponds to the
More informationProblem 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
More informationNumerisches 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
More informationConcentration inequalities for order statistics Using the entropy method and Rényi s representation
Concentration inequalities for order statistics Using the entropy method and Rényi s representation Maud Thomas 1 in collaboration with Stéphane Boucheron 1 1 LPMA Université ParisDiderot High Dimensional
More information14. Nonlinear leastsquares
14 Nonlinear leastsquares EE103 (Fall 201112) definition Newton s method GaussNewton method 141 Nonlinear leastsquares minimize r i (x) 2 = r(x) 2 r i is a nonlinear function of the nvector of variables
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
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 informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationRESULTANT AND DISCRIMINANT OF POLYNOMIALS
RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results
More informationApplications 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
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationChapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES  CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More informationCalculus. Contents. Paul Sutcliffe. Office: CM212a.
Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical
More informationLecture Notes on Elasticity of Substitution
Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 20A October 26, 205 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationInverse Functions and Logarithms
Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a onetoone function if it never takes on the same value twice; that
More informationRecall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:
Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1
More informationMINIMIZATION OF ENTROPY FUNCTIONALS UNDER MOMENT CONSTRAINTS. denote the family of probability density functions g on X satisfying
MINIMIZATION OF ENTROPY FUNCTIONALS UNDER MOMENT CONSTRAINTS I. Csiszár (Budapest) Given a σfinite measure space (X, X, µ) and a dtuple ϕ = (ϕ 1,..., ϕ d ) of measurable functions on X, for a = (a 1,...,
More informationAdvanced Microeconomics
Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions
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 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 informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
More informationSeveral 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
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationSolutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory
Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013
More informationTail inequalities for order statistics of logconcave vectors and applications
Tail inequalities for order statistics of logconcave vectors and applications Rafał Latała Based in part on a joint work with R.Adamczak, A.E.Litvak, A.Pajor and N.TomczakJaegermann Banff, May 2011 Basic
More information