UNIT 11 CONCAVE and CONVEX Functions FUNCTIONS*
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SOME CONSEQUENCES of the MEAN-VALUE THEOREM Below
SOME CONSEQUENCES OF THE MEAN-VALUE THEOREM PO-LAM YUNG Below we establish some important theoretical consequences of the mean-value theorem. First recall the mean value theorem: Theorem 1 (Mean value theorem). Suppose f :[a; b] ! R is a function defined on a closed interval [a; b] where a; b 2 R. If f is continuous on the closed interval [a; b], and f is differentiable on the open interval (a; b), then there exists c 2 (a; b) such that f(b) − f(a) = f 0(c)(b − a): This has some important corollaries. 1. Monotone functions We introduce the following definitions: Definition 1. Suppose f : I ! R is defined on some interval I. (i) f is said to be constant on I, if and only if f(x) = f(y) for any x; y 2 I. (ii) f is said to be increasing on I, if and only if for any x; y 2 I with x < y, we have f(x) ≤ f(y). (iii) f is said to be strictly increasing on I, if and only if for any x; y 2 I with x < y, we have f(x) < f(y). (iv) f is said to be decreasing on I, if and only if for any x; y 2 I with x < y, we have f(x) ≥ f(y). (v) f is said to be strictly decreasing on I, if and only if for any x; y 2 I with x < y, we have f(x) > f(y). (Can you draw some examples of such functions?) Corollary 2. -
Locally Solid Riesz Spaces with Applications to Economics / Charalambos D
http://dx.doi.org/10.1090/surv/105 alambos D. Alipr Lie University \ Burkinshaw na University-Purdue EDITORIAL COMMITTEE Jerry L. Bona Michael P. Loss Peter S. Landweber, Chair Tudor Stefan Ratiu J. T. Stafford 2000 Mathematics Subject Classification. Primary 46A40, 46B40, 47B60, 47B65, 91B50; Secondary 28A33. Selected excerpts in this Second Edition are reprinted with the permissions of Cambridge University Press, the Canadian Mathematical Bulletin, Elsevier Science/Academic Press, and the Illinois Journal of Mathematics. For additional information and updates on this book, visit www.ams.org/bookpages/surv-105 Library of Congress Cataloging-in-Publication Data Aliprantis, Charalambos D. Locally solid Riesz spaces with applications to economics / Charalambos D. Aliprantis, Owen Burkinshaw.—2nd ed. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 105) Rev. ed. of: Locally solid Riesz spaces. 1978. Includes bibliographical references and index. ISBN 0-8218-3408-8 (alk. paper) 1. Riesz spaces. 2. Economics, Mathematical. I. Burkinshaw, Owen. II. Aliprantis, Char alambos D. III. Locally solid Riesz spaces. IV. Title. V. Mathematical surveys and mono graphs ; no. 105. QA322 .A39 2003 bib'.73—dc22 2003057948 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Advanced Calculus: MATH 410 Functions and Regularity Professor David Levermore 6 August 2020
Advanced Calculus: MATH 410 Functions and Regularity Professor David Levermore 6 August 2020 5. Functions, Continuity, and Limits 5.1. Functions. We now turn our attention to the study of real-valued functions that are defined over arbitrary nonempty subsets of R. The subset of R over which such a function f is defined is called the domain of f, and is denoted Dom(f). We will write f : Dom(f) ! R to indicate that f maps elements of Dom(f) into R. For every x 2 Dom(f) the function f associates the value f(x) 2 R. The range of f is the subset of R defined by (5.1) Rng(f) = f(x): x 2 Dom(f) : Sequences correspond to the special case where Dom(f) = N. When a function f is given by an expression then, unless it is specified otherwise, Dom(f) will be understood to be all x 2 R forp which the expression makes sense. For example, if functions f and g are given by f(x) = 1 − x2 and g(x) = 1=(x2 − 1), and no domains are specified explicitly, then it will be understood that Dom(f) = [−1; 1] ; Dom(g) = x 2 R : x 6= ±1 : These are natural domains for these functions. Of course, if these functions arise in the context of a problem for which x has other natural restrictions then these domains might be smaller. For example, if x represents the population of a species or the amountp of a product being manufactured then we must further restrict x to [0; 1). -
Matrix Convex Functions with Applications to Weighted Centers for Semidefinite Programming
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Research Papers in Economics Matrix Convex Functions With Applications to Weighted Centers for Semidefinite Programming Jan Brinkhuis∗ Zhi-Quan Luo† Shuzhong Zhang‡ August 2005 Econometric Institute Report EI-2005-38 Abstract In this paper, we develop various calculus rules for general smooth matrix-valued functions and for the class of matrix convex (or concave) functions first introduced by L¨ownerand Kraus in 1930s. Then we use these calculus rules and the matrix convex function − log X to study a new notion of weighted centers for semidefinite programming (SDP) and show that, with this definition, some known properties of weighted centers for linear programming can be extended to SDP. We also show how the calculus rules for matrix convex functions can be used in the implementation of barrier methods for optimization problems involving nonlinear matrix functions. ∗Econometric Institute, Erasmus University. †Department of Electrical and Computer Engineering, University of Minnesota. Research supported by National Science Foundation, grant No. DMS-0312416. ‡Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. Research supported by Hong Kong RGC Earmarked Grant CUHK4174/03E. 1 Keywords: matrix monotonicity, matrix convexity, semidefinite programming. AMS subject classification: 15A42, 90C22. 2 1 Introduction For any real-valued function f, one can define a corresponding matrix-valued function f(X) on the space of real symmetric matrices by applying f to the eigenvalues in the spectral decomposition of X. Matrix functions have played an important role in scientific computing and engineering. -
Lecture 1: Basic Terms and Rules in Mathematics
Lecture 7: differentiation – further topics Content: - additional topic to previous lecture - derivatives utilization - Taylor series - derivatives utilization - L'Hospital's rule - functions graph course analysis Derivatives – basic rules repetition from last lecture Derivatives – basic rules repetition from last lecture http://www.analyzemath.com/calculus/Differentiation/first_derivative.html Lecture 7: differentiation – further topics Content: - additional topic to previous lecture - derivatives utilization - Taylor series - derivatives utilization - L'Hospital's rule - functions graph course analysis Derivatives utilization – Taylor series Definition: The Taylor series of a real or complex-valued function ƒ(x) that is infinitely differentiable at a real or complex number a is the power series: In other words: a function ƒ(x) can be approximated in the close vicinity of the point a by means of a power series, where the coefficients are evaluated by means of higher derivatives evaluation (divided by corresponding factorials). Taylor series can be also written in the more compact sigma notation (summation sign) as: Comment: factorial 0! = 1. Derivatives utilization – Taylor series When a = 0, the Taylor series is called a Maclaurin series (Taylor series are often given in this form): a = 0 ... and in the more compact sigma notation (summation sign): a = 0 graphical demonstration – approximation with polynomials (sinx) nice examples: https://en.wikipedia.org/wiki/Taylor_series http://mathdemos.org/mathdemos/TaylorPolynomials/ Derivatives utilization – Taylor (Maclaurin) series Examples (Maclaurin series): Derivatives utilization – Taylor (Maclaurin) series Taylor (Maclaurin) series are utilized in various areas: 1. Approximation of functions – mostly in so called local approx. (in the close vicinity of point a or 0). Precision of this approximation can be estimated by so called remainder or residual term Rn, describing the contribution of ignored higher order terms: 2. -
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] April 20, 2017 2 3 Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations Jean Gallier Abstract: Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, com- puter vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. One of my (selfish!) motivations in writing these notes was to understand the concept of shelling and how it is used to prove the famous Euler-Poincar´eformula (Poincar´e,1899) and the more recent Upper Bound Theorem (McMullen, 1970) for polytopes. Another of my motivations was to give a \correct" account of Delaunay triangulations and Voronoi diagrams in terms of (direct and inverse) stereographic projections onto a sphere and prove rigorously that the projective map that sends the (projective) sphere to the (projective) paraboloid works correctly, that is, maps the Delaunay triangulation and Voronoi diagram w.r.t. the lifting onto the sphere to the Delaunay diagram and Voronoi diagrams w.r.t. the traditional lifting onto the paraboloid. Here, the problem is that this map is only well defined (total) in projective space and we are forced to define the notion of convex polyhedron in projective space. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Nearly Convex Sets: fine Properties and Domains Or Ranges of Subdifferentials of Convex Functions
Nearly convex sets: fine properties and domains or ranges of subdifferentials of convex functions Sarah M. Moffat,∗ Walaa M. Moursi,† and Xianfu Wang‡ Dedicated to R.T. Rockafellar on the occasion of his 80th birthday. July 28, 2015 Abstract Nearly convex sets play important roles in convex analysis, optimization and theory of monotone operators. We give a systematic study of nearly convex sets, and con- struct examples of subdifferentials of lower semicontinuous convex functions whose domain or ranges are nonconvex. 2000 Mathematics Subject Classification: Primary 52A41, 26A51, 47H05; Secondary 47H04, 52A30, 52A25. Keywords: Maximally monotone operator, nearly convex set, nearly equal, relative inte- rior, recession cone, subdifferential with nonconvex domain or range. 1 Introduction arXiv:1507.07145v1 [math.OC] 25 Jul 2015 In 1960s Minty and Rockafellar coined nearly convex sets [22, 25]. Being a generalization of convex sets, the notion of near convexity or almost convexity has been gaining popu- ∗Mathematics, Irving K. Barber School, University of British Columbia Okanagan, Kelowna, British Columbia V1V 1V7, Canada. E-mail: [email protected]. †Mathematics, Irving K. Barber School, University of British Columbia Okanagan, Kelowna, British Columbia V1V 1V7, Canada. E-mail: [email protected]. ‡Mathematics, Irving K. Barber School, University of British Columbia Okanagan, Kelowna, British Columbia V1V 1V7, Canada. E-mail: [email protected]. 1 larity in the optimization community, see [6, 11, 12, 13, 16]. This can be attributed to the applications of generalized convexity in economics problems, see for example, [19, 21]. One reason to study nearly convex sets is that for a proper lower semicontinuous convex function its subdifferential domain is always nearly convex [24, Theorem 23.4, Theorem 6.1], and the same is true for the domain of each maximally monotone operator [26, Theo- rem 12.41]. -
Convex Sets and Convex Functions 1 Convex Sets
Convex Sets and Convex Functions 1 Convex Sets, In this section, we introduce one of the most important ideas in economic modelling, in the theory of optimization and, indeed in much of modern analysis and computatyional mathematics: that of a convex set. Almost every situation we will meet will depend on this geometric idea. As an independent idea, the notion of convexity appeared at the end of the 19th century, particularly in the works of Minkowski who is supposed to have said: \Everything that is convex interests me." We discuss other ideas which stem from the basic definition, and in particular, the notion of a convex and concave functions which which are so prevalent in economic models. The geometric definition, as we will see, makes sense in any vector space. Since, for the most of our work we deal only with Rn, the definitions will be stated in that context. The interested student may, however, reformulate the definitions either, in an ab stract setting or in some concrete vector space as, for example, C([0; 1]; R)1. Intuitively if we think of R2 or R3, a convex set of vectors is a set that contains all the points of any line segment joining two points of the set (see the next figure). Here is the definition. Definition 1.1 Let u; v 2 V . Then the set of all convex combinations of u and v is the set of points fwλ 2 V : wλ = (1 − λ)u + λv; 0 ≤ λ ≤ 1g: (1.1) In, say, R2 or R3, this set is exactly the line segment joining the two points u and v. -
Multivariate Calculus∗
Chapter 3 - Multivariate Calculus∗ Justin Leducy These lecture notes are meant to be used by students entering the University of Mannheim Master program in Economics. They constitute the base for a pre-course in mathematics; that is, they summarize elementary concepts with which all of our econ grad students must be familiar. More advanced concepts will be introduced later on in the regular coursework. A thorough knowledge of these basic notions will be assumed in later coursework. Although the wording is my own, the definitions of concepts and the ways to approach them is strongly inspired by various sources, which are mentioned explicitly in the text or at the end of the chapter. Justin Leduc I have slightly restructured and amended these lecture notes provided by my predecessor Justin Leduc in order for them to suit the 2016 course schedule. Any mistakes you might find are most likely my own. Simona Helmsmueller ∗This version: 2018 yCenter for Doctoral Studies in Economic and Social Sciences. 1 Contents 1 Multivariate functions 4 1.1 Invertibility . .4 1.2 Convexity, Concavity, and Multivariate Real-Valued Functions . .6 2 Multivariate differential calculus 10 2.1 Introduction . 10 2.2 Revision: single-variable differential calculus . 12 2.3 Partial Derivatives and the Gradient . 15 2.4 Differentiability of real-valued functions . 18 2.5 Differentiability of vector-valued functions . 22 2.6 Higher Order Partial Derivatives and the Taylor Approximation Theorems . 23 2.7 Characterization of convex functions . 26 3 Integral theory 28 3.1 Univariate integral calculus . 28 3.2 The definite integral and the fundamental theorem of calculus . -
OPTIMIZATION METHODS: CLASS 3 Linearity, Convexity, ANity
OPTIMIZATION METHODS: CLASS 3 Linearity, convexity, anity The exercises are on the opposite side. D: A set A ⊆ Rd is an ane space, if A is of the form L + v for some linear space L and a shift vector v 2 Rd. By A is of the form L + v we mean a bijection between vectors of L and vectors of A given as b(u) = u + v. Each ane space has a dimension, dened as the dimension of its associated linear space L. D: A vector is an ane combination of a nite set of vectors if Pn , where x a1; a2; : : : an x = i=1 αiai are real number satisfying Pn . αi i=1 αi = 1 A set of vectors V ⊆ Rd is anely independent if it holds that no vector v 2 V is an ane combination of the rest. D: GIven a set of vectors V ⊆ Rd, we can think of its ane span, which is a set of vectors A that are all possible ane combinations of any nite subset of V . Similar to the linear spaces, ane spaces have a nite basis, so we do not need to consider all nite subsets of V , but we can generate the ane span as ane combinations of the base. D: A hyperplane is any ane space in Rd of dimension d − 1. Thus, on a 2D plane, any line is a hyperplane. In the 3D space, any plane is a hyperplane, and so on. A hyperplane splits the space Rd into two halfspaces. We count the hyperplane itself as a part of both halfspaces. -
MA 105 : Calculus Division 1, Lecture 06
MA 105 : Calculus Division 1, Lecture 06 Prof. Sudhir R. Ghorpade IIT Bombay Prof. Sudhir R. Ghorpade, IIT Bombay MA 105 Calculus: Division 1, Lecture 6 Recap of the previous lecture Rolle's theorem and its proof (using Extreme Value Property) (Lagrange's) Mean Value Theorem and its proof Consequences of MVT Monotonicity. Relation with derivatives Convexity and Concavity. Prof. Sudhir R. Ghorpade, IIT Bombay MA 105 Calculus: Division 1, Lecture 6 Convexity and Concavity Let I be an interval. A function f : I R is called convex on I if the line segment joining any two points! on the graph of f lies on or above the graph of f . If x1; x2 I , then the equation 2 of the line passing through (x1; f (x1)) and (x2; f (x2)) is given by f (x2) f (x1) y = f (x1) + − (x x1). Thus f is convex on I if x2 x1 − − f (x2) f (x1) x1; x2; x I ; x1 < x < x2 = f (x) f (x1)+ − (x x1): 2 ) ≤ x2 x1 − − Similarly, a function f : I R is called concave on I if ! f (x2) f (x1) x1; x2; x I ; x1 < x < x2 = f (x) f (x1)+ − (x x1): 2 ) ≥ x2 x1 − − Remark: It is easy to see that f f ((1 λ)x1 + λx2) (1 λ)f (x1)+λf (x2) λ [0; 1] and x1; x2 I : − ≤ − 8 2 2 Likewise for concacity (with changed to ). ≤ ≥ Prof. Sudhir R. Ghorpade, IIT Bombay MA 105 Calculus: Division 1, Lecture 6 A function f : I R is called strictly convex on I if the inequality in the! definition of a convex function can be replaced by≤ the strict inequality <.