Conjugate Convex Functions in Topological Vector Spaces
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An Asymptotical Variational Principle Associated with the Steepest Descent Method for a Convex Function
Journal of Convex Analysis Volume 3 (1996), No.1, 63{70 An Asymptotical Variational Principle Associated with the Steepest Descent Method for a Convex Function B. Lemaire Universit´e Montpellier II, Place E. Bataillon, 34095 Montpellier Cedex 5, France. e-mail:[email protected] Received July 5, 1994 Revised manuscript received January 22, 1996 Dedicated to R. T. Rockafellar on his 60th Birthday The asymptotical limit of the trajectory defined by the continuous steepest descent method for a proper closed convex function f on a Hilbert space is characterized in the set of minimizers of f via an asymp- totical variational principle of Brezis-Ekeland type. The implicit discrete analogue (prox method) is also considered. Keywords : Asymptotical, convex minimization, differential inclusion, prox method, steepest descent, variational principle. 1991 Mathematics Subject Classification: 65K10, 49M10, 90C25. 1. Introduction Let X be a real Hilbert space endowed with inner product :; : and associated norm : , and let f be a proper closed convex function on X. h i k k The paper considers the problem of minimizing f, that is, of finding infX f and some element in the optimal set S := Argmin f, this set assumed being non empty. Letting @f denote the subdifferential operator associated with f, we focus on the contin- uous steepest descent method associated with f, i.e., the differential inclusion du @f(u); t > 0 − dt 2 with initial condition u(0) = u0: This method is known to yield convergence under broad conditions summarized in the following theorem. Let us denote by the real vector space of continuous functions from [0; + [ into X that are absolutely conAtinuous on [δ; + [ for all δ > 0. -
On L−Generalized Filters and Nets
Annals of Fuzzy Mathematics and Informatics Volume x, No. x, (Month 201y), pp. 1{xx @FMI ISSN: 2093{9310 (print version) c Kyung Moon Sa Co. ISSN: 2287{6235 (electronic version) http://www.kyungmoon.com http://www.afmi.or.kr On L−generalized filters and nets G. A. Kamel Received 13 March 2017; Revised 20 April 2017; Accepted 5 May 2017 Abstract. In this paper, we consider the L−generalized filters and L−generalized nets on the non empty universal set X to be a natural generalization of the crisp generalized filters and the crisp generalized nets respectively. In each the crisp and L−generalized cases, the extended and restricted filters (nets) will be obtained respectively. The construction of convergence to limit points of the crisp generalized and L−generalized filters (nets) will be studied respectively. We also show that any crisp generalized net on the set X associates a unique L−generalized net on X; and in general the converse is not true. In addition to that we also show that for any given L−generalized filter there exists the L−generalized net, and the vice-versa is true. Finally, the relations between the convergence of the L−generalized filters and the L−generalized nets will be outlined. Moreover, to support these concepts and relations, the construction of some examples in all sections of paper will be studied. 2010 AMS Classification: 54A40 Keywords: Generalized filter, Generalized nets, L−generalized filter, L−generalized nets, Neighbourhoods generalized filter systems. Corresponding Author: G. A. Kamel ([email protected]) 1. Introduction Zadeh in his famous paper [23] introduced the concept of a fuzzy set A of the non-empty universal set X as a mapping A : X −! [0; 1]: Chang in [4] had first tried to fuzzify the concept of a topology on a non-empty set X as a subfamily of [0; 1]X , satisfying the same axioms of ordinary topology. -
ADDENDUM the Following Remarks Were Added in Proof (November 1966). Page 67. an Easy Modification of Exercise 4.8.25 Establishes
ADDENDUM The following remarks were added in proof (November 1966). Page 67. An easy modification of exercise 4.8.25 establishes the follow ing result of Wagner [I]: Every simplicial k-"complex" with at most 2 k ~ vertices has a representation in R + 1 such that all the "simplices" are geometric (rectilinear) simplices. Page 93. J. H. Conway (private communication) has established the validity of the conjecture mentioned in the second footnote. Page 126. For d = 2, the theorem of Derry [2] given in exercise 7.3.4 was found earlier by Bilinski [I]. Page 183. M. A. Perles (private communication) recently obtained an affirmative solution to Klee's problem mentioned at the end of section 10.1. Page 204. Regarding the question whether a(~) = 3 implies b(~) ~ 4, it should be noted that if one starts from a topological cell complex ~ with a(~) = 3 it is possible that ~ is not a complex (in our sense) at all (see exercise 11.1.7). On the other hand, G. Wegner pointed out (in a private communication to the author) that the 2-complex ~ discussed in the proof of theorem 11.1 .7 indeed satisfies b(~) = 4. Page 216. Halin's [1] result (theorem 11.3.3) has recently been genera lized by H. A. lung to all complete d-partite graphs. (Halin's result deals with the graph of the d-octahedron, i.e. the d-partite graph in which each class of nodes contains precisely two nodes.) The existence of the numbers n(k) follows from a recent result of Mader [1] ; Mader's result shows that n(k) ~ k.2(~) . -
Chapter 1: Bases for Topologies Contents
MA3421 2016–17 Chapter 1: Bases for topologies Contents 1.1 Review: metric and topological spaces . .2 1.2 Review: continuity . .7 1.3 Review: limits of sequences . .8 1.4 Base for a topology . 11 1.5 Sub-bases, weak and product topologies . 14 1.6 Neighbourhood bases . 21 1 2 Chapter 1: Bases for topologies 1.1 Review: metric and topological spaces From MA2223 last year, you should know what a metric space is and what the metric topology is. Here is a quick refresher. Definition 1.1.1. Given any set X of points and a function d: X × X ! [0; 1) ⊂ R with these 3 properties: (i) d(z; w) ≥ 0 with equality if and only if z = w; (ii) d(z; w) = d(w; z); (iii) d(z; w) ≤ d(z; v) + d(v; w) (triangle inequality), we say that d is a metric on the space X and we call the combination (X; d) a metric space. Examples 1.1.2. Rn will denote the usual n-dimensional space (over R) and Cn the complex version. We define the (standard) Euclidean distance between pairs of points in Rn by v u n uX 2 d(x; y) = t (xj − yj) j=1 2 3 (abstracting the distance formula from R or R ), for x = (x1; x2; : : : ; xn) and y = (y1; y2; : : : ; yn). For n = 1, recall d(x; y) = jx − yj. n In the case on n-tuples z = (z1; z2; : : : ; zn) and w = (w1; w2; : : : ; wn) 2 C , we also define the standard distance via v u n uX 2 d(z; w) = t jzj − wjj j=1 Recall that for z = x + iy 2 C (with x; y 2 R the real and imaginary parts of z and i2 = −1) the modulus (or absolute value) of such a z is jzj = px2 + y2. -
CLOSED MEANS CONTINUOUS IFF POLYHEDRAL: a CONVERSE of the GKR THEOREM Emil Ernst
CLOSED MEANS CONTINUOUS IFF POLYHEDRAL: A CONVERSE OF THE GKR THEOREM Emil Ernst To cite this version: Emil Ernst. CLOSED MEANS CONTINUOUS IFF POLYHEDRAL: A CONVERSE OF THE GKR THEOREM. 2011. hal-00652630 HAL Id: hal-00652630 https://hal.archives-ouvertes.fr/hal-00652630 Preprint submitted on 16 Dec 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. CLOSED MEANS CONTINUOUS IFF POLYHEDRAL: A CONVERSE OF THE GKR THEOREM EMIL ERNST Abstract. Given x0, a point of a convex subset C of an Euclidean space, the two following statements are proven to be equivalent: (i) any convex function f : C → R is upper semi-continuous at x0, and (ii) C is polyhedral at x0. In the particular setting of closed convex mappings and Fσ domains, we prove that any closed convex function f : C → R is continuous at x0 if and only if C is polyhedral at x0. This provides a converse to the celebrated Gale-Klee- Rockafellar theorem. 1. Introduction One basic fact about real-valued convex mappings on Euclidean spaces, is that they are continuous at any point of their domain’s relative interior (see for instance [13, Theorem 10.1]). -
Machine Learning Theory Lecture 20: Mirror Descent
Machine Learning Theory Lecture 20: Mirror Descent Nicholas Harvey November 21, 2018 In this lecture we will present the Mirror Descent algorithm, which is a common generalization of Gradient Descent and Randomized Weighted Majority. This will require some preliminary results in convex analysis. 1 Conjugate Duality A good reference for the material in this section is [5, Part E]. n ∗ n Definition 1.1. Let f : R ! R [ f1g be a function. Define f : R ! R [ f1g by f ∗(y) = sup yTx − f(x): x2Rn This is the convex conjugate or Legendre-Fenchel transform or Fenchel dual of f. For each linear functional y, the convex conjugate f ∗(y) gives the the greatest amount by which y exceeds the function f. Alternatively, we can think of f ∗(y) as the downward shift needed for the linear function y to just touch or \support" epi f. 1.1 Examples Let us consider some simple one-dimensional examples. ∗ Example 1.2. Let f(x) = cx for some c 2 R. We claim that f = δfcg, i.e., ( 0 (if x = c) f ∗(x) = : +1 (otherwise) This is called the indicator function of fcg. Note that f is itself a linear functional that obviously supports epi f; so f ∗(c) = 0. Any other linear functional x 7! yx − r cannot support epi f for any r (we have supx(yx − cx) = 1 if y 6= c), ∗ ∗ so f (y) = 1 if y 6= c. Note here that a line (f) is getting mapped to a single point (f ). 1 ∗ Example 1.3. Let f(x) = jxj. -
Lecture Slides on Convex Analysis And
LECTURE SLIDES ON CONVEX ANALYSIS AND OPTIMIZATION BASED ON 6.253 CLASS LECTURES AT THE MASS. INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASS SPRING 2014 BY DIMITRI P. BERTSEKAS http://web.mit.edu/dimitrib/www/home.html Based on the books 1) “Convex Optimization Theory,” Athena Scien- tific, 2009 2) “Convex Optimization Algorithms,” Athena Sci- entific, 2014 (in press) Supplementary material (solved exercises, etc) at http://www.athenasc.com/convexduality.html LECTURE 1 AN INTRODUCTION TO THE COURSE LECTURE OUTLINE The Role of Convexity in Optimization • Duality Theory • Algorithms and Duality • Course Organization • HISTORY AND PREHISTORY Prehistory: Early 1900s - 1949. • Caratheodory, Minkowski, Steinitz, Farkas. − Properties of convex sets and functions. − Fenchel - Rockafellar era: 1949 - mid 1980s. • Duality theory. − Minimax/game theory (von Neumann). − (Sub)differentiability, optimality conditions, − sensitivity. Modern era - Paradigm shift: Mid 1980s - present. • Nonsmooth analysis (a theoretical/esoteric − direction). Algorithms (a practical/high impact direc- − tion). A change in the assumptions underlying the − field. OPTIMIZATION PROBLEMS Generic form: • minimize f(x) subject to x C ∈ Cost function f : n , constraint set C, e.g., ℜ 7→ ℜ C = X x h1(x) = 0,...,hm(x) = 0 ∩ | x g1(x) 0,...,gr(x) 0 ∩ | ≤ ≤ Continuous vs discrete problem distinction • Convex programming problems are those for which• f and C are convex They are continuous problems − They are nice, and have beautiful and intu- − itive structure However, convexity permeates -
Topology Proceedings
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 27, No. 2, 2003 Pages 661{673 CONNECTED URYSOHN SUBTOPOLOGIES RICHARD G. WILSON∗ Abstract. We show that each second countable Urysohn space which is not Urysohn-closed can be condensed onto a connected Urysohn space and as a corollary we charac- terize those countable Urysohn spaces which have connected Urysohn subtopologies. We also answer two questions from [12] regarding condensations onto connected Hausdorff spaces. 1. Introduction and Preliminary Results Recall that a space is Urysohn if distinct points have disjoint closed neighbourhoods and a space is feebly compact if every locally finite family of open sets is finite. During the 1960's and 70's many papers appeared in which countable connected Urysohn spaces were constructed (see for example [6],[7],[8] and [9]); such spaces were in some sense considered oddities. In a previous paper [12], we proved that each disconnected Hausdorff space with a countable network can be condensed (that is to say, there is a continuous bijection) onto a connected Hausdorff space if and only if it is not feebly compact. Here we prove an analogous result for Urysohn spaces: 2000 Mathematics Subject Classification. Primary 54D05; Secondary 54D10, 54D25. Key words and phrases. Second countable connected Urysohn space, count- able network, condensation, Urysohn family, Urysohn filter, Urysohn-closed space. ∗Research supported by Consejo Nacional de Ciencia y Tecnolog´ıa(CONA- CYT), Mexico, grant no. -
Discrete Convexity and Log-Concave Distributions in Higher Dimensions
Discrete Convexity and Log-Concave Distributions In Higher Dimensions 1 Basic Definitions and notations n 1. R : The usual vector space of real n− tuples n 2. Convex set : A subset C of R is said to be convex if λx + (1 − λ)y 2 C for any x; y 2 C and 0 < λ < 1: n 3. Convex hull : Let S ⊆ R . The convex hull denoted by conv(S) is the intersection of all convex sets containing S: (or the set of all convex combinations of points of S) n 4. Convex function : A function f from C ⊆ R to (−∞; 1]; where C is a n convex set (for ex: C = R ). Then, f is convex on C iff f((1 − λ)x + λy) ≤ (1 − λ)f(x) + λf(y); 0 < λ < 1: 5. log-concave A function f is said to be log-concave if log f is concave (or − log f is convex) n 6. Epigraph of a function : Let f : S ⊆ R ! R [ {±∞}: The set f(x; µ) j x 2 S; µ 2 R; µ ≥ f(x)g is called the epigraph of f (denoted by epi(f)). 7. The effective domain (denoted by dom(f)) of a convex function f on S n is the projection on R of the epi(f): dom(f) = fx j 9µ, (x; µ) 2 epi(f)g = fx j f(x) < 1g: 8. Lower semi-continuity : An extended real-valued function f given n on a set S ⊂ R is said to be lower semi-continuous at a point x 2 S if f(x) ≤ lim inf f(xi) for every sequence x1; x2; :::; 2 S s.t xi ! x and limit i!1 of f(x1); f(x2); ::: exists in [−∞; 1]: 1 2 Convex Analysis Preliminaries In this section, we recall some basic concepts and results of Convex Analysis. -
Discrete Convexity and Equilibria in Economies with Indivisible Goods and Money
Mathematical Social Sciences 41 (2001) 251±273 www.elsevier.nl/locate/econbase Discrete convexity and equilibria in economies with indivisible goods and money Vladimir Danilova,* , Gleb Koshevoy a , Kazuo Murota b aCentral Institute of Economics and Mathematics, Russian Academy of Sciences, Nahimovski prospect 47, Moscow 117418, Russia bResearch Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan Received 1 January 2000; received in revised form 1 October 2000; accepted 1 October 2000 Abstract We consider a production economy with many indivisible goods and one perfectly divisible good. The aim of the paper is to provide some light on the reasons for which equilibrium exists for such an economy. It turns out, that a main reason for the existence is that supplies and demands of indivisible goods should be sets of a class of discrete convexity. The class of generalized polymatroids provides one of the most interesting classes of discrete convexity. 2001 Published by Elsevier Science B.V. Keywords: Equilibrium; Discrete convex sets; Generalized polymatroids JEL classi®cation: D50 1. Introduction We consider here an economy with production in which there is only one perfectly divisible good (numeraire or money) and all other goods are indivisible. In other words, the commodity space of the model takes the following form ZK 3 R, where K is a set of types of indivisible goods. Several models of exchange economies with indivisible goods and money have been considered in the literature. In 1970, Henry (1970) proved the existence of a competitive equilibrium in an exchange economy with one indivisible good and money. He also gave an example of an economy with two indivisible goods in which no competitive *Corresponding author. -
Generalisations of Filters and Uniform Spaces
Rhodes University Department Of Mathematics Generalisations of Filters and Uniform Spaces Murugiah Muraleetharan A thesis submitted in ful¯lment of the requirements for the Degree of Master of Science In Mathematics Abstract X X The notion of a ¯lter F 2 22 has been extended to that of a : pre¯lter F 2 2I , X X generalised ¯lter f 2 I2 and fuzzy ¯lter ' 2 II . A uniformity is a ¯lter with some X£X other conditions and the notion of a uniformity D 2 22 has been extended to that of X£X X£X a : fuzzy uniformity D 2 2I , generalised uniformity d 2 I2 and super uniformity X£X ± 2 II . We establish categorical embeddings from the category of uniform spaces into the categories of fuzzy uniform spaces, generalised uniform spaces and super uniform spaces and also categorical embeddings into the category of super uniform spaces from the categories of fuzzy uniform spaces and generalised uniform spaces. KEYWORDS: Pre¯ter, Generalised ¯lter, Fuzzy ¯lter, Fuzzy uniform space, Generalised uniform space, Super uniform space, Embedding functor and Functor isomorphim. A. M. S. SUBJECT CLASSIFICATION: 03E72, 04A20, 18A22, 54A40, 54E15. Contents 1 Fuzzy Sets 1 1.1 Introduction . 1 1.2 Crisp Subsets of X Associated With a Fuzzy Set . 3 1.3 Fuzzy Sets Induced by Maps . 6 2 Fuzzy Topology 9 2.1 De¯nitions and Fundamental Properties . 9 2.2 Fuzzy Closure Operator . 11 2.3 Continuous Functions . 12 3 Filters 14 3.1 Introduction . 14 3.2 Ultra¯lters . 15 3.3 Topological Notions in Terms of Filters . -
Conjugate Duality and Optimization CBMS-NSF REGIONAL CONFERENCE SERIES in APPLIED MATHEMATICS
Conjugate Duality and Optimization CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS A series of lectures on topics of current research interest in applied mathematics under the direction of the Conference Board of the Mathematical Sciences, supported by the National Science Foundation and published by SIAM. GARRETT BIRKHOFF, The Numerical Solution of Elliptic Equations D. V. LINDLEY, Bayesian Statistics, A Review R. S. VARGA, Functional Analysis and Approximation Theory in Numerical Analysis R. R. BAHADUR, Some Limit Theorems in Statistics PATRICK BILLINGSLEY, Weak Convergence of Measures: Applications in Probability J. L. LIONS, Some Aspects of the Optimal Control of Distributed Parameter Systems ROGER PENROSE, Techniques of Differential Topology in Relativity HERMAN CHERNOFF, Sequential Analysis and Optimal Design J. DURBIN, Distribution Theory for Tests Based on the Sample Distribution Function SOL I. RUBINOW, Mathematical Problems in the Biological Sciences P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves I. J. SCHOENBERG, Cardinal Spline Interpolation IVAN SINGER, The Theory of Best Approximation and Functional Analysis WERNER C. RHEINBOLDT, Methods of Solving Systems of Nonlinear Equations HANS F. WEINBERGER, Variational Methods for Eigenvalue Approximation R. TYRRELL ROCKAFELLAR, Conjugate Duality and Optimization SIR JAMES LIGHTHILL, Mathematical Biofluiddynamics GERARD SALTON, Theory of Indexing CATHLEEN S. MORAWETZ, Notes on Time Decay and Scattering for Some Hyperbolic Problems F. HOPPENSTEADT, Mathematical Theories of Populations: Demographics, Genetics and Epidemics RICHARD ASKEY, Orthogonal Polynomials and Special Functions L. E. PAYNE, Improperly Posed Problems in Partial Differential Equations S. ROSEN, Lectures on the Measurement and Evaluation of the Performance of Computing Systems HERBERT B.