A Set of Lecture Notes on Convex Optimization with Some Applications to Probability Theory Incomplete Draft
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Lecture 7: Convex Analysis and Fenchel-Moreau Theorem
Lecture 7: Convex Analysis and Fenchel-Moreau Theorem The main tools in mathematical finance are from theory of stochastic processes because things are random. However, many objects are convex as well, e.g. collections of probability measures or trading strategies, utility functions, risk measures, etc.. Convex duality methods often lead to new insight, computa- tional techniques and optimality conditions; for instance, pricing formulas for financial instruments and characterizations of different types of no-arbitrage conditions. Convex sets Let X be a real topological vector space, and X∗ denote the topological (or algebraic) dual of X. Throughout, we assume that subsets and functionals are proper, i.e., ;= 6 C 6= X and −∞ < f 6≡ 1. Definition 1. Let C ⊂ X. We call C affine if λx + (1 − λ)y 2 C 8x; y 2 C; λ 2] − 1; 1[; convex if λx + (1 − λ)y 2 C 8x; y 2 C; λ 2 [0; 1]; cone if λx 2 C 8x 2 C; λ 2]0; 1]: Recall that a set A is called algebraic open, if the sets ft : x + tvg are open in R for every x; v 2 X. In particular, open sets are algebraically open. Theorem 1. (Separating Hyperplane Theorem) Let A; B be convex subsets of X, A (algebraic) open. Then the following are equal: 1. A and B are disjoint. 2. There exists an affine set fx 2 X : f(x) = cg, f 2 X∗, c 2 R; such that A ⊂ fx 2 X : f(x) < cg and B ⊂ fx 2 X : f(x) ≥ cg. -
Conjugate Convex Functions in Topological Vector Spaces
Matematisk-fysiske Meddelelser udgivet af Det Kongelige Danske Videnskabernes Selskab Bind 34, nr. 2 Mat. Fys. Medd . Dan. Vid. Selsk. 34, no. 2 (1964) CONJUGATE CONVEX FUNCTIONS IN TOPOLOGICAL VECTOR SPACES BY ARNE BRØNDSTE D København 1964 Kommissionær : Ejnar Munksgaard Synopsis Continuing investigations by W. L . JONES (Thesis, Columbia University , 1960), the theory of conjugate convex functions in finite-dimensional Euclidea n spaces, as developed by W. FENCHEL (Canadian J . Math . 1 (1949) and Lecture No- tes, Princeton University, 1953), is generalized to functions in locally convex to- pological vector spaces . PRINTP_ll IN DENMARK BIANCO LUNOS BOGTRYKKERI A-S Introduction The purpose of the present paper is to generalize the theory of conjugat e convex functions in finite-dimensional Euclidean spaces, as initiated b y Z . BIRNBAUM and W. ORLICz [1] and S . MANDELBROJT [8] and developed by W. FENCHEL [3], [4] (cf. also S. KARLIN [6]), to infinite-dimensional spaces . To a certain extent this has been done previously by W . L . JONES in his Thesis [5] . His principal results concerning the conjugates of real function s in topological vector spaces have been included here with some improve- ments and simplified proofs (Section 3). After the present paper had bee n written, the author ' s attention was called to papers by J . J . MOREAU [9], [10] , [11] in which, by a different approach and independently of JONES, result s equivalent to many of those contained in this paper (Sections 3 and 4) are obtained. Section 1 contains a summary, based on [7], of notions and results fro m the theory of topological vector spaces applied in the following . -
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. -
Techniques of Variational Analysis
J. M. Borwein and Q. J. Zhu Techniques of Variational Analysis An Introduction October 8, 2004 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo To Tova, Naomi, Rachel and Judith. To Charles and Lilly. And in fond and respectful memory of Simon Fitzpatrick (1953-2004). Preface Variational arguments are classical techniques whose use can be traced back to the early development of the calculus of variations and further. Rooted in the physical principle of least action they have wide applications in diverse ¯elds. The discovery of modern variational principles and nonsmooth analysis further expand the range of applications of these techniques. The motivation to write this book came from a desire to share our pleasure in applying such variational techniques and promoting these powerful tools. Potential readers of this book will be researchers and graduate students who might bene¯t from using variational methods. The only broad prerequisite we anticipate is a working knowledge of un- dergraduate analysis and of the basic principles of functional analysis (e.g., those encountered in a typical introductory functional analysis course). We hope to attract researchers from diverse areas { who may fruitfully use varia- tional techniques { by providing them with a relatively systematical account of the principles of variational analysis. We also hope to give further insight to graduate students whose research already concentrates on variational analysis. Keeping these two di®erent reader groups in mind we arrange the material into relatively independent blocks. We discuss various forms of variational princi- ples early in Chapter 2. We then discuss applications of variational techniques in di®erent areas in Chapters 3{7. -
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. -
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(~) . -
A Tutorial on Convex Optimization
A Tutorial on Convex Optimization Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California email: [email protected] Abstract— In recent years, convex optimization has be- quickly deduce the convexity (and hence tractabil- come a computational tool of central importance in engi- ity) of many problems, often by inspection; neering, thanks to it’s ability to solve very large, practical 2) to review and introduce some canonical opti- engineering problems reliably and efficiently. The goal of this tutorial is to give an overview of the basic concepts of mization problems, which can be used to model convex sets, functions and convex optimization problems, so problems and for which reliable optimization code that the reader can more readily recognize and formulate can be readily obtained; engineering problems using modern convex optimization. 3) to emphasize modeling and formulation; we do This tutorial coincides with the publication of the new book not discuss topics like duality or writing custom on convex optimization, by Boyd and Vandenberghe [7], who have made available a large amount of free course codes. material and links to freely available code. These can be We assume that the reader has a working knowledge of downloaded and used immediately by the audience both linear algebra and vector calculus, and some (minimal) for self-study and to solve real problems. exposure to optimization. Our presentation is quite informal. Rather than pro- I. INTRODUCTION vide details for all the facts and claims presented, our Convex optimization can be described as a fusion goal is instead to give the reader a flavor for what is of three disciplines: optimization [22], [20], [1], [3], possible with convex optimization. -
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]). -
Conjugate of Some Rational Functions and Convex Envelope of Quadratic Functions Over a Polytope
Conjugate of some Rational functions and Convex Envelope of Quadratic functions over a Polytope by Deepak Kumar Bachelor of Technology, Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, 2012 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE COLLEGE OF GRADUATE STUDIES (Computer Science) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) January 2019 c Deepak Kumar, 2019 The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis/dissertation en- titled: Conjugate of some Rational functions and Convex Envelope of Quadratic functions over a Polytope submitted by Deepak Kumar in partial fulfilment of the requirements of the degree of Master of Science. Dr. Yves Lucet, I. K. Barber School of Arts & Sciences Supervisor Dr. Heinz Bauschke, I. K. Barber School of Arts & Sciences Supervisory Committee Member Dr. Warren Hare, I. K. Barber School of Arts & Sciences Supervisory Committee Member Dr. Homayoun Najjaran, School of Engineering University Examiner ii Abstract Computing the convex envelope or biconjugate is the core operation that bridges the domain of nonconvex analysis with convex analysis. For a bi- variate PLQ function defined over a polytope, we start with computing the convex envelope of each piece. This convex envelope is characterized by a polyhedral subdivision such that over each member of the subdivision, it has an implicitly defined rational form(square of a linear function over a linear function). Computing this convex envelope involves solving an expo- nential number of subproblems which, in turn, leads to an exponential time algorithm. -
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. -
EECS 127/227AT Fall 2020 Homework 5 Homework 5 Is Due on Gradescope by Friday 10/09 at 11.59 P.M
EECS 127/227AT Fall 2020 Homework 5 Homework 5 is due on Gradescope by Friday 10/09 at 11.59 p.m. 1 A combinatorial problem formulated as a convex optimization problem. In this question we will see how combinatorial optimization problems can also sometimes be solved via related convex optimization problems. We also saw this in a different context in problem 5 on Homework 3 when we related λ2 to φ(G) for a graph. This problem does not have any connection with the earlier problem in Homework 3. The relevant sections of the textbooks are Sec. 8.3 of the textbook of Calafiore and El Ghaoui and Secs. 4.1 and 4.2 (except 4.2.5) of the textbook of Boyd and Vandenberghe. We have to decide which items to sell among a collection of n items. The items have given selling prices si > 0, i = 1; : : : ; n, so selling the item i results in revenue si. Selling item i also implies a transaction cost ci > 0. The total transaction cost is the sum of transaction costs of the items sold, plus a fixed amount, which we normalize to 1. We are interested in the following decision problem: decide which items to sell, with the aim of maximizing the margin, i.e. the ratio of the total revenue to the total transaction cost. Note that this is a combinatorial optimization problem, because the set of choices is a discrete set (here it is the set of all subsets of f1; : : : ; ng). In this question we will see that, nevertheless, it is possible to pose this combinatorial optimization problem as a convex optimization problem. -
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