CONVEX ANALYSIS and ITS APPLICATION to QUANTUM INFORMATION THEORY by BEN LI Submitted in Partial Fulfillment of the Requirements
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1 Overview 2 Elements of Convex Analysis
AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 2 | Wednesday, January 27th 1 Overview In our previous lecture we discussed several applications of optimization, introduced basic terminol- ogy, and proved Weierstass' theorem (circa 1830) which gives a sufficient condition for existence of an optimal solution to an optimization problem. Today we will discuss basic definitions and prop- erties of convex sets and convex functions. We will then prove the separating hyperplane theorem and see an application of separating hyperplanes in machine learning. We'll conclude by describing the seminal perceptron algorithm (1957) which is designed to find separating hyperplanes. 2 Elements of Convex Analysis We will primarily consider optimization problems over convex sets { sets for which any two points are connected by a line. We illustrate some convex and non-convex sets in Figure 1. Definition. A set S is called a convex set if any two points in S contain their line, i.e. for any x1; x2 2 S we have that λx1 + (1 − λ)x2 2 S for any λ 2 [0; 1]. In the previous lecture we saw linear regression an example of an optimization problem, and men- tioned that the RSS function has a convex shape. We can now define this concept formally. n Definition. For a convex set S ⊆ R , we say that a function f : S ! R is: • convex on S if for any two points x1; x2 2 S and any λ 2 [0; 1] we have that: f (λx1 + (1 − λ)x2) ≤ λf(x1) + 1 − λ f(x2): • strictly convex on S if for any two points x1; x2 2 S and any λ 2 [0; 1] we have that: f (λx1 + (1 − λ)x2) < λf(x1) + (1 − λ) f(x2): We illustrate a convex function in Figure 2. -
CORE View Metadata, Citation and Similar Papers at Core.Ac.Uk
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Bulgarian Digital Mathematics Library at IMI-BAS Serdica Math. J. 27 (2001), 203-218 FIRST ORDER CHARACTERIZATIONS OF PSEUDOCONVEX FUNCTIONS Vsevolod Ivanov Ivanov Communicated by A. L. Dontchev Abstract. First order characterizations of pseudoconvex functions are investigated in terms of generalized directional derivatives. A connection with the invexity is analysed. Well-known first order characterizations of the solution sets of pseudolinear programs are generalized to the case of pseudoconvex programs. The concepts of pseudoconvexity and invexity do not depend on a single definition of the generalized directional derivative. 1. Introduction. Three characterizations of pseudoconvex functions are considered in this paper. The first is new. It is well-known that each pseudo- convex function is invex. Then the following question arises: what is the type of 2000 Mathematics Subject Classification: 26B25, 90C26, 26E15. Key words: Generalized convexity, nonsmooth function, generalized directional derivative, pseudoconvex function, quasiconvex function, invex function, nonsmooth optimization, solution sets, pseudomonotone generalized directional derivative. 204 Vsevolod Ivanov Ivanov the function η from the definition of invexity, when the invex function is pseudo- convex. This question is considered in Section 3, and a first order necessary and sufficient condition for pseudoconvexity of a function is given there. It is shown that the class of strongly pseudoconvex functions, considered by Weir [25], coin- cides with pseudoconvex ones. The main result of Section 3 is applied to characterize the solution set of a nonlinear programming problem in Section 4. The base results of Jeyakumar and Yang in the paper [13] are generalized there to the case, when the function is pseudoconvex. -
Applications of Convex Analysis Within Mathematics
Noname manuscript No. (will be inserted by the editor) Applications of Convex Analysis within Mathematics Francisco J. Arag´onArtacho · Jonathan M. Borwein · Victoria Mart´ın-M´arquez · Liangjin Yao July 19, 2013 Abstract In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis. Keywords Adjoint · Asplund averaging · autoconjugate representer · Banach limit · Chebyshev set · convex functions · Fenchel duality · Fenchel conjugate · Fitzpatrick function · Hahn{Banach extension theorem · infimal convolution · linear relation · Minty surjectivity theorem · maximally monotone operator · monotone operator · Moreau's decomposition · Moreau envelope · Moreau's max formula · Moreau{Rockafellar duality · normal cone operator · renorming · resolvent · Sandwich theorem · subdifferential operator · sum theorem · Yosida approximation Mathematics Subject Classification (2000) Primary 47N10 · 90C25; Secondary 47H05 · 47A06 · 47B65 Francisco J. Arag´onArtacho Centre for Computer Assisted Research Mathematics and its Applications (CARMA), -
Convex) Level Sets Integration
Journal of Optimization Theory and Applications manuscript No. (will be inserted by the editor) (Convex) Level Sets Integration Jean-Pierre Crouzeix · Andrew Eberhard · Daniel Ralph Dedicated to Vladimir Demjanov Received: date / Accepted: date Abstract The paper addresses the problem of recovering a pseudoconvex function from the normal cones to its level sets that we call the convex level sets integration problem. An important application is the revealed preference problem. Our main result can be described as integrating a maximally cycli- cally pseudoconvex multivalued map that sends vectors or \bundles" of a Eu- clidean space to convex sets in that space. That is, we are seeking a pseudo convex (real) function such that the normal cone at each boundary point of each of its lower level sets contains the set value of the multivalued map at the same point. This raises the question of uniqueness of that function up to rescaling. Even after normalising the function long an orienting direction, we give a counterexample to its uniqueness. We are, however, able to show uniqueness under a condition motivated by the classical theory of ordinary differential equations. Keywords Convexity and Pseudoconvexity · Monotonicity and Pseudomono- tonicity · Maximality · Revealed Preferences. Mathematics Subject Classification (2000) 26B25 · 91B42 · 91B16 Jean-Pierre Crouzeix, Corresponding Author, LIMOS, Campus Scientifique des C´ezeaux,Universit´eBlaise Pascal, 63170 Aubi`ere,France, E-mail: [email protected] Andrew Eberhard School of Mathematical & Geospatial Sciences, RMIT University, Melbourne, VIC., Aus- tralia, E-mail: [email protected] Daniel Ralph University of Cambridge, Judge Business School, UK, E-mail: [email protected] 2 Jean-Pierre Crouzeix et al. -
On the Ekeland Variational Principle with Applications and Detours
Lectures on The Ekeland Variational Principle with Applications and Detours By D. G. De Figueiredo Tata Institute of Fundamental Research, Bombay 1989 Author D. G. De Figueiredo Departmento de Mathematica Universidade de Brasilia 70.910 – Brasilia-DF BRAZIL c Tata Institute of Fundamental Research, 1989 ISBN 3-540- 51179-2-Springer-Verlag, Berlin, Heidelberg. New York. Tokyo ISBN 0-387- 51179-2-Springer-Verlag, New York. Heidelberg. Berlin. Tokyo No part of this book may be reproduced in any form by print, microfilm or any other means with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005 Printed by INSDOC Regional Centre, Indian Institute of Science Campus, Bangalore 560012 and published by H. Goetze, Springer-Verlag, Heidelberg, West Germany PRINTED IN INDIA Preface Since its appearance in 1972 the variational principle of Ekeland has found many applications in different fields in Analysis. The best refer- ences for those are by Ekeland himself: his survey article [23] and his book with J.-P. Aubin [2]. Not all material presented here appears in those places. Some are scattered around and there lies my motivation in writing these notes. Since they are intended to students I included a lot of related material. Those are the detours. A chapter on Nemyt- skii mappings may sound strange. However I believe it is useful, since their properties so often used are seldom proved. We always say to the students: go and look in Krasnoselskii or Vainberg! I think some of the proofs presented here are more straightforward. There are two chapters on applications to PDE. -
Optimization Over Nonnegative and Convex Polynomials with and Without Semidefinite Programming
OPTIMIZATION OVER NONNEGATIVE AND CONVEX POLYNOMIALS WITH AND WITHOUT SEMIDEFINITE PROGRAMMING GEORGINA HALL ADISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF OPERATIONS RESEARCH AND FINANCIAL ENGINEERING ADVISER:PROFESSOR AMIR ALI AHMADI JUNE 2018 c Copyright by Georgina Hall, 2018. All rights reserved. Abstract The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, con- trol, machine learning, game theory, and combinatorics, among others. A number of break- through papers in the early 2000s showed that this problem, long thought to be out of reach, could be tackled by using sum of squares programming. This technique however has proved to be expensive for large-scale problems, as it involves solving large semidefinite programs (SDPs). In the first part of this thesis, we present two methods for approximately solving large- scale sum of squares programs that dispense altogether with semidefinite programming and only involve solving a sequence of linear or second order cone programs generated in an adaptive fashion. We then focus on the problem of finding tight lower bounds on polyno- mial optimization problems (POPs), a fundamental task in this area that is most commonly handled through the use of SDP-based sum of squares hierarchies (e.g., due to Lasserre and Parrilo). In contrast to previous approaches, we provide the first theoretical framework for constructing converging hierarchies of lower bounds on POPs whose computation simply requires the ability to multiply certain fixed polynomials together and to check nonnegativ- ity of the coefficients of their product. -
Sum of Squares and Polynomial Convexity
CONFIDENTIAL. Limited circulation. For review only. Sum of Squares and Polynomial Convexity Amir Ali Ahmadi and Pablo A. Parrilo Abstract— The notion of sos-convexity has recently been semidefiniteness of the Hessian matrix is replaced with proposed as a tractable sufficient condition for convexity of the existence of an appropriately defined sum of squares polynomials based on sum of squares decomposition. A multi- decomposition. As we will briefly review in this paper, by variate polynomial p(x) = p(x1; : : : ; xn) is said to be sos-convex if its Hessian H(x) can be factored as H(x) = M T (x) M (x) drawing some appealing connections between real algebra with a possibly nonsquare polynomial matrix M(x). It turns and numerical optimization, the latter problem can be re- out that one can reduce the problem of deciding sos-convexity duced to the feasibility of a semidefinite program. of a polynomial to the feasibility of a semidefinite program, Despite the relative recency of the concept of sos- which can be checked efficiently. Motivated by this computa- convexity, it has already appeared in a number of theo- tional tractability, it has been speculated whether every convex polynomial must necessarily be sos-convex. In this paper, we retical and practical settings. In [6], Helton and Nie use answer this question in the negative by presenting an explicit sos-convexity to give sufficient conditions for semidefinite example of a trivariate homogeneous polynomial of degree eight representability of semialgebraic sets. In [7], Lasserre uses that is convex but not sos-convex. sos-convexity to extend Jensen’s inequality in convex anal- ysis to linear functionals that are not necessarily probability I. -
Topics in Algorithmic, Enumerative and Geometric Combinatorics
Thesis for the Degree of Doctor of Philosophy Topics in algorithmic, enumerative and geometric combinatorics Ragnar Freij Division of Mathematics Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg G¨oteborg, Sweden 2012 Topics in algorithmic, enumerative and geometric combinatorics Ragnar Freij ISBN 978-91-7385-668-3 c Ragnar Freij, 2012. Doktorsavhandlingar vid Chalmers Tekniska H¨ogskola Ny serie Nr 3349 ISSN 0346-718X Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg SE-412 96 GOTEBORG,¨ Sweden Phone: +46 (0)31-772 10 00 [email protected] Printed in G¨oteborg, Sweden, 2012 Topics in algorithmic, enumerative and geometric com- binatorics Ragnar Freij ABSTRACT This thesis presents five papers, studying enumerative and extremal problems on combinatorial structures. The first paper studies Forman’s discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, we determine the S2 × Sn−2-homotopy type of the complex of non-connected graphs on n nodes. In the introduction, connections are drawn between the first paper and the evasiveness conjecture for monotone graph properties. In the second paper, we investigate Hansen polytopes of split graphs. By applying a partitioning technique, the number of nonempty faces is counted, and in particular we confirm Kalai’s 3d-conjecture for such polytopes. Further- more, a characterization of exactly which Hansen polytopes are also Hanner polytopes is given. -
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. -
Optimal Area-Sensitive Bounds for Polytope Approximation
Optimal Area-Sensitive Bounds for Polytope Approximation Sunil Arya∗ Guilherme D. da Fonseca† David M. Mount‡ Department of Computer Dept. de Informática Aplicada Department of Computer Science and Engineering Universidade Federal do Science and Institute for The Hong Kong University of Estado do Rio de Janeiro Advanced Computer Studies Science and Technology (UniRio) University of Maryland Clear Water Bay, Kowloon Rio de Janeiro, Brazil College Park, Maryland 20742 Hong Kong [email protected] [email protected] [email protected] ABSTRACT Mahler volume, it is possible to achieve the desired width- Approximating convex bodies is a fundamental question in based sampling. geometry and has applications to a wide variety of optimiza- tion problems. Given a convex body K in Rd for fixed d, Categories and Subject Descriptors the objective is to minimize the number of vertices or facets F.2.2 [Analysis of Algorithms and Problem Complex- of an approximating polytope for a given Hausdorff error ity]: Nonnumerical Algorithms and Problems—Geometrical ε. The best known uniform bound, due to Dudley (1974), problems and computations shows that O((diam(K)/ε)(d−1)/2) facets suffice. While this bound is optimal in the case of a Euclidean ball, it is far General Terms from optimal for skinny convex bodies. We show that, under the assumption that the width of the Algorithms, Theory body in any direction is at least ε, it is possible to approx- imate a convex body using O( area(K)/ε(d−1)/2) facets, Keywords where area(K) is the surface area of the body. -
Abstract Adaptive Sampling for Geometric
ABSTRACT Title of dissertation: ADAPTIVE SAMPLING FOR GEOMETRIC APPROXIMATION Ahmed Abdelkader Abdelrazek, Doctor of Philosophy, 2020 Dissertation directed by: Professor David M. Mount Computer Science Geometric approximation of multi-dimensional data sets is an essential algorith- mic component for applications in machine learning, computer graphics, and scientific computing. This dissertation promotes an algorithmic sampling methodology for a number of fundamental approximation problems in computational geometry. For each problem, the proposed sampling technique is carefully adapted to the geometry of the input data and the functions to be approximated. In particular, we study proximity queries in spaces of constant dimension and mesh generation in 3D. We start with polytope membership queries, where query points are tested for inclusion in a convex polytope. Trading-off accuracy for efficiency, we tolerate one-sided errors for points within an "-expansion of the polytope. We propose a sampling strategy for the placement of covering ellipsoids sensitive to the local shape of the polytope. The key insight is to realize the samples as Delone sets in the intrinsic Hilbert metric. Using this intrinsic formulation, we considerably simplify state-of-the-art techniques yielding an intuitive and optimal data structure. Next, we study nearest-neighbor queries which retrieve the most similar data point to a given query point. To accommodate more general measures of similarity, we consider non-Euclidean distances including convex distance functions and Bregman divergences. Again, we tolerate multiplicative errors retrieving any point no farther than (1 + ") times the distance to the nearest neighbor. We propose a sampling strategy sensitive to the local distribution of points and the gradient of the distance functions. -
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.