On a Conic Approach to Convex Analysis 1 Introduction
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On Choquet Theorem for Random Upper Semicontinuous Functions
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector International Journal of Approximate Reasoning 46 (2007) 3–16 www.elsevier.com/locate/ijar On Choquet theorem for random upper semicontinuous functions Hung T. Nguyen a, Yangeng Wang b,1, Guo Wei c,* a Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA b Department of Mathematics, Northwest University, Xian, Shaanxi 710069, PR China c Department of Mathematics and Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372, USA Received 16 May 2006; received in revised form 17 October 2006; accepted 1 December 2006 Available online 29 December 2006 Abstract Motivated by the problem of modeling of coarse data in statistics, we investigate in this paper topologies of the space of upper semicontinuous functions with values in the unit interval [0,1] to define rigorously the concept of random fuzzy closed sets. Unlike the case of the extended real line by Salinetti and Wets, we obtain a topological embedding of random fuzzy closed sets into the space of closed sets of a product space, from which Choquet theorem can be investigated rigorously. Pre- cise topological and metric considerations as well as results of probability measures and special prop- erties of capacity functionals for random u.s.c. functions are also given. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Choquet theorem; Random sets; Upper semicontinuous functions 1. Introduction Random sets are mathematical models for coarse data in statistics (see e.g. [12]). A the- ory of random closed sets on Hausdorff, locally compact and second countable (i.e., a * Corresponding author. -
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. -
The Hahn-Banach Theorem and Infima of Convex Functions
The Hahn-Banach theorem and infima of convex functions Hajime Ishihara School of Information Science Japan Advanced Institute of Science and Technology (JAIST) Nomi, Ishikawa 923-1292, Japan first CORE meeting, LMU Munich, 7 May, 2016 Normed spaces Definition A normed space is a linear space E equipped with a norm k · k : E ! R such that I kxk = 0 $ x = 0, I kaxk = jajkxk, I kx + yk ≤ kxk + kyk, for each x; y 2 E and a 2 R. Note that a normed space E is a metric space with the metric d(x; y) = kx − yk: Definition A Banach space is a normed space which is complete with respect to the metric. Examples For 1 ≤ p < 1, let p N P1 p l = f(xn) 2 R j n=0 jxnj < 1g and define a norm by P1 p 1=p k(xn)k = ( n=0 jxnj ) : Then lp is a (separable) Banach space. Examples Classically the normed space 1 N l = f(xn) 2 R j (xn) is boundedg with the norm k(xn)k = sup jxnj n is an inseparable Banach space. However, constructively, it is not a normed space. Linear mappings Definition A mapping T between linear spaces E and F is linear if I T (ax) = aTx, I T (x + y) = Tx + Ty for each x; y 2 E and a 2 R. Definition A linear functional f on a linear space E is a linear mapping from E into R. Bounded linear mappings Definition A linear mapping T between normed spaces E and F is bounded if there exists c ≥ 0 such that kTxk ≤ ckxk for each x 2 E. -
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 Order Preserving and Order Reversing Mappings Defined on Cones of Convex Functions MANUSCRIPT
ON ORDER PRESERVING AND ORDER REVERSING MAPPINGS DEFINED ON CONES OF CONVEX FUNCTIONS MANUSCRIPT Lixin Cheng∗ Sijie Luo School of Mathematical Sciences Yau Mathematical Sciences Center Xiamen University Tsinghua University Xiamen 361005, China Beijing 100084, China [email protected] [email protected] ABSTRACT In this paper, we first show that for a Banach space X there is a fully order reversing mapping T from conv(X) (the cone of all extended real-valued lower semicontinuous proper convex functions defined on X) onto itself if and only if X is reflexive and linearly isomorphic to its dual X∗. Then we further prove the following generalized “Artstein-Avidan-Milman” representation theorem: For every fully order reversing mapping T : conv(X) → conv(X) there exist a linear isomorphism ∗ ∗ ∗ U : X → X , x0, ϕ0 ∈ X , α> 0 and r0 ∈ R so that ∗ (Tf)(x)= α(Ff)(Ux + x0)+ hϕ0, xi + r0, ∀x ∈ X, where F : conv(X) → conv(X∗) is the Fenchel transform. Hence, these resolve two open questions. We also show several representation theorems of fully order preserving mappings de- fined on certain cones of convex functions. For example, for every fully order preserving mapping S : semn(X) → semn(X) there is a linear isomorphism U : X → X so that (Sf)(x)= f(Ux), ∀f ∈ semn(X), x ∈ X, where semn(X) is the cone of all lower semicontinuous seminorms on X. Keywords Fenchel transform · order preserving mapping · order reversing mapping · convex function · Banach space 1 Introduction An elegant theorem of Artstein-Avidan and Milman [5] states that every fully order reversing (resp. -
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. -
EE364 Review Session 2
EE364 Review EE364 Review Session 2 Convex sets and convex functions • Examples from chapter 3 • Installing and using CVX • 1 Operations that preserve convexity Convex sets Convex functions Intersection Nonnegative weighted sum Affine transformation Composition with an affine function Perspective transformation Perspective transformation Pointwise maximum and supremum Minimization Convex sets and convex functions are related via the epigraph. • Composition rules are extremely important. • EE364 Review Session 2 2 Simple composition rules Let h : R R and g : Rn R. Let f(x) = h(g(x)). Then: ! ! f is convex if h is convex and nondecreasing, and g is convex • f is convex if h is convex and nonincreasing, and g is concave • f is concave if h is concave and nondecreasing, and g is concave • f is concave if h is concave and nonincreasing, and g is convex • EE364 Review Session 2 3 Ex. 3.6 Functions and epigraphs. When is the epigraph of a function a halfspace? When is the epigraph of a function a convex cone? When is the epigraph of a function a polyhedron? Solution: If the function is affine, positively homogeneous (f(αx) = αf(x) for α 0), and piecewise-affine, respectively. ≥ Ex. 3.19 Nonnegative weighted sums and integrals. r 1. Show that f(x) = i=1 αix[i] is a convex function of x, where α1 α2 αPr 0, and x[i] denotes the ith largest component of ≥ ≥ · · · ≥ ≥ k n x. (You can use the fact that f(x) = i=1 x[i] is convex on R .) P 2. Let T (x; !) denote the trigonometric polynomial T (x; !) = x + x cos ! + x cos 2! + + x cos(n 1)!: 1 2 3 · · · n − EE364 Review Session 2 4 Show that the function 2π f(x) = log T (x; !) d! − Z0 is convex on x Rn T (x; !) > 0; 0 ! 2π . -
NP-Hardness of Deciding Convexity of Quartic Polynomials and Related Problems
NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems Amir Ali Ahmadi, Alex Olshevsky, Pablo A. Parrilo, and John N. Tsitsiklis ∗y Abstract We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time. 1 Introduction The role of convexity in modern day mathematical programming has proven to be remarkably fundamental, to the point that tractability of an optimization problem is nowadays assessed, more often than not, by whether or not the problem benefits from some sort of underlying convexity. In the famous words of Rockafellar [39]: \In fact the great watershed in optimization isn't between linearity and nonlinearity, but convexity and nonconvexity." But how easy is it to distinguish between convexity and nonconvexity? Can we decide in an efficient manner if a given optimization problem is convex? A class of optimization problems that allow for a rigorous study of this question from a com- putational complexity viewpoint is the class of polynomial optimization problems. These are op- timization problems where the objective is given by a polynomial function and the feasible set is described by polynomial inequalities.