Log-Log Convex Programming
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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. -
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. -
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Arxiv:2103.15804V4 [Math.AT] 25 May 2021
Decorated Merge Trees for Persistent Topology Justin Curry, Haibin Hang, Washington Mio, Tom Needham, and Osman Berat Okutan Abstract. This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces. DMTs combine both π0 and Hn information into a single data structure that distinguishes filtrations that merge trees and persistent homology cannot distinguish alone. Three variants on DMTs, which empha- size category theory, representation theory and persistence barcodes, respectively, offer different advantages in terms of theory and computation. Two notions of distance|an interleaving distance and bottleneck distance|for DMTs are defined and a hierarchy of stability results that both refine and generalize existing stability results is proved here. To overcome some of the computational complexity inherent in these dis- tances, we provide a novel use of Gromov-Wasserstein couplings to compute optimal merge tree alignments for a combinatorial version of our interleaving distance which can be tractably estimated. We introduce computational frameworks for generating, visualizing and comparing decorated merge trees derived from synthetic and real data. Example applications include comparison of point clouds, interpretation of persis- tent homology of sliding window embeddings of time series, visualization of topological features in segmented brain tumor images and topology-driven graph alignment. Contents 1. Introduction 1 2. Decorated Merge Trees Three Different Ways3 3. Continuity and Stability of Decorated Merge Trees 10 4. Representations of Tree Posets and Lift Decorations 16 5. Computing Interleaving Distances 20 6. Algorithmic Details and Examples 24 7. Discussion 29 References 30 Appendix A. Interval Topology 34 Appendix B. Comparison and Existence of Merge Trees 35 Appendix C. -
Linear Algebra Review
CSE 203B: Convex Optimization Week 4 Discuss Session 1 Contents • Convex functions (Ref. Chap.3) • Review: definition, first order condition, second order condition, operations that preserve convexity • Epigraph • Conjugate function • Dual norm 2 Review of Convex Function • Definition ( often simplified by restricting to a line) • First order condition: a global underestimator • Second order condition • Operations that preserve convexity • Nonnegative weighted sum • Composition with affine function • Pointwise maximum and supremum • Composition • Minimization See Chap 3.2, try to prove why the convexity • Perspective is preserved with those operations. 3 Epigraph • 훼-sublevel set of 푓: 푅푛 → 푅 퐶훼 = 푥 ∈ 푑표푚 푓 푓 푥 ≤ 훼} sublevel sets of a convex function are convex for any value of 훼. • Epigraph of 푓: 푅푛 → 푅 is defined as 퐞퐩퐢 푓 = (푥, 푡) 푥 ∈ 푑표푚 푓, 푓 푥 ≤ 푡} ⊆ 푅푛+1 • A function is convex iff its epigraph is a convex set. 4 Relation between convex sets and convex functions • A function is convex iff its epigraph is a convex set. • Consider a convex function 푓 and 푥, 푦 ∈ 푑표푚 푓 푡 ≥ 푓 푦 ≥ 푓 푥 + 훻푓 푥 푇(푦 − 푥) epi 풇 First order condition for convexity • The hyperplane supports epi 풇 at (푥, 푓 푥 ), for any 푡 푥 푦, 푡 ∈ 퐞퐩퐢 푓 ⇒ 훻푓 푥 푇 푦 − 푥 + 푓 푥 − 푡 ≤ 0 훻푓 푥 푇 푦 푥 ⇒ − ≤ 0 −1 푡 푓 푥 Supporting hyperplane, derived from first order condition 5 Pointwise Supremum • If for each 푦 ∈ 푈, 푓(푥, 푦): 푅푛 → 푅 is convex in 푥, then function 푔(푥) = sup 푓(푥, 푦) 푦∈푈 is convex in 푥. -
Lecture 3: Convex Sets and Functions 3.1 Convex Sets
EE 227A: Convex Optimization and Applications January 24, 2012 Lecture 3: Convex Sets and Functions Lecturer: Laurent El Ghaoui Reading assignment: Chapters 2 (except x2.6) and sections 3.1, 3.2, 3.3 of BV. You can also look at section 3.1 of the web textbook. 3.1 Convex Sets Definition. A subset C of Rn is convex if and only if it contains the line segment between any two points in it: 8 x1; x2 2 C; 8 θ1 ≥ 0; θ2 ≥ 0; θ1 + θ2 = 1 : θ1x1 + θ2x2 2 C: Some important special cases of convex sets are the following. • The set is said to be an affine subspace if it contains the entire line passing through any two points. This corresponds to the condition above, with θ1; θ2 arbitrary. Subspaces and affine subspaces are convex. • The set is said to be a convex cone if the condition above holds, but with the restriction θ1 + θ2 = 1 removed. Examples. • The convex hull of a set of points fx1; : : : ; xmg is defined as ( m m ) X m X Co(x1; : : : ; xm) := λixi : λ 2 R+ ; λi = 1 ; i=1 i=1 and is convex. The conic hull: ( m ) X m λixi : λ 2 R+ i=1 is a convex cone. • For a 2 Rn, and b 2 R, the hyperplane H = fx : aT x = bg is affine. The half-space fx : aT x ≤ bg is convex. 3-1 EE 227A Lecture 3 | January 24, 2012 Sp'12 n×n n • For a square, non-singular matrix R 2 R , and xc 2 R , the ellipsoid fxc + Ru : kuk2 ≤ 1g is convex. -
Convex Optimization Scai Lab Study Group
Convex Optimization ScAi Lab Study Group Zhiping (Patricia) Xiao University of California, Los Angeles Winter 2020 Outline 2 Introduction: Convex Optimization Convexity Convex Functions' Properties Definition of Convex Optimization Convex Optimization General Strategy Learning Algorithms Convergence Analysis Examples Source of Materials 3 Textbook: I Convex Optimization and Intro to Linear Algebra by Prof. Boyd and Prof. Vandenberghe Course Materials: I ECE236B, ECE236C offered by Prof. Vandenberghe I CS260 Lecture 12 offered by Prof. Quanquan Gu Notes: I My previous ECE236B notes and ECE236C final report. I My previous CS260 Cheat Sheet. Related Papers: I Accelerated methods for nonconvex optimization I Lipschitz regularity of deep neural networks: analysis and efficient estimation Introduction: Convex Optimization q Notations 5 I iff: if and only if I R+ = fx 2 R j x ≥ 0g I R++ = fx 2 R j x > 0g I int K: interior of set K, not its boundary. I Generalized inequalities (textbook 2.4), based on a proper cone K (convex, closed, solid, pointed | if x 2 K and −x 2 K then x = 0): I x K y () y − x 2 K I x ≺K y () y − x 2 int K n n T I Positive semidefinite matrix X 2 S+, 8y 2 R , y Xy ≥ 0 () X 0. Convexity of Set 6 Set C is convex iff the line segment between any two points in C lies in C, i.e. 8x1; x2 2 C and 8θ 2 [0; 1], we have: θx1 + (1 − θ)x2 2 C Both convex and nonconvex sets have convex hull, which is defined as: k k X X conv C = f θixi j xi 2 C; θi ≥ 0; i = 1; 2; : : : ; k; θi = 1g i=1 i=1 Convexity of Set (Examples from Textbook) 7 Figure: Left: convex, middle & right: nonconvex. -
Convex Sets and Functions (Part III)
Convex Sets and Functions (part III) Prof. Dan A. Simovici UMB 1 / 14 Outline 1 The epigraph and hypograph of functions 2 Constructing Convex Functions 2 / 14 The epigraph and hypograph of functions Definition Let (S; O) be a topological space and let f : S −! R^ be a function. Its epigraph is the set epi(f ) = f(x; y) 2 S × R j f (x) 6 yg: The hypograph of f is the set hyp(f ) = f(x; y) 2 S × R j y 6 f (x)g: 3 / 14 The epigraph and hypograph of functions 2 The epigraph of a function f : R −! R is the dotted area in R located above the graph of the function f ; the hypograph of f is the dotted area below the graph. y y x x (a) (b) 4 / 14 The epigraph and hypograph of functions Note that the intersection epi(f ) \ hyp(f ) = f(x; y) 2 S × R j y = f (x)g is the graph of the function f . If f (x) = 1, then (x; 1) 62 epi(f ). Thus, for the function f1 defined by f1(x) = 1 for x 2 S we have epi(f1) = ;. 5 / 14 The epigraph and hypograph of functions Theorem Let f : S −! R be a function defined on the convex subset S of a real linear space L. Then, f is convex on S if and only if its epigraph is a convex subset of S × R; f is concave if and only if its hypograph is a convex subset of S × R. -
Convex Analysis
Convex Analysis Taught by Professor Adrian Lewis Notes by Mateo D´ıaz [email protected] Cornell University Spring, 2018 Last updated May 18, 2018. The latest version is online here. Notes template by David Mehrle. Contents 1 Convex Sets .................................................. 4 2 Convex functions............................................... 6 2.1 Convexity and calculus......................................... 7 2.1.1 Gradients ............................................ 7 2.1.2 Recognizing smooth convexity................................ 8 2.2 Continuity of convex functions.................................... 8 2.3 Subgradients............................................... 9 3 Duality..................................................... 10 3.1 Fenchel Duality............................................. 10 3.2 Conic programs............................................. 12 3.3 Convex calculus............................................. 14 3.4 Lagrangian duality........................................... 15 4 Algorithms................................................... 19 4.1 Augmented Lagrangian Method ................................... 19 4.2 Proximal Point method and fixed points............................... 20 4.3 Smooth Minimization.......................................... 25 4.4 Splitting Problems and Alternating projections........................... 26 4.5 Proximal Gradient ........................................... 28 4.6 Douglas-Rachford............................................ 29 4.6.1 Consensus optimization................................... -
On a Conic Approach to Convex Analysis 1 Introduction
On a Conic Approach to Convex Analysis Jan Brinkhuis February 2008 Econometric Institute Report EI 2008-05 Abstract The aim of this paper is to make an attempt to justify the main results from Convex Analysis by one elegant tool, the conification method, which consists of three steps: conify, work with convex cones, deconify. It is based on the fact that the standard operations and facts (‘the calculi’) are much simpler for special convex sets, convex cones. By considering suitable classes of convex cones, we get the standard operations and facts for all situations in the complete generality that is required. The main advantages of this conification method are that the standard operations— linear image, inverse linear image, closure, the duality operator, the binary operations and the inf- operator—are defined on all objects of each class of convex objects—convex sets, convex functions, convex cones and sublinear functions—and that moreover the standard facts—such as the duality theorem—hold forall closed convex objects. This requires that the analysis is carried out in the context of convex objects over cosmic space, the space that is obtained from ordinary space by adding a horizon, representing the directions of ordinary space. 1 Introduction Reduction to conic calculus. This paper presents some progress on joint work with V.M.Tikhomirov that was published in [7]. It is an attempt to develop a general and universal tool for justifying the main results of Convex Analysis. The idea is as follows. Many standard operations and facts are much simpler and more complete for special convex objects, convex cones. -
Generalized Convexity in Mathematical Programming
BULL. AUSTRAL. MATH. SOC. 90C25, 90C30 VOL. 27 (1983), 185-202. GENERALIZED CONVEXITY IN MATHEMATICAL PROGRAMMING B. MOND The role of convexity in the development of mathematical programming is reviewed. Many recent generalizations of convexity and their applications to optimization theory are discussed. 1. Introduction As is well known, convexity plays a key role in mathematical programming. However, functions that are, in some sense, close to, but not quite, convex often have many of the important properties of convex functions. This, plus the fact that many practical problems involve functions that are 'almost1 convex has led to many generalizations of the notion of convex functions. Here we survey some of these extensions of convexity and indicate briefly the role they play in optimization theory. Most of the results in Sections 2 to 5 can be found in the books by Avriel [7], Mangasarian [76] and Martos [77] while the results in Section 6 on cone-convexity can be found in the book by Craven [JS]. In view of this, no additional references or original sources will be given for results in these sections except for the new dual for non-linear programs in Section 5 that does not appear in any of the above-mentioned books. More detailed references will be given in the later sections. Proofs will only be given where they are extremely brief and not generally available in the literature. It should be noted that most proofs Received 7 October 1982. An invited lecture delivered to the Annual Congress of the South African Mathematical Society, October 1982. -
GEOMETRIC APPROACH to CONVEX SUBDIFFERENTIAL CALCULUS October 10, 2018
GEOMETRIC APPROACH TO CONVEX SUBDIFFERENTIAL CALCULUS October 10, 2018 BORIS S. MORDUKHOVICH1 and NGUYEN MAU NAM2 Dedicated to Franco Giannessi and Diethard Pallaschke with great respect Abstract. In this paper we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic results of convex subdifferential calculus in full generality and also derive new results of convex analysis concerning optimal value/marginal func- tions, normals to inverse images of sets under set-valued mappings, calculus rules for coderivatives of single-valued and set-valued mappings, and calculating coderivatives of solution maps to param- eterized generalized equations governed by set-valued mappings with convex graphs. Key words. convex analysis, generalized differentiation, geometric approach, convex separation, normal cone, subdifferential, coderivative, calculus rules, maximum function, optimal value function AMS subject classifications. 49J52, 49J53, 90C31 1 Introduction The notion of subdifferential (collection of subgradients) for nondifferentiable convex func- tions was independently introduced and developed by Moreau [15] and Rockafellar [19] who were both influenced by Fenchel [6]. Since then this notion has become one of the most central concepts of convex analysis and its various applications, including first of all convex optimization. The underlying difference between the standard derivative of a differentiable function and the subdifferential of a convex function at a given point is that the subdif- ferential is a set (of subgradients) which reduces to a singleton (gradient) if the function is differentiable. Due to the set-valuedness of the subdifferential, deriving calculus rules for it is a significantly more involved task in comparison with classical differential calculus.