On Complex Convexity
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New Hermite–Hadamard and Jensen Inequalities for Log-H-Convex
International Journal of Computational Intelligence Systems (2021) 14:155 https://doi.org/10.1007/s44196-021-00004-1 RESEARCH ARTICLE New Hermite–Hadamard and Jensen Inequalities for Log‑h‑Convex Fuzzy‑Interval‑Valued Functions Muhammad Bilal Khan1 · Lazim Abdullah2 · Muhammad Aslam Noor1 · Khalida Inayat Noor1 Received: 9 December 2020 / Accepted: 23 July 2021 © The Author(s) 2021 Abstract In the preset study, we introduce the new class of convex fuzzy-interval-valued functions which is called log-h-convex fuzzy-interval-valued functions (log-h-convex FIVFs) by means of fuzzy order relation. We have also investigated some properties of log-h-convex FIVFs. Using this class, we present Jensen and Hermite–Hadamard inequalities (HH-inequalities). Moreover, some useful examples are presented to verify HH-inequalities for log-h-convex FIVFs. Several new and known special results are also discussed which can be viewed as an application of this concept. Keywords Fuzzy-interval-valued functions · Log-h-convex · Hermite–Hadamard inequality · Hemite–Hadamard–Fejér inequality · Jensen’s inequality 1 Introduction was independently introduced by Hadamard, see [24, 25, 31]. It can be expressed in the following way: The theory of convexity in pure and applied sciences has Let Ψ∶K → ℝ be a convex function on a convex set K become a rich source of inspiration. In several branches of and , ∈ K with ≤ . Then, mathematical and engineering sciences, this theory had not only inspired new and profound results, but also ofers a + 1 Ψ ≤ Ψ()d coherent and general basis for studying a wide range of prob- 2 − � lems. Many new notions of convexity have been developed Ψ()+Ψ() and investigated for convex functions and convex sets. -
Lecture 5: Maxima and Minima of Convex Functions
Advanced Mathematical Programming IE417 Lecture 5 Dr. Ted Ralphs IE417 Lecture 5 1 Reading for This Lecture ² Chapter 3, Sections 4-5 ² Chapter 4, Section 1 1 IE417 Lecture 5 2 Maxima and Minima of Convex Functions 2 IE417 Lecture 5 3 Minimizing a Convex Function Theorem 1. Let S be a nonempty convex set on Rn and let f : S ! R be ¤ convex on S. Suppose that x is a local optimal solution to minx2S f(x). ² Then x¤ is a global optimal solution. ² If either x¤ is a strict local optimum or f is strictly convex, then x¤ is the unique global optimal solution. 3 IE417 Lecture 5 4 Necessary and Sufficient Conditions Theorem 2. Let S be a nonempty convex set on Rn and let f : S ! R be convex on S. The point x¤ 2 S is an optimal solution to the problem T ¤ minx2S f(x) if and only if f has a subgradient » such that » (x ¡ x ) ¸ 0 8x 2 S. ² This implies that if S is open, then x¤ is an optimal solution if and only if there is a zero subgradient of f at x¤. 4 IE417 Lecture 5 5 Alternative Optima Theorem 3. Let S be a nonempty convex set on Rn and let f : S ! R be convex on S. If the point x¤ 2 S is an optimal solution to the problem minx2S f(x), then the set of alternative optima are characterized by the set S¤ = fx 2 S : rf(x¤)T (x ¡ x¤) · 0 and rf(x¤) = rf(x) Corollaries: 5 IE417 Lecture 5 6 Maximizing a Convex Function Theorem 4. -
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. -
Robustly Quasiconvex Function
Robustly quasiconvex function Bui Thi Hoa Centre for Informatics and Applied Optimization, Federation University Australia May 21, 2018 Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 1 / 15 Convex functions 1 All lower level sets are convex. 2 Each local minimum is a global minimum. 3 Each stationary point is a global minimizer. Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 2 / 15 Definition f is called explicitly quasiconvex if it is quasiconvex and for all λ 2 (0; 1) f (λx + (1 − λ)y) < maxff (x); f (y)g; with f (x) 6= f (y): Example 1 f1 : R ! R; f1(x) = 0; x 6= 0; f1(0) = 1. 2 f2 : R ! R; f2(x) = 1; x 6= 0; f2(0) = 0. 3 Convex functions are quasiconvex, and explicitly quasiconvex. 4 f (x) = x3 are quasiconvex, and explicitly quasiconvex, but not convex. Generalised Convexity Definition A function f : X ! R, with a convex domf , is called quasiconvex if for all x; y 2 domf , and λ 2 [0; 1] we have f (λx + (1 − λ)y) ≤ maxff (x); f (y)g: Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 3 / 15 Example 1 f1 : R ! R; f1(x) = 0; x 6= 0; f1(0) = 1. 2 f2 : R ! R; f2(x) = 1; x 6= 0; f2(0) = 0. 3 Convex functions are quasiconvex, and explicitly quasiconvex. 4 f (x) = x3 are quasiconvex, and explicitly quasiconvex, but not convex. Generalised Convexity Definition A function f : X ! R, with a convex domf , is called quasiconvex if for all x; y 2 domf , and λ 2 [0; 1] we have f (λx + (1 − λ)y) ≤ maxff (x); f (y)g: Definition f is called explicitly quasiconvex if it is quasiconvex and -
Convexity I: Sets and Functions
Convexity I: Sets and Functions Ryan Tibshirani Convex Optimization 10-725/36-725 See supplements for reviews of basic real analysis • basic multivariate calculus • basic linear algebra • Last time: why convexity? Why convexity? Simply put: because we can broadly understand and solve convex optimization problems Nonconvex problems are mostly treated on a case by case basis Reminder: a convex optimization problem is of ● the form ● ● min f(x) x2D ● ● subject to gi(x) 0; i = 1; : : : m ≤ hj(x) = 0; j = 1; : : : r ● ● where f and gi, i = 1; : : : m are all convex, and ● hj, j = 1; : : : r are affine. Special property: any ● local minimizer is a global minimizer ● 2 Outline Today: Convex sets • Examples • Key properties • Operations preserving convexity • Same for convex functions • 3 Convex sets n Convex set: C R such that ⊆ x; y C = tx + (1 t)y C for all 0 t 1 2 ) − 2 ≤ ≤ In words, line segment joining any two elements lies entirely in set 24 2 Convex sets Figure 2.2 Some simple convexn and nonconvex sets. Left. The hexagon, Convex combinationwhich includesof x1; its : :boundary : xk (shownR : darker), any linear is convex. combinationMiddle. The kidney shaped set is not convex, since2 the line segment between the twopointsin the set shown as dots is not contained in the set. Right. The square contains some boundaryθ1x points1 + but::: not+ others,θkxk and is not convex. k with θi 0, i = 1; : : : k, and θi = 1. Convex hull of a set C, ≥ i=1 conv(C), is all convex combinations of elements. Always convex P 4 Figure 2.3 The convex hulls of two sets in R2. -
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. -
Convex Functions
Convex Functions Prof. Daniel P. Palomar ELEC5470/IEDA6100A - Convex Optimization The Hong Kong University of Science and Technology (HKUST) Fall 2020-21 Outline of Lecture • Definition convex function • Examples on R, Rn, and Rn×n • Restriction of a convex function to a line • First- and second-order conditions • Operations that preserve convexity • Quasi-convexity, log-convexity, and convexity w.r.t. generalized inequalities (Acknowledgement to Stephen Boyd for material for this lecture.) Daniel P. Palomar 1 Definition of Convex Function • A function f : Rn −! R is said to be convex if the domain, dom f, is convex and for any x; y 2 dom f and 0 ≤ θ ≤ 1, f (θx + (1 − θ) y) ≤ θf (x) + (1 − θ) f (y) : • f is strictly convex if the inequality is strict for 0 < θ < 1. • f is concave if −f is convex. Daniel P. Palomar 2 Examples on R Convex functions: • affine: ax + b on R • powers of absolute value: jxjp on R, for p ≥ 1 (e.g., jxj) p 2 • powers: x on R++, for p ≥ 1 or p ≤ 0 (e.g., x ) • exponential: eax on R • negative entropy: x log x on R++ Concave functions: • affine: ax + b on R p • powers: x on R++, for 0 ≤ p ≤ 1 • logarithm: log x on R++ Daniel P. Palomar 3 Examples on Rn • Affine functions f (x) = aT x + b are convex and concave on Rn. n • Norms kxk are convex on R (e.g., kxk1, kxk1, kxk2). • Quadratic functions f (x) = xT P x + 2qT x + r are convex Rn if and only if P 0. -
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. -
Subgradients
Subgradients Ryan Tibshirani Convex Optimization 10-725/36-725 Last time: gradient descent Consider the problem min f(x) x n for f convex and differentiable, dom(f) = R . Gradient descent: (0) n choose initial x 2 R , repeat (k) (k−1) (k−1) x = x − tk · rf(x ); k = 1; 2; 3;::: Step sizes tk chosen to be fixed and small, or by backtracking line search If rf Lipschitz, gradient descent has convergence rate O(1/) Downsides: • Requires f differentiable next lecture • Can be slow to converge two lectures from now 2 Outline Today: crucial mathematical underpinnings! • Subgradients • Examples • Subgradient rules • Optimality characterizations 3 Subgradients Remember that for convex and differentiable f, f(y) ≥ f(x) + rf(x)T (y − x) for all x; y I.e., linear approximation always underestimates f n A subgradient of a convex function f at x is any g 2 R such that f(y) ≥ f(x) + gT (y − x) for all y • Always exists • If f differentiable at x, then g = rf(x) uniquely • Actually, same definition works for nonconvex f (however, subgradients need not exist) 4 Examples of subgradients Consider f : R ! R, f(x) = jxj 2.0 1.5 1.0 f(x) 0.5 0.0 −0.5 −2 −1 0 1 2 x • For x 6= 0, unique subgradient g = sign(x) • For x = 0, subgradient g is any element of [−1; 1] 5 n Consider f : R ! R, f(x) = kxk2 f(x) x2 x1 • For x 6= 0, unique subgradient g = x=kxk2 • For x = 0, subgradient g is any element of fz : kzk2 ≤ 1g 6 n Consider f : R ! R, f(x) = kxk1 f(x) x2 x1 • For xi 6= 0, unique ith component gi = sign(xi) • For xi = 0, ith component gi is any element of [−1; 1] 7 n -
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. -
NMSA403 Optimization Theory – Exercises Contents
NMSA403 Optimization Theory { Exercises Martin Branda Charles University, Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Version: December 17, 2017 Contents 1 Introduction and motivation 2 1.1 Operations Research/Management Science and Mathematical Program- ming . .2 1.2 Marketing { Optimization of advertising campaigns . .2 1.3 Logistic { Vehicle routing problems . .2 1.4 Scheduling { Reparations of oil platforms . .3 1.5 Insurance { Pricing in nonlife insurance . .4 1.6 Power industry { Bidding, power plant operations . .4 1.7 Environment { Inverse modelling in atmosphere . .5 2 Introduction to optimization 6 3 Convex sets and functions 6 4 Separation of sets 8 5 Subdifferentiability and subgradient 9 6 Generalizations of convex functions 10 6.1 Quasiconvex functions . 10 6.2 Pseudoconvex functions . 12 7 Optimality conditions 13 7.1 Optimality conditions based on directions . 13 7.2 Karush{Kuhn{Tucker optimality conditions . 15 7.2.1 A few pieces of the theory . 15 7.2.2 Karush{Kuhn{Tucker optimality conditions . 17 7.2.3 Second Order Sufficient Condition (SOSC) . 19 1 Introduction and motivation 1.1 Operations Research/Management Science and Mathe- matical Programming Goal: improve/stabilize/set of a system. You can reach the goal in the following steps: • Problem understanding • Problem description { probabilistic, statistical and econometric models • Optimization { mathematical programming (formulation and solution) • Verification { backtesting, stresstesting • Implementation (Decision Support -
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.