Convex functions and their applications a contemporary. Is f concave first note that the domain of f is a convex set, so the definition of concavity can apply the functions g and f are illustrated in the following figures. This implies that cccp can be applied to almost any optimization problem. The concave convex procedure cccp is a majorizationminimization algorithm that solves d.
On convex envelopes and underestimators for bivariate. International journal of scientific and research publications ijsrp 8. The inequalities in the definition of concave and convex functions are weak. Developing a working knowledge of convex optimization can be mathematically demanding, especially for the reader interested primarily in applications. C f x 2 dom f j f x g sublevel sets of convex functions are convex converse is fa lse epigraph of f. It is a wellknown fact from convex analysis27 that the convex envelope of. Convex sets separation concave and convex functions quasiconcave functions other useful properties a. A concave surface curves inward, like the mouth of a cave. Concave and convex functions1 washington university. A convex lens gives a real, inverted image if the object is outside the focal length of the lens. Conversely, any function that is convex and concave is a. Uniformly convex functions edit a uniformly convex function, 9 10 with modulus. The concave convex procedure cccp is a way to construct discretetime iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically. If this definition is followed, then yes a constant function, indeed any linear function, would be concave.
If a convex functional is finite at some point, it has a finite or infinite derivative in any direction at that point. Convex and concave functions of singular values of. In particular, if the domain is a closed interval in r, then concave functions can jump down at end points and convex functions can jump up. Can a function be both convex and concave, or neither. If f is concave and f is concave and increasing, then ux ffx is concave. Like strictly convex functions, strongly convex functions have unique minima on compact sets. The cone of convex functions, implied by membership relation 507, provides foundation for what is known as a lagrangian function. If you have trouble remembering whether a surface is convex or concave, there is an easy way to find out. Roughly speaking concavity of a function means that the graph is above chord. Another relevant result about polyhedral convex envelopes is presented in 16 and will be discussed later. It is strictly concave if and only if the inequality is strict for x x0.
Today well talk about characterizations of convex functions, show simple ways to test whether a function is convex, and importantly present the gradient ascent algorithm for minimizing a convex function. Firstorder methods of smooth convex optimization with inexact oracle. Our book aims to be a thorough introduction to the contemporary convex functions theory. A spoon is a kind of a curved mirror with both concave and convex surfaces. Equivalently, a function is convex if its epigraph the set of points on or above the graph of the function is a convex set. I have a quick silly question about a statement given without proof in convex optimization by boyd and vandenberghe page 87. Take a look at a function that is both convex and concave on math\mathbb rmath.
In mathematics, a quasiconvex function is a realvalued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form. Concave and convex functions 1concaveandconvexfunctions 1. Examples of convex functions and classi cations of normed. But avoid asking for help, clarification, or responding to other answers. In 15, 23 edgeconcave functions over a polytope p i. So are the log convex functions, the multiplicatively convex functions, the subharmonic functions, and the functions which are convex with respect to a subgroup of the linear group. When you plot a function in the cartesian system, if the curve opens towards the positive y axis then it is said to be concave up or convex function while if it opens downwards then the curve is said to be concave down or concave function. Y denote real vector spaces, unless otherwise speci ed. In machine learning, cccp is extensively used in many learning algorithms like sparse support vector machines svms, transductive svms, sparse principal component analysis, etc. Lenses concave and convex apparatus selection of lenses, collimated light source, e. The concaveconvex procedure cccp is a way to construct discretetime iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically. Convex sets and concave functions carmen astornefigari. A concave function uy is continuous, di erentiable from the left and from the right.
Convex lenses are converging lenses, and concave lenses are diverging lenses. A convex functional bounded from above in a neighbourhood of some point is continuous at that point. Thanks for contributing an answer to mathematics stack exchange. Theorem 6 let f be a differentiable function on the open convex set c. The class of dc problems includes many difficult problems such as the traveling salesman problem. However, the mathematical foundations of duality theory rest on the theory of convex sets and concave and convex functions.
The \proper characterization of convex functions in terms of convex sets is given by proposition 3. New york university department of economics econua 6 mathematics for economists c. Quasi concave functions have nice properties for maximization. The certain regularity conditions in the statement below will be speci. Critics of this account often rely on assumptions about preferences over wide ranges of. We derive closedform expressions for convex envelopes of various.
Theorem 1 shows that any function, subject to weak conditions, can be expressed as the sum of a convex and concave part this decomposition is not unique. R is convex, if and only if the socalled epigraph of f, which is the subset of rn r consisting of all points x. Necessary and sufficient conditions such that extended mean values are schurconvex or schurconcave chu, yuming and zhang, xiaoming, journal of mathematics of kyoto university, 2008 a note on integral inequalities of hadamard type for logconvex and logconcave functions yang, gousheng, tseng, kueilin, and wang, hungta, taiwanese journal of. Note that the convexity of level sets does not characterize convex functions. The dotted line is the set of convex combinations of these two points. Concave functions havenice characterizationwhen theyaredifferentiable. It is also possible to characterize concavity or convexity of functions in terms of the convexity of particular sets. Why is this composition of concave and convex functions. It appears you dont have a pdf plugin for this barrpostingser. Curvedspherical mirror a curved mirror is a mirror with a curved reflecting surface. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. Variations and extension of the convexconcave procedure. The concaveconvex procedure cccp is a majorizationminimization algorithm that solves d.
If f is convex and f is convex and increasing, then ux ffx is convex. A concave function that has no linear parts is said to be strictly concave. In mathematics, a realvalued function defined on an ndimensional interval is called convex or convex downward or concave upward if the line segment between any two points on the graph of the function lies above or on the graph. According to the wikipedia defintion, for a function to be concave on the interval between a and b, the following must hold.
Finally, for convex f, fis concave, hence fis continuous, and fis continuous i fis continuous. The surface may be either convex bulging outward or concave bulging inward. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Firstorder methods of smooth convex optimization with.
On convex envelopes and underestimators for bivariate functions. For simplicity, assume mathf \colon \mathbb r \to \mathbb rmath, and take mathx,y \in \mathbb r2math and math\lambda \in 0,1math. Katarina nordblomconcave and convex functionsseptember 6, 20191 11. Necessary and sufficient conditions such that extended mean values are schur convex or schur concave chu, yuming and zhang, xiaoming, journal of mathematics of kyoto university, 2008 a note on integral inequalities of hadamard type for log convex and log concave functions yang, gousheng, tseng, kueilin, and wang, hungta, taiwanese journal of. Concave and convex maintain their status as adjectives when used in this context. There is a natural way to embed the set of convex bodies in that of logconcave functions, and there are surprisingly many analogies between the theory of convex bodies and that of logconcave date. Abstract concave functions play a central role in optimization. Convex envelopes of products of convex and componentwise. Curved mirrors concave and convex linkedin slideshare. C f x 2 dom f j f x g sublevel sets of convex functions.
Why is this composition of concave and convex functions concave. We investigate the convexconcave procedure, a local heuristic that utilizes the tools of convex optimization to find local optima of difference of convex dc programming problems. This is a minmax relation between a pair of convex and concave functions and their conjugate functions. Convex functions basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions logconcave and logconvex functions convexity with respect to generalized inequalities 31. Convex functions whose subdi erentials have range with nonconvex interior are constructed on nonre exive spaces, and we exhibit examples of convex functions on in nite dimensional spaces whose subdi erentials have sparse domains. They should try to describe any images they form in terms of magnification, upright or inverted, real or. Note that the union of convex sets is not necessarily convex. In fact, a ne functions are the only functions that are both convex and concave. We say that f is concave if the line segment joining any two points on the graph of f is never above the graph. There is a very close connection between convex sets and convex functions. Because results on concave functions can easily translated for convex functions we will only consider concave functions in the sequel. Socalled exponentially concave functions are of similar importance in information theory. It covers a large variety of subjects, from one real variable.
It will give a virtual upright image if the object is within the focal length. The negative of a quasiconvex function is said to be quasiconcave. In this paper, we comprehensively discuss mathematical properties of the class of exponentially concave functions, like closedness under linear and convex combi. Concave and convex functions let f be a function of many variables, defined on a convex set s. The axes for g are shown in perspective, like those for f, to make. A 5 a cartesian product of convex functions remains convex, so we.
On exponentially concave functions and their impact in. We extend the standard procedure in two major ways and describe several variations. Today well talk about characterizations of convex functions, show simple ways to test whether a function is convex, and importantly present the. Convexconcave programming is an organized heuristic for solving nonconvex problems that involve objective and constraint functions that are a sum of a convex and a concave term. Concave production function implies convex cost function. If the function u gx is convex downward, and the function y fu is convex downward and nondecreasing, then the composite function y fgx is also convex downward. Convex function, bounded subdi erential, domain, range, mackey convergence, support points. How to determine whether a function is concave, convex, quasiconcave and quasiconvex. Concave converging mirror it is a curved mirror in which the reflective. Thanks for contributing an answer to economics stack exchange.
880 701 719 304 268 1187 1026 311 244 96 1276 1377 21 917 894 1391 1043 1583 778 987 1106 1331 731 196 592 1222 680 677 1061 695 238