Motivated by the problem of sampling from a logconcave density function, we study their geometry and introduce a technique for \smoothing them out. Sum of convex and concave function can have non global minima. If f is both convex and concave, then f is said to be affine. We consider best uniform approximation by nconvex and nconcave functions and, by considering alternation. Can a function be both convex and concave, or neither. Equivalently, a function is convex if its epigraph the set of points on or above the graph of the function is a convex set.
Take a look at a function that is both convex and concave on math\mathbb rmath. Note that the convexity of level sets does not characterize convex functions. The concaveconvex procedure cccp is a majorizationminimization algorithm that solves d. Ying cui department ofelectrical engineering shanghai jiaotonguniversity 2018 sjtu yingcui 142. Oct 22, 2014 in my part of europe, people would ask you ok, what do you call convex concave. For instance, finding the projection on the convex functions in h k. A set m is called convex if for any two points vectors x1,x2. M is called convex if for any two x1 x2, any convex. Mar 08, 2016 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. Inequalities of general convex functions and applications. Andrea colesanti and daniel hug abstract this paper originates from the investigation of support measures of convex bodies sets of positive reach, which form a central subject in convex geometry and also represent an important tool in related. The algorithm generates linear programs whose solutions minimize the convex envelope of the original function over successively tighter polytopes enclosing the feasible region. Keywords convex optimization convex concave procedure sequential.
Because results on concave functions can easily translated for convex functions we will only consider concave functions in the sequel. In my part of europe, people would ask you ok, what do you call convexconcave. Lets go over what it means for a shape to be concave and what it means for a function to be concave. Kawohl b minimum problems over sets of concave functions and. Discover the concept and criteria of convex and concave functions. Concave functions article about concave functions by the.
On functions and inverses, both positive, decreasing and convex. A concave function that has no linear parts is said to be strictly concave. 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. Taskovic 39 we notice that the set of all jconvex and jconcave functions can be a propersubsetofthesetofallgeneraljinnerfunctions. In this paper, an integral inequality and an application of it, that imply the chebyshev functional for two 3convex 3concave functions, are given. Concave functions of two variables while we will not provide a proof here, the following three definitions are equivalent if the function f is differentiable. Roughly speaking concavity of a function means that the graph is above chord. Outline basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions logconcave and logconvex functions convexity with respect to generalized.
Convex functions on real banach spaces springerlink. I think the sufficient condition to have at most two stationary points is to show that the first derivatives itself are strictly convexconcave increasingdecreasing monotonic convex or monotonic concave. It is shown that in an n dimensional space this problem is equivalent to the problem of minimizing a concave function on a convex set. Curved mirrors concave and convex linkedin slideshare. In mathematics, a concave function is the negative of a convex function. On selfconcordant convexconcave functions arkadi nemirovski april 19, 1999 abstract in this paper, we introduce the notion of a selfconcordant convexconcave function, estab lish basic properties of these functions and develop a pathfollowing interior point method.
Summary from the viewpoint of optimisation, the main results concerning convex functions that we will needrefer to during this class are. Because the name concave includes the word cave in it, its easy to remember that a concave shape is a shape that looks. Observe that the shape in the image above looks as though its caved in. This implies that cccp can be applied to almost any optimization problem. If all f i s are convex, then f is if all f i s are concave, then f is example. The class of logconcave functions in rn is a common generalization of gaussians and of indicator functions of convex sets. R is convex concave, if and only if xh1x is convex concave. Pdf the concaveconvex procedure cccp is a way to construct discretetime iterative dynamical systems.
Variations and extension of the convexconcave procedure. The inequalities in the definition of concave and convex functions are weak. Concave function article about concave function by the. Convex and concave relaxations of implicit functions matthew d. A function f is concave over a convex set if and only if the function.
Bee1020basicmathematicaleconomics week7,lecturetuesday17. The first two conditions ensure that the probability density function will map a concave function onto another concave function. Hartl and alexander mehlmann institut fur 6konometrie and operations research, university of technology, argentinierstrasse 84a, vienna, a1040 wien, austria received december 1981. In this paper, an integral inequality and an application of it, that imply the chebyshev functional for two 3 convex 3 concave functions, are given. Linear functions are concave, but not strictly concave.
Springer nature is making coronavirus research free. All results, derived in the sequel for convex functions f, can be reformulated for concave functions g, by setting g. Decomposing a function into convex and concave parts. In mathematics, the term concave can be applied to shapes and functions. 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. Hessian measures of semiconvex functions and applications to.
Why is this composition of concave and convex functions concave. A method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear. For simplicity, assume mathf \colon \mathbb r \to \mathbb rmath, and take mathx,y \in \mathbb r2math and math\lambda \in 0,1math. Our motivation for studying such func tions is their interest in two classes of applica tions. Pick any two points xy00 and xy11 on the graph of the function. Convex and concave relaxations of implicit functions. Study the intervals of concavity and convexity of the following function.
Quadrilaterals can be categorized as convex or concave. They should try to describe any images they form in terms of magnification, upright or inverted, real or. Difference of concave functions mathematics stack exchange. Which functions are the composition of convex functions. Concave and convex functions let f be a function of many variables, defined on a convex set s. Note that the union of convex sets is not necessarily convex. Download it in pdf format by simply entering your email. What is composition of convex and concave function. We consider here the problem of minimizing a particular subclass of quasidifferentiable functions. 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. 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. The inverse of a positive, decreasing convex function is positive, decreasing. John riley minor corrections 25 july 2016 ucla economics.
C fx 2dom fjfx g sublevel sets of convex functions are convex converse is false epigraph of f. Concave functions of one variable consider a function fx with a graph as depicted below. A quadrilateral is a four sided polygon that can be classified in several categories based on its properties. Lenses concave and convex apparatus selection of lenses, collimated light source, e. Barton 2015 convex and concave relaxations of implicit functions, optimization methods and software, 30. The axes for g are shown in perspective, like those for f, to make. A concave function uy is continuous, di erentiable from the left and from the right.
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. We say that f is concave if the line segment joining any two points on the graph of f is never above the graph. This video explains what are concave functions and convex functions. Finally, for convex f, fis concave, hence fis continuous, and fis continuous i fis continuous. Concave function the function f is concave on x if, for any x x x01. Concave function the function f is concave on if for any and any definition 2. Hessian measures of semiconvex functions and applications. We notice that the set of all jconvex and jconcave functions can be a. Sums of increasing convex and increasing concave functions. On minimizing the sum of a convex function and a concave. Curvedspherical mirror a curved mirror is a mirror with a curved reflecting surface. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i. R is convex, if and only if the socalled epigraph of f, which is the subset of rn r consisting of all points x.
Approximating optimization problems over convex functions. There is a very close connection between convex sets and convex functions. In machine learning, cccp is extensively used in many learning algorithms like sparse support vector machines svms, transductive svms, sparse principal component analysis, etc. Quasi concave functions have nice properties for maximization. Notice this does not guarantee that a solution exists. These results are applied to analyze two ecient algorithms for sam. A method for globally minimizing concave functions over. That is because in some russian, yugoslav and i assume other eastern european textbooks, concavityconvexity was defined in a manner opposite to the one in davids a. The \proper characterization of convex functions in terms of convex sets is given by proposition 3. Roberts and varberg 3 give sever,d results and references. Hessian measures of semiconvex functions and applications to support measures of convex bodies. Concave function the differentiable function f is concave on x if for any x x x01, and any. The algorithm does not involve cuts of the feasible region, requires only simplex pivot.
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