Keywords convex optimization convex concave procedure sequential optimization difference of convex programming 1 introduction in this paper we present several extensions of and variations on the convexconcave procedure ccp, a powerful heuristic method used to. If f is strictly convex, then x is the unique global minimum. The concaveconvex procedure cccp is a way to construct discretetime iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically. Simon and blume on concave and quasiconcave functions. The \proper characterization of convex functions in terms of convex sets is given by proposition 3. Hence, we will study a few aspects of this theory in the present chapter before studying duality theory in the following chapter.
This is a minmax relation between a pair of convex and concave functions and their conjugate functions. Then check the conditions of the above theorem for. They should try to describe any images they form in terms of magnification, upright or inverted, real or. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex. This implies that cccp can be applied to almost any optimization problem. A quadratic functionxtqxis convex if qis positive semidefinite, or concave if qis negative semidefinite. Minimizing differences of convex functionsthe dca fermattorricelli problem in the early of the 17th century, at the end of his book, treatise on minima and maxima, the french mathematician fermat 16011665 proposed the following problem. In this paper, we comprehensively discuss mathematical properties of the class of exponentially concave functions, like. In this paper, we comprehensively discuss mathematical properties of the class of exponentially concave functions, like closedness under linear and convex combi. In mathematics, a concave function is the negative of a convex function. Concave and convex functions let f be a function of many variables, defined on a convex set s. Extendedvalue extensions the extendedvalue extension of is. Milman school of mathematical sciences tel aviv university tel aviv 69978, israel abstract we present a view of log concave measures, which enables one to build an isomorphic theory for high dimensional logconcave measures, analogous to the corresponding theory for convex bodies. We deal with the convex functions on the bounded closed convex sets with common barycenter.
The concave convex procedure cccp is a majorizationminimization algorithm that solves d. Note that the union of convex sets is not necessarily convex. Convex sets separation concave and convex functions quasiconcave functions convex sets in rn 2 another ways of writing down the convex combination of two points is. Definition 11 concave function let c % r be a convex set, and let. Minimizing differences of convex functionsthe dca convex functions theorem let fi. In machine learning, cccp is extensively used in many learning algorithms like sparse support vector machines svms, transductive svms, sparse principal component analysis, etc. Concave and convex functions1 washington university. The question we pose is whether it is possible to obtain fast rates for expconcave functions using more computationallye cient algorithms. 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.
We focus on convex optimization problems in this course. If f is concave and f is concave and increasing, then ux ffx is concave. Milman school of mathematical sciences tel aviv university tel aviv 69978, israel abstract we present a view of log concave measures, which enables one to build an isomorphic theory for high dimensional log concave measures, analogous to the corresponding theory for convex bodies. Variations and extension of the convexconcave procedure. The axes for g are shown in perspective, like those for f, to make. Socalled exponentially concave functions are of similar importance in information theory. 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. We say that f is concave if the line segment joining any two points on the graph of f is never above the graph. However, the mathematical foundations of duality theory rest on the theory of convex sets and concave and convex functions. Quasiconvex and log concave functions convexity with respect to generalized inequalities convex functions a supplementary note to chapter 3 of convex optimization by s. C f x 2 dom f j f x g sublevel sets of convex functions. For convexity, r2fx q and use secondorder condition.
C f x 2 dom f j f x g sublevel sets of convex functions are convex converse is fa lse epigraph of f. 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. Note that the convexity of level sets does not characterize convex functions. Affine functions are both convex and concave, and vice versa.
Taskovic 39 we notice that the set of all jconvex and jconcave functions can be a propersubsetofthesetofallgeneraljinnerfunctions. It covers a large variety of subjects, from one real variable. Is the statement the product of convex functions is. Finally, for convex f, fis concave, hence fis continuous, and fis continuous i fis continuous. 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. C2 function can be expressed as a difference of convex functions hartman 1959. Convex optimization page 1 daniel guetta convex optimization chapter 2 convex sets basics o a set is affine if it contains any line through two of its point. Students first meet concave and convex functions in their study of functions of. Abstractconcave functions play a central role in optimization.
Cs257 linear and convex optimization lecture 4 convex functions. Algorithms for minimizing differences of convex functions and. A concave function uy is continuous, di erentiable from the left and from the right. Bis convex since it is a translation of the strictly positive orthant in r2. Like strictly convex functions, strongly convex functions have unique minima on compact sets.
Is the statement the product of convex functions is convex true or false. In a paper published wednesday in the journal of optics, mr. Almost every convex function can be expressed as the pointwise supremum of a family of affine functions. Convex functions a supplementary note to chapter 3 of convex. Ie department seoul national university 19th august 2009. Cs257 linear and convex optimization lecture 5 convex functions. 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. A set s in rn euclidean n dimensional space is convex iff if and only if. We notice that the set of all jconvex and jconcave functions can be a. Convex, concave, strictly convex, and strongly convex functions first and second order characterizations of convex functions optimality conditions for convex problems 1 theory of convex functions 1. Inequalities of general convex functions and applications. F filly f is 0 for all x, y e w, then f is quasiconcave on w. Calibrating probability distributions with convexconcaveconvex functions table 1 market quotes for 5, 7, 10year itraxx on 20 december, 2006 obtained from halperin 2009 quotes for the 0 to 3 % tranche are the percent of the principal that must be paid up front in addition to 500 basis points per year. Convex sets and concave functions carmen astornefigari.
Consider the problem of minimizing a convex function f over a convex set k rn where our only access to fis via a stochastic gradient oracle, that given a point x2kreturns a random vector g x for which eg. If f is convex and f is convex and increasing, then ux ffx is convex. Cs257 linear and convex optimization lecture 4 convex. Proving that multiplication of convex function is convex. Calibrating probability distributions with convex concave convex functions table 1 market quotes for 5, 7, 10year itraxx on 20 december, 2006 obtained from halperin 2009 quotes for the 0 to 3 % tranche are the percent of the principal that must be paid up front in addition to 500 basis points per year. In ordinary usage, concave and convex are typically used when referring to glass surfaces, like the lenses of optical viewing equipment. Algorithms for minimizing differences of convex functions. Equivalently, a function is convex if its epigraph the set of points on or above the graph of the function is a convex set. The certain regularity conditions in the statement below will be speci.
On exponentially concave functions and their impact in. The graph of a concave function will always lie below the tangent line at a given point as. 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. Convex sets separation concave and convex functions quasiconcave functions other useful properties a. More precisely, the integral arithmetic means of a convex function f are compared on two sets a and b. Our book aims to be a thorough introduction to the contemporary convex functions theory. In contrast to logarithmically convex logconvex functions, their. Moreover, convexity and concavity allow for very elegant proofs in inequality theory, cf. To see this, suppose there is a xsuch that such that fx fx and hence there is an a 1 such that fx rbe a partial differentiable mapping on a,b. To see this, we can cast the boolean linear program lp. It is a wellknown fact from convex analysis27 that the convex envelope of. Cs257 linear and convex optimization lecture 5 convex. On the convergence of the concave convex procedure.
The inequalities in the definition of concave and convex functions are weak. Uniformly convex functions edit a uniformly convex function, 9 10 with modulus. A dc program is not convex unless the functions g i are af. Because results on concave functions can easily translated for convex functions we will only consider concave functions in the sequel. 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. In particular, ais convex since f is concave and gis convex hence gis concave. The question we pose is whether it is possible to obtain fast rates for exp concave functions using more computationallye cient algorithms. Calibrating probability distributions with convex concave.
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