If all f i s are convex, then f is if all f i s are concave, then f is example. It is shown that nonnegative, increasing, convex respectively, concave functions are superadditive respectively, subadditive with respect to submajorisation on the positive cone of the space of all. Select any convex function f x with positive definite hessian. Convex and concave functions are extremely important in nonlinear programming because they are among the few functions for which sufficient optimality criteria can be given chaps. Then we can always decompose it into the sum of a convex function and a concave function. Uniformly convex functions edit a uniformly convex function, 9 10 with modulus. Leastsquares, linear and quadratic programs, semidefinite programming, minimax, extremal volume, and other problems. A note on schurconcave functions journal of inequalities. Ris concave convex on x if and only if for all x, y. Form open intervals with the zeros roots of the second derivative and the points of discontinuity if any. In mathematics, the term concave can be applied to shapes and functions. Submajorisation inequalities for convex and concave. Strictly quasiconvex concave functions and duality in. A mirror is a surface which reflects a clear image.
Concentrates on recognizing and solving convex optimization problems that arise in engineering. The focal length of a convex lens is positive, while that of a concave lens is negative. This duality makes it possible to relate to each convex functional a dual object, the conjugate functional. This implies that cccp can be applied to almost any optimization problem. 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. To study the concavity and convexity, perform the following steps. Such a functional is the least upper bound of the affine functions which it dominates. Pick any two points xy00 and xy11 on the graph of the function. 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 derive closedform expressions for convex envelopes of various. Other families of schurconvex functions are studied in 17, 2024, 27. Questions about saddlevalues and saddlepoints of concaveconvex functions. 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.
Convex envelopes of products of convex and componentwise. 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. We consider the problem of minimizing a class of quasiconcave functions over a convex set. John riley minor corrections 25 july 2016 ucla economics. Take a look at a function that is both convex and concave on math\mathbb rmath. An important source of schurconvex functions can be found in merkle. Decomposing a function into convex and concave parts. Quasiconcave functions are generalizations of concave functions and nphard to minimize in general. Concave mirrors and convex mirrors image formation, ray. Images of emptiness in women article pdf available in the american journal of psychoanalysis 601. The class of continuous convex concave functions is extended to the class of lower semicontinuous strictly quasiconvex upper semi continuous strictly quasiconcave functions respectively. The functions f and g are both concave functions of a single variable. 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. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.
The following theoremgives acharacterizationof concave convexfunctions. Convex functions have another obvious property, which. Barton 2015 convex and concave relaxations of implicit functions, optimization methods and software, 30. Lets go over what it means for a shape to be concave and what it means for a function to be concave. On andos inequalities for convex and concave functions. Sum of convex and concave functions when one is greater than the other. The certain regularity conditions in the statement below will be speci. These extensions permit us to get, among other results, a direct strict duality theorem when ifi is restricted to the classical lagrangian function.
Central to our discussion are the concepts of logcon. The inequalities in the definition of concave and convex functions are weak. The \proper characterization of convex functions in terms of convex sets is given by proposition 3. 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. 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. Curved mirrors concave and convex linkedin slideshare. The dotted line is the set of convex combinations of these two points. Difference between convex and concave lens with figure. Fromnow on we will assume thatx is aconvex subset of rn. The convex combinations of two vectors are most commonly written as follows.
A sharp bound for the cebysev functional of convex or concave functions is proved. If f is both convex and concave, then f is said to be affine. A function f is concave over a convex set if and only if the function. Convex and concave functions of singular values of. Equivalently, a function is convex if its epigraph the set of points on or above the graph of the function is a convex set. We present a simple fully polynomial time approximation scheme fptas for minimizing a class of lowrank quasi concave functions. Find the second derivative and calculate its roots. An fptas for minimizing a class of lowrank quasiconcave. Convex and concave relaxations of implicit functions matthew d. Like strictly convex functions, strongly convex functions have unique minima on compact sets. Concave function the function f is concave on x if, for any x x x01. The concaveconvex procedure cccp is a way to construct discretetime iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically. Can a function be both convex and concave, or neither. 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.
Dec 08, 2018 the focal length of a convex lens is positive, while that of a concave lens is negative. We consider the problem of minimizing a class of quasi concave functions over a convex set. Concave and convex functions 1concaveandconvexfunctions 1. This is a minmax relation between a pair of convex and concave functions and their conjugate functions. Guan 10, 11 proved that all symmetric elementary functions and the symmetric means of order k are schurconcave 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. For simplicity, assume mathf \colon \mathbb r \to \mathbb rmath, and take mathx,y \in \mathbb r2math and math\lambda \in 0,1math. On the contrary, the image formed by the concave lens is erect, virtual and smaller, than the object.
Optimality conditions, duality theory, theorems of alternative, and applications. A concave function that has no linear parts is said to be strictly concave. A convex functional bounded from above in a neighbourhood of some point is continuous at that point. We present a simple fully polynomial time approximation scheme fptas for minimizing a class of lowrank quasiconcave functions. Because results on concave functions can easily translated for convex functions we will only consider concave functions in the sequel. Convex functions basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions log concave and log convex functions convexity with respect to generalized inequalities 31. Note that the convexity of level sets does not characterize convex functions. If a convex functional is finite at some point, it has a finite or infinite derivative in any direction at that point. These methods give possibility to combine linear trend patterns with convexconcave patterns and to precisiate linguistic. Pdf the concaveconvex procedure cccp is a way to construct. Concave functions of one variable consider a function fx with a graph as depicted below. What can we say about the other function in the product. Quasi concave functions are generalizations of concave functions and nphard to minimize in general. However, it remains an intriguing research challenge how local optimal points are defined and.
A concave function uy is continuous, di erentiable from the left and from the right. Submajorisation inequalities for convex and concave functions. Study the intervals of concavity and convexity of the following function. Convergence to a saddle point for convexconcave functions has been studied for decades, while the last few years have seen a surge of interest in nonconvexnonconcave minmax optimization due to the rise of deep learning. C f x 2 dom f j f x g sublevel sets of convex functions. On the contrary, in a concave mirror, the reflecting surface bulges inwards the main difference between a convex and concave mirror lies in the image formed by the two mirrors, i. Convex and concave functions nonlinear programming. Concave mirror convex mirror image formation by concave mirror concave mirror ray diagram image formation by convex mirror. Generally, a convex lens forms a real image, but it can also create a virtual image when the object is in the middle of the focus and optical centre. Finally, for convex f, fis concave, hence fis continuous, and fis continuous i fis continuous. In this chapter we introduce convex, concave, strictly convex, and strictly concave functions defined on subsets of r n. A sharp bound for the cebysev functional of convex or.
Others, like taking the convex hull of a collection of functions, are geometrically motivated. Download fulltext pdf download fulltext pdf convex and concave, part i. Some of the operations, like pointwise addition of functions, are familiar from ordinary analysis. Download fulltext pdf download fulltext pdf convex bodies and concave functions article pdf available in proceedings of the american mathematical society 1232. The convex functions are left unchanged and all concave functions are replaced by linearizations around the current solution point x k, i. To download right click on to be added take a look at sheet1. Discover the concept and criteria of convex and concave functions. Concave andquasiconcave functions 1 concaveandconvexfunctions 1. An image which can be formed on the screen is known as real image and the one which cannot be formed on screen is known as a virtual image. Optimality and stability in nonconvexnonconcave minmax. Convex and concave relaxations of implicit functions. With concave functions, solving maximization problems is so much easier. Concave mirrors and convex mirrors image formation, ray diagram. However, it remains an intriguing research challenge how local optimal points are defined and which algorithm can converge to such points.
1229 1050 1344 212 1093 305 386 1224 598 394 614 174 507 300 429 1150 524 1437 683 1022 1187 470 201 1492 167 1286 917 436 313 174 937 862 953 294 345 1149 769 316 54