Convex function - Wikipedia, the free encyclopedia. Convex function on an interval. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. Well- known examples of convex functions include the quadratic functionx. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. Simulation and Analysis of Various Routing Algorithms for Optical Networks Research supported in part by: June, 2004 LIDS Publication # 2615 NSF Grant ECS-0218328 Meli, A. This book, developed through class instruction at MIT over the last 15 years, provides an accessible, concise, and intuitive presentation of algorithms for solving convex optimization problems. It relies on rigorous. Convex Optimization – Boyd and Vandenberghe : Convex Optimization Stephen Boyd and Lieven Vandenberghe Cambridge University Press. A MOOC on convex optimization, CVX101, was run from 1/21/14 to 3/14/14. Convex Optimization Theory Athena Scienti. Bertsekas Massachusetts Institute of Technology Supplementary Chapter 6 on Convex Optimization Algorithms This chapter aims to supplement the book Convex. Model Library Author Index This is an listing of authors of the models available in the on-line model library. There is also an alphabetical index, a subject index, and a chronological index. Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense 'easier' than the general case - for example, any. Bertsekas, Dimitri (2003). Convex Analysis and Optimization. Borwein, Jonathan, and Lewis, Adrian. Convex Analysis and Nonlinear Optimization. For instance, a (strictly) convex function on an open set has no more than one minimum. Even in infinite- dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and, as a result, they are the most well- understood functionals in the calculus of variations. In probability theory, a convex function applied to the expected value of a random variable is always less than or equal to the expected value of the convex function of the random variable. This result, known as Jensen's inequality, underlies many important inequalities (including, for instance, the arithmetic. Exponential growth narrowly means . Suppose f is a function of one real variable defined on an interval, and let. R(x. 1,x. 2)=f(x. This characterization of convexity is quite useful to prove the following results. A convex function f defined on some open interval. C is continuous on C and Lipschitz continuous on any closed subinterval. As a consequence, f is differentiable at all but at most countably many points. If C is closed, then f may fail to be continuous at the endpoints of C (an example is shown in the examples' section). A function is midpoint convex on an interval C if. For example, a real valued Lebesgue measurable function that is midpoint convex will be convex by Sierpinski Theorem. A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non- decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable. For the basic case of a differentiable function from (a subset of) the real numbers to the real numbers, . A continuously differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents. A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non- negative there; this gives a practical test for convexity. Visually, a twice differentiable convex function . If its second derivative is positive at all points then the function is strictly convex, but the converse does not hold. For example, the second derivative of f(x) = x. Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum. For a convex function f, the sublevel sets . However, a function whose sublevel sets are convex sets may fail to be a convex function. A function whose sublevel sets are convex is called a quasiconvex function. Jensen's inequality applies to every convex function f. If X is a random variable taking values in the domain of f, then E(f(X)) . If a function f is convex, and f(0) . As an example, if f(x) is convex, then so is ef(x). A strongly convex function is also strictly convex, but not vice versa. A differentiable function f is called strongly convex with parameter m > 0 if the following inequality holds for all points x, y in its domain. Some authors, such as . Despite this, functions exist that are strictly convex but are not strongly convex for any m > 0 (see example below). If the function f is twice continuously differentiable, then f is strongly convex with parameter m if and only if . This is equivalent to requiring that the minimum eigenvalue of . If the domain is just the real line, then . If m = 0, then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that f. Start by using Taylor's Theorem: f(y)=f(x)+. If f is twice continuously differentiable and the domain is the real line, then we can characterize it as follows: f convex if and only if f. The proof of this statement follows from the extreme value theorem, which states that a continuous function on a compact set has a maximum and minimum. Strongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class. Like strictly convex functions, strongly convex functions have unique minima on compact sets. Uniformly convex functions. This is a generalization of the concept of strongly convex function; by taking . It is also strongly convex (and hence strictly convex too), with strong convexity constant 2. The function f(x)=x. It is strictly convex, even though the second derivative is not strictly positive at all points. It is not strongly convex. The absolute value function f(x)=. It is not strictly convex. The function f(x)=. It is also strictly convex, since f. More generally, the function g(x)=ef(x). It is concave on the interval (. Convex Analysis and Optimization. Borwein, Jonathan, and Lewis, Adrian. Convex Analysis and Nonlinear Optimization. Springer. Donoghue, William F. Distributions and Fourier Transforms. Hiriart- Urruty, Jean- Baptiste, and Lemar. Fundamentals of Convex analysis. Berlin: Springer. Krasnosel'skii M. A., Rutickii Ya. B. Convex Functions and Orlicz Spaces. Groningen: P. Noordhoff Ltd. Lauritzen, Niels (2. Undergraduate Convexity. World Scientific Publishing. Luenberger, David (1. Linear and Nonlinear Programming. Luenberger, David (1. Optimization by Vector Space Methods. Princeton: Princeton University Press. Thomson, Brian (1. Symmetric Properties of Real Functions. Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc.
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