Search results
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.
Apr 18, 2023 · In this mathematics article, we will study what is convex functions, strictly convex function, proper convex function, techniques for identifying convexity, and properties of convex functions through worked-out examples.
Sep 22, 2024 · In mathematics, convex and concave functions are functions with different curvature of graphs. A convex function curves upwards, meaning that any line segment connecting two points on the curve will lie above or on the graph.
What is a Convex Function? Closed Convex Function; Jensen’s Inequality; Convex Function Definition. A convex function has a very distinct ‘smiley face’ appearance. A line drawn between any two points on the interval will never dip below the graph.
A function \(f: I \rightarrow \mathbb{R}\) is convex if and only if for every \(\lambda_{i} \geq 0, i=1, \ldots, n\), with \(\sum_{i=1}^{n} \lambda_{i}=1\) \((n \geq 2)\) and for every \(x_{i} \in I\), \(\i=1, \dots, n\), \[f\left(\sum_{i=1}^{n} \lambda_{i} x_{i}\right) \leq \sum_{i=1}^{n} \lambda_{i} f\left(x_{i}\right) .\]
Definition of Convexity of a Function. Consider a function y = f (x), which is assumed to be continuous on the interval [a, b]. The function y = f (x) is called convex downward (or concave upward) if for any two points x1 and x2 in [a, b], the following inequality holds:
In this lecture, we shift our focus to the other important player in convex optimization, namely, convex functions. Here are some of the topics that we will touch upon: Convex, concave, strictly convex, and strongly convex functions. First and second order characterizations of convex functions.
A function is called convex if the line segment connecting any two points on the graph lies above the graph. Formally, a function f : Rn ! R is called convex i. f( x + (1. )y) f(x) + (1. )f(y); 8x; y 2 Rn; 0 < < 1. We have assumed the domain to be the entire Rn space.
Convex functions. Let f : I ! R be a real valued function on an interval I. De nition. The function f is weakly convex if, for every a; b 2 I and every t 2 (0; 1), f((1 t)a + tb) (1 t)f(a) + tf(b): The function f is strictly convex if, for every a; b 2 I and every t 2 (0; 1), f((1 t)a + tb) < (1.
Convexity is a term that pertains to both sets and functions. For functions, there are di erent degrees of convexity, and how convex a function is tells us a lot about its minima: do they exist, are they unique, how quickly can we nd them using optimization algorithms, etc.
