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. …

In the previous couple of lectures, we’ve been focusing on the theory of convex sets. In this lecture, we shift our focus to the other important player in convex optimization, namely, convex …

Apr 18, 2023 · A convex function is a mathematical function that satisfies a particular property known as convexity. A function is convex if for any two points on its graph, the line segment …

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 …

What is a Convex Function? 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.

Apr 30, 2025 · A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.

Definition 1: A function f : Rn ! R is called convex if for any pair of non-equal x1; x2 in the domain of f (which is assumed to be a closed convex set) and any pair of real positive numbers 1; 2, …

Feb 9, 2018 · An extended real-valued function f defined on a subset Ω of a vector space V over the reals is said to be convex if its epigraph is a convex subset of V × ℝ. With this definition, …

Function $$$ {y}={f{{\left({x}\right)}}} $$$ that is defined and continuous on interval $$$ {X} $$$ is called concave (or convex upward or concave downward) if for any $$$ {a} $$$ and $$$ {b} …

CONVEX FUNCTION Theorem 1: A function 𝑓𝑋is convex if for any two points 𝑋1 and 𝑋2, we have 𝑓𝑋2 R𝑓𝑋1 +𝛻𝑓𝑇𝑋1 𝑋2−𝑋1 Proof: If 𝑓𝑋is convex, we have 𝑓𝜆𝑋2+1−𝜆𝑋1 Q𝜆𝑓𝑋2 +1−𝜆𝑓𝑋1 𝑓𝑋1+𝜆𝑋2−𝑋1 Q𝑓𝑋1 +𝜆𝑓𝑋2 −𝑓𝑋1 𝜆𝑓𝑋2 −𝑓𝑋1