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1.1 Definitions. Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have:
It is also possible to characterize concavity or convexity of functions in terms of the convexity of particular sets. Given the graph of a function, the hypograph of f,
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May 22, 2024 · Concavity provides valuable insights into how a function curves, distinguishing between concave upward and concave downward shapes, while points of inflection mark locations where the curvature changes sign.
Concavity of a function is a sufficient condition for this property, but not a necessary one. We define a family of functions by the convexity of their upper-level sets. Such functions are called quasi-concave functions. They are general-ized concave functions, since it is easy to show that every concave function is quasiconcave, but not ...
Sep 22, 2024 · Definition of Concave Functions. A function g (x) is called concave on an interval if, for any two points x_1 x1 and x_2 x2 in the interval and any \lambda \in [0, 1] λ∈ [0,1], the following holds: g (\lambda x_1 + (1-\lambda) x_2) \geq \lambda g (x_1) + (1-\lambda) g (x_2) g(λx1 +(1−λ)x2) ≥ λg(x1)+(1−λ)g(x2)
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.
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A single variable function f is said to be convex if f λx + (1−λ)y ≤ λf (x) +(1− λ) f (y), (1.1) for 0 < λ < 1. Problem 1.1. Execute the following python script. Is lnx convex or concave? import numpy as np import matplotlib . pyplot as plt # Plotting log ( x ) x = np . linspace (1 ,8 ,50)# points on the x axis f=np . log (x ...
