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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: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.
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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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 ...
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.
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)
A function f is strictly convex if the line segment connecting any two points on the graph of f lies strictly above the graph (excluding the endpoints). Consequences for optimization:
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CONCAVITY. The following notions of concavity are used to describe the increase and decrease of the slope of the tangent to a curve. Concavity If the function f(x) is differentiable on the interval a x b, then the graph of f is. concave upward on a x b if f is increasing on the interval concave downward on a x b if f is decreasing on the interval.
