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Each atomic function has a sign and curvature. Using composition rules from convex analysis, DCP tries to propagate the curvature and sign from atomic functions to the complete expression.
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)
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) + +.
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 ...
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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The function f is convex if for all x; y 2 X and for all 2 [0; 1], we have: f (x) is concave. If f (x) is convex, then af (x) is convex if a > 0. If f (x) and g (x) are convex, then h (x) = f (x) + g (x) is convex. If f (x) and g (x) are convex, then h (x) = f (x) g (x) is not necessar-ily convex.
