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A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
ne functions are the only functions that are both convex and concave. Some quadratic functions: f(x) = xT Qx + cT x + d. { Convex if and only if Q 0. { Strictly convex if and only if Q 0. { Concave if and only if Q. 0; strictly concave if and only if Q 0.
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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) + +.
f(b)(c a) f(a)(c b) + f(c)(b a); which (since c a > 0) holds i. f(b)baf(a) +aaf(c):Take = (c b)=(c a) 2 (0; 1) and verify. that, indeed, b = a + (1 )c. Then the last inequal. ty holds since f is concave. Conversely, the preceding argument shows that if the rst inequality in (1) holds then f is.
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A function f : Rn ! R is convex if dom f is a convex set and. ( x + (1. )y) f (x) + (1. )f (y) for all x, y 2 dom f and . 2 [0, 1] f is strictly convex if the above holds with “ ” replaced by “<”. f is concave if.
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When f'' (x) \textcolor {purple} {> 0}, we have a portion of the graph where the gradient is increasing, so the graph is convex at this section. When f'' (x) \textcolor {red} {< 0}, we have a portion of the graph where the gradient is decreasing, so the graph is concave at this section.
Convex functions Definition f : Rn → R is convex if dom f is a convex set and f(θx +(1−θ)y) ≤ θf (x) +(1−θ)f (y) for all x,y ∈ dom f, and θ ∈ [0,1]. f is concave if −f is convex f is strictly convex if dom f is convex and f(θx +(1−θ)y) < θf (x) +(1−θ)f (y) for all x,y ∈ dom f, x 6= y, θ ∈ (0,1).
