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Theorem 3. Let C R be an open interval. 1. f: C!R is concave i for any a;b;c2C, with a<b<c, f(b) f(a) b a f(c) f(b) c b; and, f(b) f(a) b a f(c) f(a) c a: For strict concavity, the inequalities are strict. 2. f: C!R is convex i for any a;b;c2C, with a<b<c, f(b) f(a) b a f(c) f(b) c b; and, f(b) f(a) b a f(c) f(a) c a:
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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) + +.
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.
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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Concavity. Definition. Let f be a differentiable function on the open interval I. Let Ta(x) = f′(a)(x − a) + f(a), ∀ a ∈ I. This is the tangent line to. at x = a. If ∀a ∈ I, f(x) > Ta(x) ∀x ∈ I − {a}, then we say f is concave up. If ∀a ∈ I, f(x) < Ta(x) ∀x ∈ I − {a}, then we say f is concave down. Theorem.
Definition. An inflection point of a function f is a point where f changes the direction of concavity. Theorem 8: c is an inflection point =⇒ f′′(c) = 0. Example 3. For the function f(x) = x3, we have f′(x) = 3x2 and f′′(x) = 6x, so this function is concave up on (0, +∞), and is concave down on (−∞, 0). y.
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You are probably familiar with the notion of concavity of functions. Given a twice-di erentiable function ’: R !R, We say that ’is convex (or concave up) if ’00(x) 0 for all x2R. We say that ’is concave (or concave down) if ’00(x) 0 for all x2R. For example, a quadratic function ’(x) = ax2 + bx+ c is convex if a 0, and is concave if ...
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