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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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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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1 Concave and convex functions. Definition 1 A function f defined on the convex set C ⊂ Rn is called con-cave if for every x1, x2 ∈ C and 0 ≤ t ≤ 1, we have. f(tx1 + (1 − t)x2) ≥ tf(x1) + (1 − t)f(x2). Definition 2 A function f defined on the convex set C ⊂ Rn is called strictly concave if for every x1 6= x2,and 0 < t < 1, we have.
Convexity, Concavity and Points of Inflexion . 12.1 Introduction . In the plane, we consider a curve , which is the graph of a single - valued differentiable function . Definition 12.1: We say that the curve is convex downward bending up on the interval . if all points of the curve lie above the tangent at any point on the interval.
Why do we need concavity and convexity? We will make the following important assumptions, denoted by CC: 1. The set Z is convex; 2. The function g is concave; 3. The function h is convex. Recall the de–nition of the set B : B = f(k;v) : k h(z);v g(z) for some z 2 Zg: Proposition under CC, the set B is convex Proof: suppose that (k 1;v 1) and ...
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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.
