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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) + +.
5.6 Concave and convex functions An extended real-valued function f on a convex set C is concave if its hypograph {(x,α) ∈ Rm: f(x) ⩾ α} is a convex set, or equivalently if f (1−λ)x+λy ⩾ (1−λ)f(x)+λf(y), (0 < λ < 1). KC Border src: L05 v. 2020.09.30::14.29
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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 ...
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.
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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A Calculus of Convex Functions Practical methods for establishing convexity of a function: - verify definition (often simplified by restricting to a line) - for twice di↵erentiable function, show its Hessian is PSD - show that f is obtained from simple convex functions by operations that preserve convexity – nonnegative weighted sum
