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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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This is just a quick and condensed note on the basic definitions and characterizations of concave, convex, quasiconcave and (to some extent) quasiconvex functions, with some examples. Contents. Concave and convex functions 1. 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.
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.
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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Anatomy of a Function def main(): mid = average(10.6, 7.2) print(mid) def average(a, b): sum = a + b return sum / 2 Think/Pair/Share: Find the function definition, function name, parameter(s), and return value in average.
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When you have to compute an angle, you need to divide, or use some approximation (anything involving Pi, for example) or some trigonometric function. When you have to compute an angle in code , you’ll almost always be approximating.
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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 ...
