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1. pareto.uab.cat › xvila › Micro_INotes on Concavity, Convexity, Quasiconcavity and Quasiconvexity

   pareto.uab.cat › xvila › Micro_I

   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) + +.

2. math.ucr.edu › ~res › math153-2019Concave and Convex Functions - Department of Mathematics

   math.ucr.edu › ~res › math153-2019

   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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3. www.mitrikitti.fi › concavity1 Concave and convex functions - Mitri Kitti

   www.mitrikitti.fi › concavity

   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.

4. web.stanford.edu › 6-Python_FunctionsPython Functions - Stanford University

   web.stanford.edu › 6-Python_Functions

   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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5. www.princeton.edu › S16 › ORF523_S16_Lec7_gh1 Theory of convex functions - Princeton University

   www.princeton.edu › S16 › ORF523_S16_Lec7_gh

   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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6. www.cse.iitd.ac.in › slides › 5-Python-FunctionsPython: Functions - IIT Delhi

   www.cse.iitd.ac.in › slides › 5-Python-Functions

   Python: Functions. Mathematical functions f(x) = x2. f(x,y) = x2 + y2. In programming functions also help creating better structure with decomposition . Consider. f(x) = x2. def square(x): return x*x. square(4) 16 #defining function. #invoking function. # output. Example: Functions may not have arguments, and return statement .

7. People also ask

   What is the concavity of a function?

   • This notion is called the concavity of the function. Figure 4.34(a) shows a function f with a graph that curves upward. As x increases, the slope of the tangent line increases. Thus, since the derivative increases as x increases, f′ is an increasing function. We say this function f is concave up.

   Concavity and Points of Inflection - University of North Georgia

   faculty.ung.edu/mkim/Teaching/Math 2040/Project/Higher Derivatives, Concavity, and the Second Derivative Test .pdf

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   Why are concave functions important?

   • The importance of concave (and convex) functions from the optimization point of view lies in some properties of concave functions with regard to their extrema. Let f be a concave function defined on a convex set C ⊂ Rn. Then every local maximum of f at x∗ ∈ C is a global maximum of f over all C.

   1 Concave and convex functions - Mitri Kitti

   www.mitrikitti.fi/concavity.pdf

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   How do you analyze concavity if n 1 is separable?

   • the de nition of concavity.special case in which N > 1 is e ectively as easy to analyze as N = 1 is when the function is separable in the sense that it is the sum of univariate functions. For example, the function h(x1; 2) = ex1 + ex2 is separable. More generally, suppose Cn ar intervals a d let C = QCn. Then h : C ! R is separable i

   Concave and Convex Functions | Department of Mathematics

   math.ucr.edu/~res/math153-2019/concavity.pdf

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   Is F a concave function?

   • f(x) ≤ f(x0) + f0(x0)(x − x0). It is strictly concave if and only if the inequality is strict for x 6= x0. Theorem 6 Let f be a differentiable function on the open convex set C ⊂ R. It is concave (strictly concave) if and only if f0 is a nonincreasing (decreasing) function. C. Then f is a concave function if and only if f ”(x) ≤ 0 for every x ∈ C.

   1 Concave and convex functions - Mitri Kitti

   www.mitrikitti.fi/concavity.pdf

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   How do we generalize concave functions?

   • We begin generalizing concave functions by recalling from the preceding chap-ter that the upper-level sets of concave functions are convex sets. Concavity of a function is a sufficient condition for this property, but not a necessary one. We define a family of functions by the convexity of their upper-level sets.

   1 Concave and convex functions - Mitri Kitti

   www.mitrikitti.fi/concavity.pdf

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   How do you prove a concave function?

   • Any afine function is both concave and convex. Proof. The proof follows from Theorem 6 above and from the fact that f(x) xi, f(x) −xi, and f(x) Theorem 9. Let f : S ⊂ Rn → R be a concave (convex) function, and let g : R → R be concave (convex) and increasing. Then (f ◦ g) : S ⊂ Rn → R is a concave (convex) function. Proof.

   Notes on Concavity, Convexity, Quasiconcavity and Quasiconvexity

   pareto.uab.cat/xvila/Micro_I/ConcavityConvexityQuasiconcaviyQuasiconvexity.pdf

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8. faculty.ung.edu › mkim › TeachingConcavity and Points of Inflection - University of North Georgia

   faculty.ung.edu › mkim › Teaching

   This notion is called the concavity of the function. Figure 4.34(a) shows a function f with a graph that curves upward. As x increases, the slope of the tangent line increases. Thus, since the derivative increases as x increases, f′ is an increasing function. We say this function f is concave up.