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

   pareto.uab.cat › xvila › Micro_I

   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.

3. personal.lse.ac.uk › RAZIN › lecture3Lecture 3- Concavity and convexity - London School of Economics

   personal.lse.ac.uk › RAZIN › lecture3

   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 ...

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

   www.mitrikitti.fi › concavity

   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. Such functions are called quasi-concave functions. They are general-ized concave functions, since it is easy to show that every concave function is quasiconcave, but not ...

5. statmath.wu.ac.at › courses › MEC_MMConvex and Concave - WU

   statmath.wu.ac.at › courses › MEC_MM

   A function f can be convex in some interval and concave in some other interval. For two times continuously differentiable functions(i.e., when f 00(x) is continuous)we can use the following procedure: 1. Compute second derivative f 00(x). 2. Determine all roots of f 00(x). 3. We thus obtain intervals where f 00(x) does not change sign. 4.

6. 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.

7. People also ask

   What is an example of a concave function?

   • Example Concave Function Logarithm function: (x>0) f(x) =ln(x) f0(x) =1 x f00(x) =1 x2<0for allx>0 ln(x)is (strictly) concave. 1 1 e Josef Leydold Foundations of Mathematics WS 2023/2413 Convex and Concave 20 / 45 Locally Convex Functions A functionfcan be convex in some interval and concave in some other interval.

   Convex and Concave - WU

   statmath.wu.ac.at/courses/MEC_MM/download/handouts/MMEcon-handouts-13-Convex_and_Concave.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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   What is generalized concavity?

   • Generalized Concavity deals with the many nonconcave functions that have properties similar to those of concave functions.

   Generalized Concavity | SIAM Publications Library

   epubs.siam.org/doi/book/10.1137/1.9780898719437

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   How do we define a family of functions based on convexity?

   • We define a family of functions by the convexity of their upper-level sets. Such functions are called quasi-concave functions. They are general-ized concave functions, since it is easy to show that every concave function is quasiconcave, but not conversely. Definition 21 Let f be defined on the convex set C ⊂ Rn.

   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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   Is f g a concave function?

   • where the first inequality holds since f is concave and g increasing, and the second inequility follows from g being concave. We have thus proved that Thus, (f ◦ g) is a concave function. (with a similar proof for convexity.) Remark 10. An increasing transformation of a concave (convex) function is not necessarily concave (convex).

   Notes on Concavity, Convexity, Quasiconcavity and Quasiconvexity

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

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8. epubs.siam.org › doi › bookGeneralized Concavity - SIAM Publications Library

   epubs.siam.org › doi › book

   Various generalizations of concavity (studied in Chapter 3) preserve properties (i) and (ii), respectively. In Chapter 2, we also study two classes of functions that are more restrictive than the class of concave functions: strictly concave and strongly concave functions.