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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) + +.

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     Concavity of Functions EXPLAINED with Examples

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     Concavity, Inflection Points, and Second Derivative

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     Calculus I - Concavity and Inflection Points - Example 1

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3. 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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4. 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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5. 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 ...

6. 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.

7. ecoursesonline.iasri.res.in › pluginfile › 74Lesson 12 Convexity, Concavity and Points of Inflexion

   ecoursesonline.iasri.res.in › pluginfile › 74

   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.

8. People also ask

   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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   Is concavity a necessary condition for a function?

   • 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.

   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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   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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   Is F a convex function?

   • f is convex. Note: linear functions are convex and concave. Concave and convex functions need to have convex sets as their domain. Otherwise, we cannot use the conditions above. Why do we need concavity and convexity? The function h is convex. B is convex.

   Lecture 3- Concavity and convexity - London School of Economics and

   personal.lse.ac.uk/RAZIN/lecture3.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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9. kircher.mae.cornell.edu › cvxaConvex Analysis Notes - MIT

   kircher.mae.cornell.edu › cvxa

   They cover the basic theory of convex sets and functions, several avors of duality, a variety of optimization algorithms (with a focus on nonsmooth problems), and an introduction to variational analysis building up to the Karush-Kuhn-Tucker conditions. Proofs are mostly omitted.