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1. mycourses.aalto.fi › content › 2CVX Tutorial (Python), - Lecture 6 - Aalto

   mycourses.aalto.fi › content › 2

   f(expr1,expr2,...,expr3) is convex if f is a convex function and for each expri one of the following conditions hold: f is increasing in argument i and expri is convex. f is decreasing in argument i and expri is concave. expri is affine or constant. Similar logic is applied to establishing concavity or affinity of a function. If a function

2. docs.scipy.org › doc › scipyOptimization (scipy.optimize) — SciPy v1.14.1 Manual

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   The implementations shown in the following sections provide examples of how to define an objective function as well as its jacobian and hessian functions. Objective functions in scipy.optimize expect a numpy array as their first parameter which is to be optimized and must return a float value.

3. web.stanford.edu › lectures › cvx_lecture_slidesDisciplined Convex Programming and CVX - Stanford University

   web.stanford.edu › lectures › cvx_lecture_slides

   Disciplined convex programming • describe objective and constraints using expressions formed from – a set of basic atoms (affine, convex, concave functions)

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4. 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) + +.

5. www.geeksforgeeks.org › concavity-and-points-ofConcavity and Points of Inflection - GeeksforGeeks

   www.geeksforgeeks.org › concavity-and-points-of

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   May 22, 2024 · Concavity provides valuable insights into how a function curves, distinguishing between concave upward and concave downward shapes, while points of inflection mark locations where the curvature changes sign.

6. davidrosenberg.github.io › mlcourse › ArchiveLagrangian Duality and Convex Optimization - GitHub Pages

   davidrosenberg.github.io › mlcourse › Archive

   The Lagrange dual function (or just dual function ) is g( )=inf x L(x, )=inf x f0 (x)+ Xm i=1 if i(x)!. The dual function may take on the value -1(e.g. f0 (x)=x). The dual function is always concave since pointwise min of a ne functions Julia Kempe & David Rosenberg (CDS, NYU) DS-GA 1003 February 19, 201921/31

7. People also ask

   Why is concavity important in mathematics?

   • Concavity refers to the shape of a curve or function, indicating whether the curve curves upward (convex) or downward (concave). It is crucial in mathematics as it provides insights into the behavior of functions, helping analyze the curvature changes and determine key properties such as maxima, minima, and inflection points.

   Concavity and Points of Inflection - GeeksforGeeks

   www.geeksforgeeks.org/concavity-and-points-of-inflection/

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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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   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 a dual function concave or convex?

   • The dual function may take on the value -1(e.g. f0(x)=x). The dual function is always concave since pointwise min of ane functions Julia Kempe & David Rosenberg (CDS, NYU) DS-GA 1003 February 19, 201921/31 Lagrangian Duality: Convexity not required The Lagrange Dual Problem: Search for Best Lower Bound

   Lagrangian Duality and Convex Optimization - GitHub Pages

   davidrosenberg.github.io/mlcourse/Archive/2019/Lectures/04a.convex-optimization.pdf

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   How do you know if a function is concave or convex?

   • A function is concave (convex) if the graph of the function is always below (above) the graph of the tangent to the function. Theorem 29. Let Let f : S ⊂ Rn → R be a C2 function. Then, f is strictly convex if the Hessian matrix D2 f(x) is positive definite for all x ∈ S . with similar implications for convexity and positive definiteness.

   Notes on Concavity, Convexity, Quasiconcavity and Quasiconvexity

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

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   What is a concave downward function?

   • In concave downward functions, the slope decreases as you move from left to right. They often have a maximum point, where the slope changes from positive to negative, representing the peak of the curve. Examples: g (x)=-x2 and g (x)=log (x). Mathematical Definition of Concavity

   Concavity and Points of Inflection - GeeksforGeeks

   www.geeksforgeeks.org/concavity-and-points-of-inflection/

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8. web.stanford.edu › ~boyd › papersDisciplined Convex-Concave Programming - Stanford University

   web.stanford.edu › ~boyd › papers

   Convex-concave programming is an organized heuristic for solving nonconvex problems that involve objective and constraint functions that are a sum of a convex and a concave term.