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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. 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.

3. 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) + +.

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

   davidrosenberg.github.io › mlcourse › Archive

   A function f is strictly convex if the line segment connecting any two points on the graph of f lies strictly above the graph (excluding the endpoints). Consequences for optimization:

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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

   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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   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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   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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   Which quadratic functions are convex and concave?

   • ne functions are the only functions that are both convex and concave. Some quadratic functions: f(x) = xT Qx + cT x + d. { Convex if and only if Q 0. { Strictly convex if and only if Q 0. 0; strictly concave if and only if Q 0. { The proofs are easy if we use the second order characterization of convexity (com-ing up).

   1 Theory of convex functions - Princeton University

   www.princeton.edu/~aaa/Public/Teaching/ORF523/S16/ORF523_S16_Lec7_gh.pdf

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   Are NE functions convex or concave?

   • They are convex, but not strictly convex; they are also concave: ne functions are the only functions that are both convex and concave. Some quadratic functions: f(x) = xT Qx + cT x + d. { Convex if and only if Q 0. { Strictly convex if and only if Q 0. 0; strictly concave if and only if Q 0.

   1 Theory of convex functions - Princeton University

   www.princeton.edu/~aaa/Public/Teaching/ORF523/S16/ORF523_S16_Lec7_gh.pdf

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   What is convex-concave programming?

   • 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. DCP is a structured way to de ne convex optimization problems, based on a family of basic convex and concave functions and a few rules for combining them.

   Disciplined Convex-Concave Programming - Stanford University

   web.stanford.edu/~boyd/papers/pdf/dccp.pdf

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8. 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,