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- A concave function can be quasiconvex. For example, x ↦ log (x) {displaystyle xmapsto log (x)} is both concave and quasiconvex. Any monotonic function is both quasiconvex and quasiconcave. More generally, a function which decreases up to a point and increases from that point on is quasiconvex (compare unimodality).
en.wikipedia.org/wiki/Quasiconvex_function
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Is a concave function a quasiconvex function?
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Is 0 1 A quasiconcave function?
A function $f(x)$ is said to be quasi-concave if its domain and all its $\alpha$-superlevel sets defined as $$\mathcal{S}_\alpha \triangleq\{x|x\in dom f, f(x)\geq\alpha\}$$ are convex for every $\alpha$. Another representation of quasi-concave functions: $f(x)$ is called quasi- concave if and only if
- convex analysis
Every monotone function is both quasiconvex and...
- convex analysis
In mathematics, a quasiconvex function is a real -valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints.
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) + +.
A weaker condition to describe a function is quasiconvexity (or quasiconcav-ity). Functions which are quasiconvex maintain this quality under monotonic transformations; moreover, every monotonic transformation of a concave func-tion is quasiconcave (although it is not true that every quasiconcave function
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• A function f: C → R is explicitly quasiconvex if it is quasiconvex and satisfies (f(y) < f(x) and 0 < λ < 1) =⇒ f ((1−λ)x+λy) < f(x). (E) • A function f: C → R is strictly quasiconcave if for every x,y ∈ C with x ̸= y. and every 0 < λ < 1, f(y) ⩾ f(x) =⇒ f ((1−λ)x+λy) > f(x). Equivalently, if for every x ̸= y and 0 ...
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Every monotone function is both quasiconvex and quasiconcave. Indeed, the definition of quasiconvexity amounts to saying that on every closed interval, the function attains its maximum at an endpoint.
Yes! Both linear and affine functions are both convex and concave. Example 21.1.2: Convex Functions. Any linear function f(x) = p·x is both concave and convex, as is the generic affine function f(x) = a+p·x.
