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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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A function is concave (convex) if the graph of the function is always above (below) any chord (line segment between two points in the graph). Remark 4. f concave ⇔−f convex. Example 5. Let S = [0,∞) and consider f(x) = √ x and g(x) = −f(x) = − √ x f is a concave function and g is a convex function. 1.2 Selected properties of ...
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.
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
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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convex function is quasiconvex. Every c. Prove that the following statements are equivalent. The function f : C R is quasiconvex. →. (b) For all α R, the sublevel set x C : f(x) α is convex. ∈ { ∈ ⩽ } 7–2. (c) For all α R, the strict sublevel set x C : f(x) < α is convex. ∈ { ∈ }
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A positive scalar multiple of a concave (convex) function is concave (convex). The sum of two concave (convex) functions is concave (convex). If φ is concave (convex) and weakly increasing on R and f is a concave (convex) function, then φ f is concave (convex).
The function f of many variables defined on a convex set S is quasiconvex if every lower level set of f is convex. (That is, P a = { x ∈ S : f ( x ) ≤ a } is convex for every value of a .)
