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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 $f(x)$ 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
Abstract. This is just a quick and condensed note on the basic definitions and characterizations of concave, convex, quasiconcave and (to some extent) quasiconvex functions, with some examples. Contents. Concave and convex functions 1. 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.
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 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 can be written as a monotonic ...
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KC Border Quasiconvex Functions I 7–3 7.2.4 Example Note that strict quasiconvexity implies Condition (E). The con-verse does not hold. In fact, Condition (E) alone does not guarantee even quasi-convexity. For instance, the function f: Rm → R defined by f(x) = 1 x = 0, 0 x ̸= 0 . is not quasiconvex, since the sublevel set {f < 1} = Rm \{0 ...
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This function is quasiconcave (its upper level sets are the sets of points above rectangular hyperbolae), but is not concave (for example, f(0, 0) = 0, f(1, 1) = 1, and f(2, 2) = 4, so that f((1/2)(0, 0) + (1/2)(2, 2)) = f(1, 1) = 1 < 2 = (1/2)f(0, 0) + (1/2)f(2, 2)).
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. The function f(x) = Pm l=1 x2 l is convex while f(x) = Pm l=1 x1/2 is concave. l The function f(x) = ex is convex, while f(x) = ln x is concave. .
