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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
For example, f(x) transformation, but g(f(x)) = = −x2 is concave, and g(x) 2 = ex is a monotonic e−x2. 2 is not concave. This is problematic when we want to analyze things like utility which we consider to be ordinal concepts. A weaker condition to describe a function is quasiconvexity (or quasiconcav-ity).
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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.
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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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.
examples. 2 piecewise-linear function: f(x) = maxi=1;:::;m(aT i x + bi) is convex 2 sum of r largest components of x 2 Rn: f(x) = x[1] + x[2] + ¢ ¢ ¢ + x[r] is convex (x[i] is ith largest component of x) proof: f(x) = maxfxi1+ xi2 + ¢ ¢ ¢ + xir j 1. · i1 < i2 < ¢ ¢ ¢ < ir · ng. Convex functions.
