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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 ...
• The function f is quasiconvex if for all x,y ∈ C and 0 ⩽ λ ⩽ 1, we have f ((1−λ)x+λy) ⩽ max{f(x),f(y)}. • The function is quasiconcave if for all x,y ∈ C and 0 ⩽ λ ⩽ 1, we have f ((1−λ)x+λy) ⩾ min{f(x),f(y)}. 7.2.2 Exercise Prove the following. 1. Every convex function is quasiconvex. Every concave function is ...
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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.
transformation of f is concave, then f is concave. I Example: Check whether the f(x;y) = xy + x2y2 + x3y3 de ned on <2 + is quasiconcave. Note that f(x) = g(u(x;y)) where u(x;y) = xy and g(z) = z + z2 + z3. Since g0 > 0, f is quasi-concave if and only if u is quasi-concave. But u(x;y) = ev(x;y) where v(x;y) = lnx + lny. The function v is easily ...
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Definition: A function is strictly quasiconvex if all of its lower contour sets are strictly convex sets and none of its level sets have any width (i.e., no interior). The first condition rules out straight-line level sets while the second rules out flat spots.
