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  1. What's the intuitive difference between quasi-concavity and concavity?
  2. Can you give an example of a quasi-concave function that is not concave?
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  • $\begingroup$ Suppose $f$ is $C^1(\mathbb{R})$. A nice additional way of viewing the difference (and the reason it's cared about in economics, say) is that $f'$ is monotone decreasing if $f$ is concave, and is single crossing for the zero line if $f$ is quasi-concave (the importance lying in the ubiquitousness of first-order conditions). So in terms of getting a unique (or at least convex) set of solutions to the FOC, concavity is 'global', whereas quasi-concavity is only 'local'. $\endgroup$ Nov 1, 2017 at 17:29

2 Answers 2

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enter image description here

A function with this graph is quasiconcave, but not concave. It can be proved that a function $f$ is quasiconcave if and only there exists $x_0$ s.t. $f$ is nondecreasing for $x<x_0$ while $f$ is nonincreasing for $x>x_0$. I don't write precisely about a domain. It should be interval on a real line. Of course, quasiconcavity could be also defined on a plane or on any linear space.

Another characterization of quasiconcavity: $f:D\to\Bbb R$ (where $D$ is a convex subset of a linear space) is quasiconcave iff $$ f\bigl(tx+(1-t)y\bigr)\ge \min\{f(x),f(y)\} $$ for any $x,y\in D$, $t\in[0,1]$.

This graph consists of two convex pieces. Nevertheless, the whole function is quasiconcave. enter image description here

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  • $\begingroup$ Thank you for your help! If I understand quasi-concavity correctly, then every stricty increasing function must be quasi-concave, right? $\endgroup$
    – stollenm
    Nov 4, 2017 at 17:42
  • $\begingroup$ And also quasi-convex. :) Functions $f:\Bbb R\to\Bbb R$ which are both quasiconcave and quasiconvex are precisely monotonic functions. In higher dimensions they are called quasi-monotonic. $\endgroup$
    – szw1710
    Nov 4, 2017 at 19:48
  • $\begingroup$ Perfect, :) Thanks! $\endgroup$
    – stollenm
    Nov 4, 2017 at 20:41
  • $\begingroup$ Is there an example of an f(x,y) which is convex but also quasi concave? $\endgroup$
    – PGupta
    Jul 15, 2020 at 9:46
  • $\begingroup$ @Pgupta Any monotone convex function works for the example. Observe that any convex function is quasiconvex. So, in particular, we look for a function which is both quasiconvex and quasiconcave. In the case of one-dimensional domain, this is a monotone function (see my previous comments). $\endgroup$
    – szw1710
    Jul 15, 2020 at 17:31
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  1. 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

$$f(\theta x+(1-\theta)y) \geq \min\{f(x),f(y)\}$$

for all $x,y \in dom f$ and $\theta\in[0,1]$.

  1. A function $f(x)$ is said to be concave if the two conditions below satisfied:

    (i) $dom f$ is convex.

    (ii) For all $x,y\in dom f$ and $\theta\in[0,1],$

$$f(\theta x+(1-\theta)y) \geq \theta f(x) + (1-\theta)f(y)$$

In fact, this implies that if a function is convcave then that's also quasi-concave but not necessarily the converse is true.

  1. For example $f(\mathbf{X}) = \text{rank}(\mathbf{X})$ is a quasi-concave on $\mathbb{S}^n_{+}$.

  2. Ceiling function $f(x)= \lceil x \rceil$ is a quasi-concave function (also, it is quasi-convex which is called quasi-linear).


References for more info.:

  1. Convex Optimization for Signal Processing and Communications, by Chong-Yung Chi.
  2. Convex Optimization by Stephen Boyd.
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  • $\begingroup$ how is 2. an answer to OP's second question ? $\endgroup$ Nov 1, 2017 at 17:24
  • $\begingroup$ @GabrielRomon: you can refer to each reference above. Here, you can multiply both sides by negative sign and you will find the sublevel sets associated with the quasi-convex functions. $\endgroup$
    – Amin
    Nov 1, 2017 at 17:51

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