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?
  • $\begingroup$ 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


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

  • $\begingroup$ 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
  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.
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.