What are examples illustrating why quasilinear function is different from monotone function?

Question

A quasilinear function is a function that is both quasiconvex and quasiconcave. Are there quasilinear functions that are not monotone? Are there monotone functions that are not quasilinear? If so, an example of each would be great.

Answer 1

We say $f$ is monotone for a function $f$ defined on one dimension $\mathbb{R}$. For higher dimension $\mathbb{R}^n$ ($n\ge 2$), there is a generalization of the relation between quasilinear and monotone.

See 1, page 122, Exercise 3.46

1) For a function $f: \mathbb{R} \to \mathbb{R}$, $f$ is quasilinear if and only if $f$ is monotone.

2) For a function of $f: \mathbb{R}^n \to \mathbb{R}$ ($n\ge 2$), if $f$ is continuous quasilinear, then there is a monotone function $g: \mathbb{R} \to \mathbb{R}$ and $a \in \mathbb{R}^n$ such that $f(x) = g(a^T x)$. The converse is also true.

The proof from the solution manual:

References

1 Boyd and Vandenberghe, "Convex optimization". http://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf

Answer 2

According to Wikipedia’s references, a real-valued function $f$ defined on a convex subset $S$ of $\Bbb R$ is quasilinear iff for each $x,y\in S$ and each $\lambda\in [0,1]$ we have $$\min\{f(x),f(y)\}\le f(\lambda(x)+(1-\lambda)y)\le \max\{f(x),f(y)\}.$$ Since $\lambda(x)+(1-\lambda)y$ is a point of the segment with the endpoints $x$ and $y$ and the minimum and the maximum of a function monotone on a segment are attained at the endpoints of the segment, each monotone function on $S$ is quasilinear on $S$.

On the other hand, each quasilinear function $f$ on $S$ is monotone on $S$. Indeed, suppose to the contrary that $f$ is non-monotone on $S$. It easily follows that there exists points $x<z<y$ of $S$ such that $f(x)<f(z)>f(y)$ or $f(x)>f(z)<f(y)$. Then $z=\lambda(x)+(1-\lambda)y$ for some $\lambda\in (0,1)$. In the first case $\max\{f(x),f(z)\}<f(\lambda(x)+(1-\lambda)y)$, so $f$ is non-quasiconvex on $S$. Similarly, in the second case $f$ is non-quasiconcave on $S$.