Quasiconvex function

A function $$f:\mathbb R^n \to R$$ is called quasiconvex if its domain and all its sublevel sets

$$S_\alpha = \{x \in \textbf{dom} f : f(x) \le \alpha\}$$

for $$\alpha \in \mathbb R$$, are convex. A function is quasiconcave if $$-f$$ is quasiconcave. A function is quasilinear if it's both quasiconvex and quasiconcave.

Intuitively, quasiconvex functions are unimodal functions

Quasiconvexity criteria
The following inequality, called modified Jensen inequality, holds for all quasiconvex functions:

$$f(\theta x+(1-\theta)y) \le \max\{f(x), f(y)\}$$

Important: A function is quasiconvex iff it's quasiconvex when restricted to a line. This is an important criterion -- it can be used to quickly and roughly check if a function is not quasiconvex.

Differentiable quasiconvex functions
A differentiable function $$f$$ is quasiconvex iff $$\textbf{dom} f$$ is convex and for all $$x,y \in \textbf{dom}f$$

$$f(y) \le f(x) \implies \nabla f(x)^T (y-x) \le 0$$

If $$f$$ is twice differentiable, then it's quasiconvex iff

$$y^T \nabla f(x) 0 \implies y^T \nabla^2 f(x) y \ge 0$$

Operations that preserve quasiconvexity
A nonnegative weighted maximum of quasiconvex functions,

$$f = \max\{w_1f_1, \dots, w_mf_m\}$$,

with $$w_i\ge 0$$ and $$f_i$$ quasiconvex, is quasiconvex. Extends to supremum as well.

Examples

 * $$\log x$$ is quasilinear
 * The ceiling function $$\text{ceil}(x) = \inf \{z \in \mathbb Z : z \ge x\}$$ is quasiconvex
 * The function $$f(x_1,x_2) = x_1 x_2$$ is neither convex nor concave, but it's quasiconcave (on $$\mathbb R^2_{+}$$).
 * Linear-fractional function $$f(x) = \frac{a^T x+b}{c^Tx +d}$$ with $$\textbf{dom} f = \{c^Tx+d > 0\}$$ is quasilinear.
 * Distance ratio function, $$f(x) = \frac{||x-a||_2}{||x-b||_2}$$ is quasiconvex on the halfspace $$\{x: ||x-a||_2 \le ||x-b||_2\}$$.
 * $$\text{rank}(X+Y)$$ is quasiconcave
 * The maximum generalized eigenvalue, $$\lambda_{\max}(X,Y) = \sup \frac{u^T X u}{u^T Y u}$$ is quasiconvex on $$\textbf{dom} f = \mathbb S^n \times \mathbb S^n_{++} $$.