Quasiconvex functions

A function $f$ is quasiconvex iff $\mathbf{dom} f$ is convex and for any $x,y \in \mathbf{dom} f$ and $0\leq \theta \leq 1$
$$f(\theta x + (1-\theta) y) \leq \max\{ f(x), f(y) \}$$  A continuous function $f: \mathbf{R} \rightarrow \mathbf{R}$ is quasiconvex iff at least one of the following conditions holds

• $f$ is nondecreasing
• $f$ is nonincreasing
• there is a point $c\in \mathbf{dom} f$ such that for $t \leq c$, $f$ is nonincreasing, and for $t \ge c$, $f$ is nondecreasing.

First order conditions

A continuous differentiable function $f: \mathbf{R}^n \rightarrow \mathbf{R}$ is quasiconvex iff $\mathbf{dom} f$ is convex and for all $x,y\in \mathbf{dom}f$ 

$$f(y) \leq f(x) \Rightarrow \nabla f(x)^T (y-x) \leq 0$$

Operations that preserve quasiconvexity

• nonnegative weighted maximum
  The function $$f(x) = \sup_{y\in C} (w_y g_y(x) )$$  where $w_y \ge 0$ and $g_y(x)$ is a parameterized family of quasiconvex functions.
• composition
• if $g$ is quasiconvex and $h$ is nondecreasing then, $f = h \circ g$ is quasiconvex
• composition of a quasiconvex function with an affine or linear-fractional transformation yields a quasiconvex function.
• minimization
  if $f(x,y)$ is quasiconvex jointly in $x$ and $y$ and $C$ is a convex set, then the function
  $$g(x) = \inf_{y\in C} f(x,y)$$ is quasiconvex.