a quasiconvex function f: \mathbb{R}^{n} \to \mathbb{R} is quasiconvex if \text{dom } f is a convex set and the sublevel sets:

are convex for all \alpha. These functions are also called unimodal functions. properties of quasiconvex functions modified Jensen’s Inequality \begin{equation} 0 \leq \theta \leq 1 \implies f\left(\theta x + \left(1-\theta\right)y\right) \leq\max\left{f\left(y\right), f\left(x\right)\right} \end{equation} l first-order condition differential f with convex domain is quasiconvex IFF

second order condition \begin{equation} y^{T}\nabla f\left(x\right) = 0 \implies y^{T} \nabla^{2} f\left(x\right) y \geq 0 \end{equation} operations that preserve quasi-convexity non-negative weighted maximum minimization over a variable composition with a non-decreasing function; i.e. general composition rule that preserve convexity, but the outside thing doesn’t have to be convex/quasiconvex