a quasiconvex function \(f: \mathbb{R}^{n} \to \mathbb{R}\) is quasiconvex if \(\text{dom } f\) is a convex set and the sublevel sets:
\begin{equation} S_{\alpha} = \qty {x \in \text{dom } f \mid f\qty(x) \leq \alpha } \end{equation}
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\qty(\theta x + \qty(1-\theta)y) \leq\max\qty{f\qty(y), f\qty(x)} \end{equation}
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first-order condition
differential \(f\) with convex domain is quasiconvex IFF
\begin{equation} f\qty(y) \leq f\qty(x) \implies \nabla f\qty(x)^{T} \qty(y - x) \leq 0 \end{equation}
second order condition
\begin{equation} y^{T}\nabla f\qty(x) = 0 \implies y^{T} \nabla^{2} f\qty(x) y \geq 0 \end{equation}
operations that preserve quasi-convexity
- non-negative weighted maximum