quasiconvex function
Last edited: February 2, 2026a 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
quasiconvex optimization
Last edited: February 2, 2026\begin{align} \min_{x}\quad & f_{0}\qty(x) \\ \textrm{s.t.} \quad & f_{i}\qty(x) \leq 0, i = 1 \dots m\\ &Ax = b \end{align}
Where \(f_{0}\) is quasiconvex, and \(f_{i\geq 1}\) is convex.
solution methods
convex representation of sublevel sets
if \(f_{0}\) is quasiconvex, then there exists a family of functions \(\phi_{t}\) for which:
\begin{equation} f_{0} \leq t \iff \phi_{t}\qty(x) \leq 0 \end{equation}
where \(\phi_{t}\) is convex for fixed \(t\).
bisection method for quasiconvex optimization
We can cast all quasiconvex problems into a binary search over \(t\). Solve the following convex feasibility problem:
SU-EE364A FEB032026
Last edited: February 2, 2026Key Sequence
Notation
New Concepts
- Strong and Weak Duality
- Geometric Interperattion of the Dual
- preturbation analysis
- dual transformation
- theorem of alternatives