Suggest edit — quasiconvex optimization

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---
title: "quasiconvex optimization"
type: concept
source: https://www.jemoka.com/posts/kbhquasiconvex_optimization/
confidence: high
status: active
---

\begin{align} \min_{x}\quad &amp; f_{0}\qty(x) \\ \textrm{s.t.} \quad &amp; f_{i}\qty(x) \leq 0, i = 1 \dots m\\ &amp;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:
\begin{equation} \phi_{t}\qty(x) \leq 0; f_{i}\qty(x) \leq 0, i = 1\dots m, Ax = b \end{equation}
if its feasible, then we can say \(t \geq p^{*}\), otherwise \(t \leq p^{*}\) where \(p^{ *}\) is the optimum. So you pick \(t\) by binary searching on feasibility until you get down to certain tolerance.
example linear-fractional program
`\begin{align} minx\quad &amp; \qty(cTx + d) / \qty(eTx + f) \textrm{s.t.} \quad &amp; G x ≼ h, Ax = b \end{align}
Von-Neumann Model of the Economy &ldquo;allocate activity to maximize growth rate of slowest growing sector.&rdquo;
\begin{align} \max_{x, x^{+}}\quad &amp; \min \frac{x_{i}^{+}}{x_{i}} \\ \textrm{s.t.} \quad &amp; x^{+} \succeq 0, Bx^{+} \preceq Ax \end{align}
where \(Ax\) is the amount of good produced in the current period \(Ax^{+ }\) is the amount of good consumed in the next period. Where \(x, x^{+} \in \mathbb{R}_{++}^{n}\) is the activity levels of \(n\) economic sectors; this and next period.