quasiconvex optimization — Jemoka Knowledge Base

\begin{align} \min_{x}\quad & f_{0}\left(x\right) \\ \textrm{s.t.} \quad & f_{i}\left(x\right) \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}\left(x\right) \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}\left(x\right) \leq 0; f_{i}\left(x\right) \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 & \left(cTx + d\right) / \left(eTx + f\right) \textrm{s.t.} \quad & G x ≼ h, Ax = b \end{align} Von-Neumann Model of the Economy “allocate activity to maximize growth rate of slowest growing sector.”

\begin{align} \max_{x, x^{+}}\quad & \min \frac{x_{i}^{+}}{x_{i}} \\ \textrm{s.t.} \quad & 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.