## Monday, March 9, 2020

### Parametrized Fusion - benchmarks

We present some indicative benchmarks of the new Parametrized Fusion, to be released in MOSEK 9.2. We compare a few typical optimization models being reoptimized in two scenarios:

• the model is constructed from scratch with new data for each optimization, MOSEK version 9.1 (green bars)
• the parametrized model is constructed once and parameters are set before every optimization, MOSEK version 9.2 (blue bars)
The red bar in all cases represents the amount of time spent in the optimizer.

## 1. Markowitz portfolio optimization

Here we consider a simple model
$$\begin{array}{ll}\mbox{maximize} & \mu^T x \\ \mbox{subject to} & e^Tx=1,\\ & \gamma \geq \|Fx\|_2, \\ & x\geq 0, \end{array}$$

where
• $x\in\mathbb{R}^{2000}$,
• $F\in \mathbb{R}^{200\times 2000}$ is a dense matrix,
• $\gamma$ is the only parameter changed between optimizations,
• we resolve the model for 200 values of $\gamma$.
Note that by parametrizing we avoid inputting the same big matrix $F$ over and over again.

## 2. Various models

Next we run a few models implemented in Python only.
• MPC - Model Predictive Control similar as in this demo with 15 states, 8 inputs, horizon length $T=80$, parametrized by the initial state, 100 resolves,
• Polytopes - a geometric polytope containment problem, the parameter is a $3\times 3$ matrix $P$ which enters in many constraints of the form $Px=y$, 1000 resolves.
• Elastic - a linear regression model with main term $\|Ax-b\|_2$, where $A\in\mathbb{R}^{3000\times 100}$ is dense, parametrized by $b\in\mathbb{R}^{3000}$, 60 resolves.
• Option - a big option pricing model with complicated structure and multiple parameters, 10 resolves. 