### FMGL: a MATLAB package for Fused Multiple Graphical Lasso

#### Yangjing Zhang, Ning Zhang, Defeng Sun, Kim-Chuan Toh

The software was first released on September 2020.

The software is designed to solve fused multiple graphical lasso (FMGL) problems of the following form given input data \(S=(S^{(1)},\ldots,S^{(L)})\)

$$

\min\Big\{ \sum_{l=1}^L \Big(-\log {\rm det} \Theta^{(l)} + \langle S^{(l)}, \Theta^{(l)} \rangle\Big) + P(\Theta)\mid \Theta = \big(\Theta^{(1)},\ldots,\Theta^{(L)}\big)\in

\mathbb{S}^p\times \cdots\times \mathbb{S}^p\Big\}

$$

where \(

P(\Theta) = \lambda_1\,\sum_{l=1}^L \sum_{i\not=j} \left| \Theta^{(l)}_{ij}\right| + \lambda_2\,\sum_{l=2}^L \sum_{i\not=j}\left| \Theta^{(l)}_{ij} – \Theta^{(l-1)}_{ij} \right|

\)

and \(\lambda_1\) and \(\lambda_2\) are positive regularization parameters. The dual problem is given by

$$

-\min \Big\{\sum_{l=1}^{L} ( -\log{\rm det}(X^{(l)}) – p ) + P^*(X-S) \mid X=(X^{(1)},\ldots,X^{(L)})\Big\}

$$

where \(P^*(\cdot)\) denotes the conjugate function of \(P\).

Solver: FMGL_PPA.m

Important note: this is a research software. It is not intended nor designed to be a general purpose software at the moment.

##### Citation:

- N. Zhang, Y.J. Zhang, D.F. Sun, and K.C. Toh,
**An efficient linearly convergent regularized proximal point algorithm for fused multiple graphical Lasso problems,**arXiv:1902.06952

**Copyright:** This version of **FMGL** is distributed under the **BSD 3-clause** **License**.

###### Download here: FMGL-0.zip

###### Please read. Welcome to FMGL-0!

- Firstly, unpack the software.
- Run Matlab in the directory FMGL-0.
- After that, to see whether you have installed FMGL-0 correctly, type:

>> startup >> test_FMGL - By now, FMGL is ready for you to use.