# Fused Multiple Graphical Lasso

### 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:
1. 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