The problem of maximizing the sum of linear functional and several weighted logarithmic determinant (logdet) functions under semidefinite constraints is a generalization of the semidefinite programming (SDP) and has a number of applications in statistics and datamining, and other areas of informatics and mathematical sciences. In this paper, we extend the framework of standard primal-dual path-following algorithms for SDP to this problem. Employing this framework, we show that the long-step path-following algorithm analogous to the one in SDP has ${\cal O}(N\log(1/\varepsilon)+N)$ iteration-complexity to reduce the duality gap by a factor of $\varepsilon$, where $N = \sum N_i$, where $N_i$ is the size of the $i$-th positive semidefinite matrix block which is assumed to be an $N_i\times N_i$ matrix.

## Citation

Research Memorandum No. 980, The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, JAPAN, February 2006