# An extension of the standard polynomial-time primal-dual path-following algorithm to the weighted determinant maximization problem with semidefinite constraints

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