December 06, 2016
Semidefinite Approximations of Matrix Logarithm
Hamza FAWZI, Massachusetts Institute of Technology
(Joint work with James Saunderson (Monash University) and Pablo Parrilo (MIT))
We propose a new way to treat the exponential/relative entropy cone using symmetric cone solvers. Our approach is based on highly accurate rational (Pad\'e) approximations of the logarithm function. The key to this approach is that our rational approximations, by construction, inherit the (operator) concavity of the logarithm. Importantly, our method extends to the matrix logarithm and other derived functions such as the matrix relative entropy, giving new semidefinite optimization-based tools for convex optimization involving these functions. We compare our method to the existing successive approximation scheme in CVX, and show that it can be much faster, especially for large problems.