In this paper, we propose a new formulation for moving horizon estimation (MHE) of constrained discrete-time linear systems subject to unknown state and output disturbances. The presented MHE approach is based on the concept of relaxed logarithmic barrier functions and allows to formulate the state estimator in form of an unconstrained optimization problem. We present sufficient conditions for robust stability of the underlying estimation error dynamics and illustrate the obtained results by means of a numerical example in which the proposed MHE scheme is applied in the context of robust output feedback model predictive control.