A majorize-minimize line search algorithm for barrier function optimization

Many signal and image estimation problems such as maximum entropy reconstruction and positron emission tomography, require the minimization of a criterion containing a barrier function i.e., an unbounded function at the boundary of the feasible solution domain. This function has to be carefully handled in the optimization algorithm. When an iterative descent method is used for the minimization, a search along the line supported by the descent direction is usually performed at each iteration. However, standard line search strategies tend to be inefficient in this context. In this paper, we propose an original line search algorithm based on the majorize-minimize principle. A tangent majorant function is built to approximate a scalar criterion containing a barrier function. This leads to a simple line search ensuring the convergence of several classical descent optimization strategies, including the most classical variants of nonlinear conjugate gradient. The practical efficiency of the proposal scheme is illustrated by means of two examples of signal and image reconstruction.