A Majorize-Minimize line search algorithm for barrier functions

Criteria containing a barrier function i.e., an unbounded function at the boundary of the feasible solution domain are frequently encountered in the optimization framework. When an iterative descent method is used, a search along the line supported by the descent direction through the minimization of the underlying scalar function has to be performed at each iteration. Usual line search strategies use an iterative procedure to propose a stepsize value ensuring the fulfillment of sufficient convergence conditions. The iterative scheme is classically based on backtracking, dichotomy, polynomial interpolations or quadratic majorization of the scalar function. However, since the barrier function introduces a singularity in the criterion, classical line search procedures tend to be inefficient. In this paper we propose a majorizationbased line search strategy by deriving a nonquadratic form of a majorant function well suited to approximate a criterion containing a barrier term. Furthermore, we establish the convergence of classical descent algorithms when this strategy is employed. The efficiency of the proposed line search strategy is illustrated by means of numerical examples in the field of signal and image processing.