Descent direction algorithms are widely used for solving unconstrained minimization problems. After computing a descent direction, the algorithm performs a search along the line supported by this direction to get the value of the step size. Recently, several papers [Chen02, Labat08] have introduced a new line search strategy founded on the theory of Majorization-Minimization algorithms [Hunter04]. An iterate of the line search algorithm is the minimizer of a quadratic majorant approximation of the criterion along the search direction. This approach, very easy to implement, provides good convergence properties to the overall algorithm, namely when combining with the conjugate gradient method. However, in the case of functions having a linear barrier, i. e. tending to infinity when some linear constraints are active, quadratic tangent majorant won't be well suited to approximate such functions. Then we suggest to appropriately modify the line search strategy by choosing a new form of tangent majorant. Since the majorant function is still not quadratic, precedent results of convergence analysis do not hold anymore. A complete convergence analysis is thus performed. This approach yields to a new efficient line search method for functions having a linear barrier.