The objective of this paper is to further investigate the idea of backtracking linesearch for solving a generic nonlinear constrained problem of the form $min h(x), s.t. c(x) = b, x \in C (closed convex subset of \mathbb{R}^d$ where f and c are smooth functions. To this end, we develop an algorithmic framework based on the backtracking line search for the Lagrangrian function associate to this problem. We propose an efficient numerical method based on backtracking line search on the augmented Lagrangian function to solve this optimization problem. The convergence of the proposed method is characterized in terms of the gradient mapping, feasibility, and objective function as well as the convergence to stationary points.