Given a network modeled as a graph G with each link associated with a cost and k weights, the Constrained Shortest Path (CSP(k)) problem asks for computing a minimum cost path from a source node s to a target node t satisfying pre-specified bounds on path weights: This problem is NP-hard. In this paper we propose a new approximation algorithm called GEN-LARAC for CSP(k) problem based on Lagrangian relaxation method. For k = 1, we show that the relaxed problem can be solved by a polynomial time algorithm with time complexity O((m + n log n)2). Using this algorithm as a building block and combing it with ideas from mathematical programming, we propose an efficient algorithm for arbitrary k. We, prove the convergence of our algorithm and compare it with previously known algorithms. We point out that our algorithm is also applicable to a more general class of constrained optimization problems.