An algorithm to find the set of the critical points of a linearly constrained multi-objective optimization problem is presented. This computational method can be interpreted as an interior point method for multi-objective optimization and it is based on a descent direction for the vector objective function that reflects the projected gradient direction of the scalar optimization. The algorithm finds the critical points as limit points of a suitable dynamical system and it does not rely on an a priori scalarization of the vector objective function. A convergence analysis is proposed showing that the limit points of the solutions of the dynamical system introduced satisfy the Karush-Kuhn-Tucker (KKT) first order necessary condition for linearly constrained multi-objective optimization problems. The algorithm is validated on some test problems derived from multi-objective optimization test problems well known in the scientific literature. The numerical results obtained show that the algorithm approximates satisfactorily the whole (weakly) local optimal Pareto set.