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In this paper, we consider a projected gradient method for constrained multiobjective optimization problems. Under suitable assumptions, we show that the sequence generated by the method converges to a Pareto stationary point (or a weak Pareto optimal point) of the problem when the multiobjective function is quasiconvex (or pseudoconvex). Furthermore, in the case that the multiobjective function is convex, by using some approximate conditions that are imposed on the gradients of the objective functions and the search directions, we obtain the linear convergence result for this method. [ABSTRACT FROM AUTHOR]