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For this class of problems, inspired by the homonymous necessary optimality condition for sparse single-objective optimization, we define the concept of <jats:italic>L<\/jats:italic>-stationarity and we analyze its relationships with other existing conditions and Pareto optimality concepts. We then propose two novel algorithmic approaches: the first one is an iterative hard thresholding method aiming to find a single <jats:italic>L<\/jats:italic>-stationary solution, while the second one is a two-stage algorithm designed to construct an approximation of the whole Pareto front. Both methods are characterized by theoretical properties of convergence to points satisfying necessary conditions for Pareto optimality. 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