Lagrangian heuristics for discrete optimization work by modifying Lagrangian relaxed solutions into feasible solutions to an original problem. They are designed to identify feasible, and hopefully also near-optimal, solutions and have proven to be highly successful in many applications. Based on a primal-dual global optimality condition for non-convex optimization problems, we develop a meta-heuristic extension of the Lagrangian heuristic framework. The optimality condition characterizes (near-)optimal solutions in terms of near-optimality and near-complementarity measures for Lagrangian relaxed solutions. The meta-heuristic extension amounts to constructing a weighted combination of these measures, thus creating a parametric auxiliary objective function, which is a close relative to a Lagrangian function, and embedding a Lagrangian heuristic in a search procedure in the space of the weight parameters. We illustrate and make a first assessment of this meta-heuristic extension by applying it to the generalized assignment and set covering problems. Our computational experience show that the meta-heuristic extension of a standard Lagrangian heuristic can significantly improve upon solution quality.