We provide a new well-posedness concept for saddle-point problems. We characterize it by means of the behavior of the sublevel sets of an associated function. We then study the concave-convex case in Euclidean spaces. Applying these results in the setting of Convex Programming, we get a result on the convergence of the pair solution-Lagrange multiplier of approximating problems to the pair solution-Lagrange multiplier of the limit problem.
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