On the Strong Concavity of the Dual Function of an Optimization Problem

We provide three new proofs of the strong concavity of the dual function of some convex optimization problems. For problems with nonlinear constraints, we show that the assumption of strong convexity of the objective cannot be weakened to convexity and that the assumption that the gradients of all constraints at the optimal solution are linearly independent cannot be further weakened. Finally, we illustrate our results with several examples.