On the strong concavity of the dual function of an optimization problem

TL;DR

This paper presents three new proofs demonstrating the strong concavity of the dual function in certain convex optimization problems, highlighting the necessity of specific assumptions and illustrating with examples.

Contribution

It provides novel proofs of strong concavity and clarifies the essential assumptions needed for nonlinear constrained problems.

Findings

Strong concavity of the dual function is established with new proofs.

Weakening assumptions like strong convexity or linear independence of gradients is not possible.

Examples illustrate the applicability and limitations of the results.

Abstract

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 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.