The Lagrange multiplier method is widely used for solving constrained optimization problems. In this brief, the classic Lagrangians are generalized to a wider class of functions that satisfies the strong duality between primal and dual problems. Then the generalized Karush-Kuhn-Tucker conditions for this generalized Lagrange multiplier method are derived. This useful method has applications in optimization problems and designs of consensus protocols, which is demonstrated by proposing a new continuous-time algorithm and its distributed version for optimization. The convergence advantages of the distributed algorithm are shown in a simulation example.
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Mengmou Li (2018) studied this question.