Generalized Lagrange Multiplier Method and KKT Conditions With an Application to Distributed Optimization

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.