Modified Lagrange multiplier method and its convergence analysis

By redefining multiplier associated with inequality constraint as a positive definite function of the originally-defined multiplier, nonnegative constraint imposed on inequality constraints in Karush-Kuhn-Tucker necessary conditions is removed, and a modified Lagrange multiplier method, which may handle inequality constraints directly, is constructed. Then its convergence is analyzed rigorously. By using LaSalle invariance principle, the underlying mechanism that attains the algorithmic convergence is uncovered. Some measures for relaxing the convergence conditions and enlarging the attractive domain are discussed.