In this paper, we address three important issues in applying Lagrangian methods to solve optimization problems with inequality constraints. First, we propose a MaxQ method that transforms inequality constraints to equality constraints. It overcomes divergence and oscillations that occur in the slack-variable method. Some strategies to speed up its convergence are also examined. Second, we develop a method to monitor the balance between descents in the originalvariable space and ascents in the Lagrange-multiplier space in Lagrangian methods. During the search, we adjust this balance adaptively in order to improve convergence speed. Third, we introduce a nonlinear traveling trace to pull a search trajectory out of a local equilibrium point in a continuous fashion without restarting the search and without losing information already obtained in the local search. This strategy extends existing Lagrangian methods from a local search of equilibrium points to a global search. We implement thes..