Handling Inequality Constraints in Continuous Nonlinear Global Optimization

In this paper, we present a new method to handle inequality constraints and apply it in NOVEL (Nonlinear Optimization via External Lead), a system we have developed for solving constrained continuous nonlinear optimization problems. In general, in applying Lagrange-multiplier methods to solve these problems, inequality constraints are rst converted into equivalent equality constraints. One such conversion method adds a slack variable to each inequality constraint in order to convert it into an equality constraint. The disadvantage of this conversion is that when the search is inside a feasible region, some satissed constraints may still pose a non-zero weight in the La-grangian function, leading to possible oscillations and divergence when a local optimum lies on the boundary of a feasible region. We propose a new conversion method called the MaxQ method such that all satissed constraints in a feasible region always carry zero weight in the Lagrange function; hence, minimizing the Lagrange function in a feasible region always leads to local minima of the objective function. We demonstrate that oscillations do not happen in our method. We also propose methods to speed up convergence when a local optimum lies on the boundary of a feasible region. Finally, we show improved experimental results in applying our proposed method in NOVEL on some existing benchmark problems and compare them to those obtained by applying the method based on slack variables.