Enlarging the Region of Convergence of Newton's Method for Constrained Optimization

摘要

Newton's method for solving the system of necessary optimality conditions of optimization problems with inequality constraints is considered in this paper. The prinicpal drawbacks of the method are the need for a good starting point, the inability of distinguish between local maxima and local minima, and when inequality constraints are present, the necessity to solve a quadratic programming problem at each iteration. Extensions to inequality constraints are based on the use of penalty functions by converting inequality constraints to equality constraints through the use of squared slack variables. The resulting Newton-like methods do not require solution of quadratic programming subproblems but rather employ systems of linear equations similar to those arising in the equality constrained case and involving only the active and nearly active constraints. It is shown that there is a close relationship between the class of penalty functions of Di Pillo and Grippo and the class of Fletcher. It is also shown that the region of convergence of a variation of Newton's method can be enlarged by making use of one of Fletcher's penalty functions.