A Robust Trust Region Method for Constrained Nonlinear Programming Problems

Most of the published work on trust region algorithms for constrained optimization is derived from the original work of Fletcher on trust region algorithms for nondifferentiable exact penalty functions. These methods are restricted to applications where a reasonable estimate of the magnitude of an optimal Kuhn–Tucker multiplier vector can be given. More recently an effort has been made to extend the trust region methodology to the sequential quadratic programming (SQP) algorithm of Wilson, Han, and Powell. All of these extensions to the Wilson–Han–Powell SQP algorithm consider only the equality-constrained case and require strong global regularity hypotheses. This paper presents a general framework for trust region algorithms for constrained problems that does not require such regularity hypotheses and allows very general constraints. The approach is modeled on the one given by Powell for convex composite optimization problems and is driven by linear subproblems that yield viable estimates for the value of an exact penalty parameter. These results are applied to the Wilson–Han–Powell SQP algorithm and Fletcher’s S₁ QP algorithm. Local convergence results are also given.