Methods for Nonlinear Constraints in Optimization Calculations

Ten years ago, the broad consensus among researchers in constrained optimization was that sequential quadratic programming (SQP) methods were the methods of choice. While, in the long term, this position may be justified, the past ten years have exposed a number of difficulties with the SQP approach. Moreover, alternative methods have shown themselves capable of solving large-scale problems. In this paper, we shall outline the defects with SQP methods, and discuss the alternatives. In particular, we shall indicate how our understanding of the subproblems which inevitably arise in constrained optimization calculations has improved. We shall also consider the impact of interior-point methods for inequality constrained problems, described elsewhere in this volume, and argue that these methods likely provide a more useful Newton model for such problems than do traditional SQP methods. Finally, we shall consider trust-region methods for constrained problems, and the impact of automatic differentiation on algorithm design.