Algorithms for Large-Scale Nonlinearly Constrained Optimization

The solution of large-scale nonlinear optimization-- minimization or maximization - problems lies at the heart of scientific computation. Structures take up positions of minimal constrained potential energy, investors aim to maximize profit while controlling risk, public utilities run transmission networks to satisfy demand at least cost, and pharmaceutical companies desire minimal drug doses to target pathogens. All of these problems are large either because the mathematical model involves many parameters or because they are actually finite discretisations of some continuous problem for which the variables are functions. The purpose of this grant application is to support the design, analysis and development of new algorithms for nonlinear optimization that are particularly aimed at the large-scale case. We shall focus on methods which attempt to improve simplified (cheaper) approximations of the actual (complicated) problem. Such a procedure may be applied recursively, and the most successful ideas in this vein are known as sequential quadratic programming (SQP). Our research is directed on ways to improve on SQP particularly when the underlying problem is large, and indeed particularly in the case where SQP itself may be too expensive to contemplate. The end goal of our research is to produce high-quality, publicly available software as part of the GALAHAD library.