The objective of this research project is to extend interior-point methods so that they can be applied to smooth nonlinear optimization problems. In particular, an existing linear/quadratic solver will be enhanced so as to become an easy-to-use and efficient system for the solution of these problems. If a problem is convex, the system will find a globally optimal solution. If it is nonconvex, then the system will find a locally optimal solution in a neighborhood of a given initial solution. The algorithmic issues to be addressed include: (a) which merit function to use, (b) when and by how much to perturb indefinite Hessians, (c) how to adapt the predictor-corrector variant, (d) how and when to sparsify Hessians, (e) how much to push initial values away from bounds, (f) how to detect and handle problems in which the constraints don't satisfy a constraint qualification condition, and (g) how to ensure that the infeasible interior-point method doesn't evaluate any function outside its domain. As part of this research, a database of real-world nonlinear optimization models will be assembled. This database will be freely available via the internet. Finally, techniques will be sought that will enable one to use the interior-point-based nonlinear-programming system to solve semidefinite programming problems.
Nonlinear optimization plays a fundamental role in all branches of applied science and engineering. Whether it is maximizing the speed at which flutter destroys a wing, minimizing the use of nonrenewable resources in hydrothermal power generation, designing a structure that can bear a specified load with minimal amount of material, or minimizing side-lobe signal strength in the output of an antenna array while preserving main-lobe characteristics, there is one commonly shared theme: optimization. Interior-point methods provide a new approach to the solution of these problems, which promises to enable one to solve them more reliably and much more efficiently than before. This research project will focus on resolving the issues involved in producing an efficient and robust interior-point algorithm for nonlinear optimization.
