Numerical Optimization Techniques

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This project investigates the design and implementation algorithms for nonlinear optimization. The research focuses on the solution of large problems, and has two broad objectives. The first is to develop a new algorithm for solving large and unstructured unconstrained problems. This work is motivated by the need of solving large and nonsparse problems efficiently. The new algorithm uses ideas from both Newton methods and variable metric updating, and is tested on large problems arising in weather forecasting. The second objective is to study efficient techniques for handling inequality constraints occurring in nonlinear programming. This research focuses on large problems, and techniques are designed to be useful in that case. The use of an ellipsoidal-type method is explored. This approach has the advantage of leading readily into a sequence of equality constrained subproblems which can be solved by existing software. An alternative approach, which consists of extending the primal-dual method of linear programming to the nonlinear case, is also explored. The algorithm is posed in the framework of trust region methods, and its relationship with sequential quadratic programming is fully exploited. The two new approaches for handling constraints are tested on a set of large problems, and the treatment of ill-conditioning receives careful consideration.