Large-Scale Constrained Optimization

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The goal of this project is to develop effective numerical methods for solving complex nonlinear optimization problems. The algorithms to be developed and analyzed follow two distinct but related approaches: - Sequential quadratic programming (SQP) methods; and - Barrier methods (i.e. interior methods). This research will concentrate on large-scale problems, direct treatment of sparse linear and nonlinear constraints, and the guarantee of superlinear convergence. The research on SQP methods will extend techniques based on linearly constrained subproblems and an augmented Lagrangian merit function. Topics to be considered include use of a reduced Hessian, use of exact second derivatives, strategies for accepting inexact solutions of subproblems, and techniques for treating infeasible subproblems. Work in barrier methods have already proved remarkably successful for linear programs. Much research remains to improve their reliability and to extend their application to large-scale nonlinear (and convex) problems. Research in barrier methods emphasizes matrix factorization rather than matrix updating, and can utilize the substantial body of research on factorization methods for machines with parallel and vector architectures. We propose both direct and iterative methods for solving the indefinite systems involved.