RAL-TR-1998-069 On the solution of equality constrained quadratic programming problems arising in optimization

We consider the application of the conjugate gradient method to the solution of large equality constrained quadratic programs arising in nonlinear optimization. Our approach is based on a reduced linear system and generates iterates in the null space of the constraints. Instead of computing a basis for this null space, we choose to work directly with the matrix of constraint gradients, computing projections into the null space by either a normal equations or an augmented system approach. This can yield substantial economies in both line search and trust region methods for large-scale optimization, but can also result in signi cant rounding errors. We propose iterative re nement techniques, as well as an adaptive reformulation of the quadratic problem, that can greatly reduce these errors without incurring in high computational overhead. Numerical results illustrating the e ciency of the proposed approaches are presented. 1 Department for Computation and Information, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, EU. Email : [email protected] 2 Current reports available from from \http://www.numerical.rl.ac.uk/reports/reports.html". 3 CAAM Department, Rice University, Houston TX 77005, USA. Email : [email protected]. This author was supported by Department of Energy grant DE-FG02-87ER25047-A004. 4 Department of Electrical and Computer Engineering, Northwestern University, Evanston, IL 60208-3118, USA. Email : [email protected]. This author was supported by Science Foundation grant CCR-9625613 and by Department of Energy grant DE-FG02-87ER25047-A004. 5 Current reports available from \http://www.eecs.nwu.edu/~nocedal/PS les". Department for Computation and Information Atlas Centre Rutherford Appleton Laboratory Oxfordshire OX11 0QX September 30, 1998. Equality constrained quadratic programming problems arising in optimization 1