ANZIAM  J.  45 (2003), 91-114
Numerical algorithms for constrained maximum likelihood estimation

Z. F. Li
  National Centre for Epidemiology
  and Population Health
  Australian National University
  Canberra ACT 0200
M. R. Osborne
  Centre for Mathematics and its Applications
  School of Mathematical Sciences
  Australian National University
  Canberra ACT 0200
T. Prvan
  School of Mathematics and Statistics
  The University of Canberra
  Canberra ACT 2617

This paper describes a SQP-type algorithm for solving a constrained maximum likelihood estimation problem that incorporates a number of novel features. We call it MLESOL. MLESOL maintains the use of an estimate of the Fisher information matrix to the Hessian of the negative log-likelihood but also encompasses a secant approximation S to the second-order part of the augmented Lagrangian function along with tests for when to use this information. The local quadratic model used has a form something like that of Tapia's SQP augmented scale BFGS secant method but explores the additional structure of the objective function. The step choice algorithm is based on minimising a local quadratic model subject to the linearised constraints and an elliptical trust region centred at the current approximate minimiser. This is accomplished using the Byrd and Omojokun trust region approach, together with a special module for assessing the quality of the step thus computed. The numerical performance of MLESOL is studied by means of an example involving the estimation of a mixture density.
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