Unconstrained Minimization Techniques - Boundary Constraints

During iteration, it is not uncommon for the approximate Hessian matrix Hk to become singular, especially when the parameter estimates are still far from the desired local or global minimum. Jennrich and Sampson (1968) introduced a stepwise regression procedure which deals with this problem quite effectively in practice. During iteration, the Jennrich-Sampson procedure checks whether parameter estimates are within prescribed bounds, and constrains the estimates to remain within these bounds. This facility is quite useful in the context of structural modeling, where variances, if estimated as unconstrained free parameters, may frequently take on negative values. Browne and DuToit (1982) describe a FORTRAN implementation of the Jennrich-Sampson procedure which is especially adapted to facilitate the constrained estimation procedure (described below) for standardizing the variance of endogenous latent variables during iteration. SEPATH now uses the Jennrich-Sampson procedure to calculate dk on each iteration.