An Acceleration Method for Dogleg Methods in Simple Singular Regions
Date of Original Version
The behavior of dogleg methods in singular regions that have a one-dimensional null space is studied. A two-tier approach of identifying singular regions and accelerating convergence to a singular point is proposed. It is shown that singular regions are easily identified using a ratio of the two-norm of the Newton step to the two-norm of the Cauchy step since Newton steps tend to infinity and Cauchy steps tend to zero as a singular point is approached. Convergence acceleration is accomplished by bracketing the singular point using a projection of the gradient of the two-norm of the process model functions onto the normalized Newton direction in conjunction with bisection, thus preserving the global convergence properties of the dogleg method. Numerical examples for a continuous-stirred tank reactor and vapor-liquid equilibrium flash are used to illustrate the reliability and effectiveness of the proposed approach. Several geometric illustrations are presented.
Industrial and Engineering Chemistry Research
Lucia, Angelo, and Delong Liu. "An Acceleration Method for Dogleg Methods in Simple Singular Regions." Industrial and Engineering Chemistry Research 37, 4 (1998): 1358-1363. doi:10.1021/ie970681f.