Stability Analysis Of Continuous Conjugate Gradient Method
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Date
2008-06
Authors
HARUN, NURZALINA
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Abstract
In order to solve a large-scale unconstrained optimization, Conjugate Gradient
Method has been proven to be successful. However, the line search required in
Conjugate Gradient Method is sometimes extremely difficult and computationally
expensive. Studies conducted by Sun and Zhang [J. Sun and J. Zhang (2001), Global
convergence of conjugate gradient methods without line search], claimed that the
Conjugate Gradient Method was globally convergence using "fixed" stepsize at
determined using formula at = 8rk
T fk . The result suggested that for global
Ilpkl~
convergence of Conjugate Gradient Method, line search was not compUlsory.
Therefore, tlfts dissertation's objective is to determine the range of a and P where
this range will ensure the stability of Conjugate Gradient Method. Range for P is
obtained from research work done by Torii & Hagan (2002) and Bhaya &
Kaszkurewicz (2003). In order to establish the range for a, the coefficient matrix of
the system A~ = !!. was assumed to be symmetric positive-definite n x n
autocorrelation matrix of a Markov-l input signal for case p = O. This was done by
using the continuous realization of the Conjugate Gradient Method iteration which
took the fonn of an autonomous system of differential equation. The resulting range
of a and p was then simulated to demonstrate the convergence for the system A~ = !!.
on the stationary as well as nonstationary Conjugate Gradient Method. For
nonstationary Conjugate Gradient Method, A and b were varied with time. Based on
the simulation test, convergence of the Conjugate Gradient Method was established
for Q and p within the obtained range whi~n confirms the stability of Conjugate
Gradient Method. The simulation verify that the stability range also holds for p > 0 .
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Keywords
Stability Analysis Of Continuous , Conjugate Gradient Method