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A new line search technique
Abstract
In this paper the problem of line search, an
important step in most multidimensional optimization algorithms, is considered.
If conjugate gradient descent is to be used in producing a
descent sequence for a given functional f on a Hilbert space H,
then
at every step in the sequence on is faced with the problem of finding α,
the
step length to minimize f(x(k) + αP(k)) where x(k), P(k) ∈ H are
the k-th
elements of the descent sequence and k-th
descent direction, respectively. This is usually accomplished by a
one-dimensional search. It is the purpose of this paper to discuss a method of
search that determines, by a synergy of analytic-synthetic procedures, a
sequence {α(k)} such that the sequence { f(x(k) + α(k)P(k))}
converges to f(x*(k)), the minimum of f(x).
Specifically, the step in Fletcher-Reeve's algorithm that employs the geometric
mean of the past values of α(k) as initial
estimate is replaced by their harmonic mean to yield initially a low order
accuracy formula. The efficiency of this technique is confirmed by numerical
experimentation.
Mathematics Subject
Classification (2000): 65D15
Quaestiones Mathematicae 25 (2002), 453-464
important step in most multidimensional optimization algorithms, is considered.
If conjugate gradient descent is to be used in producing a
descent sequence for a given functional f on a Hilbert space H,
then
at every step in the sequence on is faced with the problem of finding α,
the
step length to minimize f(x(k) + αP(k)) where x(k), P(k) ∈ H are
the k-th
elements of the descent sequence and k-th
descent direction, respectively. This is usually accomplished by a
one-dimensional search. It is the purpose of this paper to discuss a method of
search that determines, by a synergy of analytic-synthetic procedures, a
sequence {α(k)} such that the sequence { f(x(k) + α(k)P(k))}
converges to f(x*(k)), the minimum of f(x).
Specifically, the step in Fletcher-Reeve's algorithm that employs the geometric
mean of the past values of α(k) as initial
estimate is replaced by their harmonic mean to yield initially a low order
accuracy formula. The efficiency of this technique is confirmed by numerical
experimentation.
Mathematics Subject
Classification (2000): 65D15
Quaestiones Mathematicae 25 (2002), 453-464
