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 one 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.