Abstract
In this paper, we propose a three–term PRP–type conjugate gradient method which always satisfies the sufficient descent condition independently of line searches employed. An important property of our method is that its direction is closest to the direction of the Newton method or satisfies conjugacy condition as the iterations evolve. In addition, under mild condition, we prove global convergence properties of the proposed method. Numerical comparison illustrates that our proposed method is efficient for solving the optimization problems.
Highlights
Consider the following unconstrained optimization problem: min f (x), x∈Rn where f : Rn → R is a smooth function
The above methods do not satisfy the conjugacy condition and the probably effective combination between them have been ignored, which motivate this paper. We propose another three–term PRP–type method by noting that our chief concern is to take advantage of the property of the Newton method or the conjugacy condition in constructing the conjugate gradient (CG) method
To eliminate the probable effect of unboundedness of ξk and establish the global convergence of our proposed method, we proposed the following strategy in constructing the search direction: dk =
Summary
Consider the following unconstrained optimization problem: min f (x), x∈Rn where f : Rn → R is a smooth function. The parameters of the PRP and HS methods are respectively given by βkHS. Both above methods are usually recommended in actual computation due to its superior computational performance. In the corresponding PRP+ method, in which the parameters βkP RP + = max{βkP RP , 0}, the current search direction will automatically adjust to the steepest descent direction when the step sk−1 is small, which prevents effectively jamming phenomenon from occurring. The HS method has the property that it can satisfy automatically the standard conjugacy condition independent of any line search used [19], i.e., dTk yk−1 = 0. The sufficient descent condition has been used in the literature to analyze the global convergence of conjugate gradient methods with inexact line searches. There exists a constant c > 0 such that dTk gk ≤ −c gk 2, ∀k ∈ N
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