Abstract
In this paper, based on the HS method and a modified version of the PRP method, a hybrid conjugate gradient (CG) method is proposed for solving large-scale unconstrained optimization problems. The CG parameter generated by the method is always nonnegative. Moreover, the search direction possesses the sufficient descent property independent of line search. Utilizing the standard Wolfe–Powell line search rule to yield the stepsize, the global convergence of the proposed method is shown under the common assumptions. Finally, numerical results show that the proposed method is promising compared with two existing methods.
Highlights
Consider the problem of minimizing f over Rn: min f(x), x∈Rn (1)where f: Rn ⟶ R is continuously differentiable. roughout, the gradient of f at x is denoted by g(x), i.e., g(x): ∇f(x)
We know that conjugate gradient (CG) methods are very popular and effective for solving unconstrained optimization problems (1), especially for large-scale case by means of their simplicity and low memory requirements. ese preferred features greatly promote their applications in various areas such as image deblurring and denoising, neural network, compressed sensing, and others
We refer the interested readers to some recent works [1,2,3] and references therein for more details. e numerical results reported in [1] reveal that the CG method has great potential in solving image restoration problems
Summary
Where f: Rn ⟶ R is continuously differentiable. roughout, the gradient of f at x is denoted by g(x), i.e., g(x): ∇f(x). When the WWP line search rule is used to compute the stepsize, the resulting search direction in [18] is a descent one and the global convergence for the hHD method is proved. The WYL method is globally convergent under the WWP line search rule and possesses superior numerical performance. Based on the above observations, it is interesting to design a hybrid CG method such that the CG parameter is nonnegative and the resulting search direction possesses the sufficient descent property independent of line search technique. Plugging the CG parameter βk: βhkHPR into (3), we can show that the resulting search direction possesses the sufficient descent property independent of line search technique (see Lemma 1 below). Some preliminary numerical results are reported to verify the efficiency of the presented method
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