Abstract
This paper presents a Wei-Yao-Liu conjugate gradient algorithm for nonsmooth convex optimization problem. The proposed algorithm makes use of approximate function and gradient values of the Moreau-Yosida regularization function instead of the corresponding exact values. Under suitable conditions, the global convergence property could be established for the proposed conjugate gradient method. Finally, some numerical results are reported to show the efficiency of our algorithm.
Highlights
We consider the unconstrained minimization problem min f (x), (1)x∈Rn where f : Rn → R is a nonsmooth convex function
By using the Moreau-Yosida regularization approach and a nonmonotone line search technique, we propose a Wei-Yao-Liu conjugate gradient algorithm for solving a nonsmooth unconstrained convex minimization problem
We perform numerical experiments to test the performance of the given algorithm, compare it with the MPRP gradient method in [29] and the proximal bundle method (PBL) in [15]
Summary
We consider the unconstrained minimization problem min f (x), (1)x∈Rn where f : Rn → R is a nonsmooth convex function. The function F has some good properties: it is a differentiable convex function, it has a Lipschitz continuous gradient even when the function f is nondifferentiable, and F is not twice differentiable in general, but the gradient function of F can be proved to be semismooth under some reasonable conditions [8, 21]. Based on these features, some algorithms have been proposed for solving (2), see [2, 8, 21, 24]. The proximal methods have been proved to be effective in dealing with evaluating the function value of F (x) and its gradient ∇F (x) exactly at a given point x
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