A Modified Hestenes and Stiefel Conjugate Gradient Algorithm for Large-Scale Nonsmooth Minimizations and Nonlinear Equations

Gonglin Yuan (), Zehong Meng () and Yong Li ()
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Gonglin Yuan: Guangxi University
Zehong Meng: Zhejiang University of Finance and Economics
Yong Li: Baise University

Journal of Optimization Theory and Applications, 2016, vol. 168, issue 1, No 7, 129-152

Abstract: Abstract It is well known that nonlinear conjugate gradient methods are very effective for large-scale smooth optimization problems. However, their efficiency has not been widely investigated for large-scale nonsmooth problems, which are often found in practice. This paper proposes a modified Hestenes–Stiefel conjugate gradient algorithm for nonsmooth convex optimization problems. The search direction of the proposed method not only possesses the sufficient descent property but also belongs to a trust region. Under suitable conditions, the global convergence of the presented algorithm is established. The numerical results show that this method can successfully be used to solve large-scale nonsmooth problems with convex and nonconvex properties (with a maximum dimension of 60,000). Furthermore, we study the modified Hestenes–Stiefel method as a solution method for large-scale nonlinear equations and establish its global convergence. Finally, the numerical results for nonlinear equations are verified, with a maximum dimension of 100,000.

Keywords: Nonsmooth; Nonlinear equations; Conjugate gradient; Large scale; Global convergence; 65K05; 90C26 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (13)

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DOI: 10.1007/s10957-015-0781-1

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