Two spectral gradient projection methods for constrained equations and their linear convergence rate.

*(English)*Zbl 1310.65066Summary: Due to its simplicity and numerical efficiency for unconstrained optimization problems, the spectral gradient method has received more and more attention in recent years. In this paper, two spectral gradient projection methods for constrained equations are proposed, which are combinations of the well-known spectral gradient method and the hyperplane projection method. The new methods are not only derivative-free, but also completely matrix-free, and consequently they can be applied to solve large-scale constrained equations. Under the condition that the underlying mapping of the constrained equations is Lipschitz continuous or strongly monotone, we establish the global convergence of the new methods. Compared with the existing gradient methods for solving such problems, the new methods possess a linear convergence rate under some error bound conditions. Furthermore, a relax factor \(\gamma\) is attached in the update step to accelerate convergence. Preliminary numerical results show that they are efficient and promising in practice.

##### MSC:

65K05 | Numerical mathematical programming methods |

90C30 | Nonlinear programming |

90C06 | Large-scale problems in mathematical programming |

##### Keywords:

constrained equations; spectral gradient method; projection method; global convergence; unconstrained optimization; large-scale constrained equations; error bound; numerical result##### Software:

MCPLIB
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\textit{J. Liu} and \textit{Y. Duan}, J. Inequal. Appl. 2015, Paper No. 8, 13 p. (2015; Zbl 1310.65066)

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