A Family of Derivative-Free Conjugate Gradient Methods for Constrained Nonlinear Equations and Image Restoration

• Published in: IEEE access Vol. 8; pp. 162714 - 162729
• Main Authors: Ibrahim, Abdulkarim Hassan, Kumam, Poom, Kumam, Wiyada
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive search techniques; Applications of mathematics; Compressed sensing; compressive sensing; Conjugate gradient method; Conjugates; Constraints; Convergence; convex constrained; Gradient methods; Image restoration; Mathematical analysis; Nonlinear equations; Operators (mathematics); projection method; Unconstrained optimization
• ISSN: 2169-3536; 2169-3536
• DOI: 10.1109/ACCESS.2020.3020969

In this paper, a derivative-free conjugate gradient method for solving nonlinear equations with convex constraints is proposed. The proposed method can be viewed as an extension of the three-term modified Polak-Ribiére-Polyak method (TTPRP) and the three-term Hestenes-Stiefel conjugate gradient method (TTHS) using the projection technique of Solodov and Svaiter [Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, 1998, 355-369]. The proposed method adopts the adaptive line search scheme proposed by Ou and Li [Journal of Applied Mathematics and Computing 56.1-2 (2018): 195-216] which reduces the computational cost of the method. Under the assumption that the underlying operator is Lipschitz continuous and satisfies a weaker condition of monotonicity, the global convergence of the proposed method is established. Furthermore, the proposed method is extended to solve image restoration problem arising in compressive sensing. Numerical results are presented to demonstrate the effectiveness of the proposed method.