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A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. (English) Zbl 1316.90050
Summary: \(\mathrm{CG}_{-}\mathrm{DESCENT}\) is a state-of-the-art algorithm to solve large-scale unconstrained minimization problems. However, research activities on \(\mathrm{CG}_{-}\mathrm{DESCENT}\) in some other scenarios are relatively fewer. In this paper, by combining with the projection method of M. V. Solodov and B. F. Svaiter [Appl. Optim. 22, 355–369 (1999; Zbl 0928.65059)], we extend \(\mathrm{CG}_{-}\mathrm{DESCENT}\) to solve large-scale nonlinear convex constrained monotone equations. The proposed method does not require the Jacobian information, even though it does not store any matrix at each iteration. It thus has the potential to solve large-scale non-smooth problems. Under some mild conditions, we show that the proposed method converges globally. Primary numerical results illustrate that the proposed method works quite well. Moreover, we also extend this method to solve the \(\ell_1\)-norm regularized problems to decode a sparse signal in compressive sensing. Performance comparisons show that the proposed method is practical, efficient and competitive with the compared ones.

90C30 Nonlinear programming
90C25 Convex programming
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