Abstract
Combining multivariate spectral gradient method with projection scheme, this paper presents an adaptive prediction-correction method for solving large-scale nonlinear systems of monotone equations. The proposed method possesses some favorable properties: (1) it is progressive step by step, that is, the distance between iterates and the solution set is decreasing monotonically; (2) global convergence result is independent of the merit function and its Lipschitz continuity; (3) it is a derivative-free method and could be applied for solving large-scale nonsmooth equations due to its lower storage requirement. Preliminary numerical results show that the proposed method is very effective. Some practical applications of the proposed method are demonstrated and tested on sparse signal reconstruction, compressed sensing, and image deconvolution problems.
Highlights
IntroductionConsidering the problem to find solutions of the following nonlinear monotone equations:
Considering the problem to find solutions of the following nonlinear monotone equations: g (x) = 0, (1)where g : Rn → Rn is a continuous and monotone, that is, ⟨g(x) − g(y), x − y⟩ ≥ 0 for all x, y ∈ Rn.Nonlinear monotone equations arise in many practical applications such as ballistic trajectory computation [1] and vibration systems [2], the first-order necessary condition of the unconstrained convex optimization problem, and the subproblems in the generalized proximal algorithms with Bregman distances [3]
We develop an adaptive prediction-correction method for solving nonlinear monotone equations
Summary
Considering the problem to find solutions of the following nonlinear monotone equations:. Zhang and Zhou [6] presented a spectral gradient projection (SG) method for solving systems of monotone equations which combines a modified spectral gradient method and projection method This method is shown to be globally convergent if the nonlinear monotone equations is Lipschitz continuous. Xiao et al [7] proposed a spectral gradient method to minimize a nonsmooth minimization problem, arising from spare solution recovery in compressed sensing, consisting of a least-squares data-fitting term and a l1norm regularization term. This problem is firstly formulated as a convex quadratic program (QP) problem and reformulated to an equivalent nonlinear monotone equation.
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