Projection method with inertial step for nonlinear equations: Application to signal recovery

Abstract

<p style='text-indent:20px;'>In this paper, using the concept of inertial extrapolation, we introduce a globally convergent inertial extrapolation method for solving nonlinear equations with convex constraints for which the underlying mapping is monotone and Lipschitz continuous. The method can be viewed as a combination of the efficient three-term derivative-free method of Gao and He [Calcolo. 55(4), 1-17, 2018] with the inertial extrapolation step. Moreover, the algorithm is designed such that at every iteration, the method is free from derivative evaluations. Under standard assumptions, we establish the global convergence results for the proposed method. Numerical implementations illustrate the performance and advantage of this new method. Moreover, we also extend this method to solve the LASSO problems to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of our algorithm.</p>

Three adaptive hybrid derivative‐free projection methods for constrained monotone nonlinear equations and their applications

“…Finally, similar to References 36‐38, we think that it could be interesting to construct some more efficient derivative‐free projection methods based on inertial extrapolation technique.…”

“…In fact, the CGM has been attached much importance as an effective numerical method for solving large‐scale unconstrained optimization problems because of its simplicity and low storage 8‐29 . Thus, using the projection technique, 30 several researchers extended these methods resulting in derivative‐free methods (see References 31‐48 and references therein). For example, based on Hestenes–Stiefel (HS) method 15 for unconstrained optimization, Wang et al 43 proposed a self‐adaptive three‐term conjugate gradient projection method (CGPM) for solving the SCMNE.…”

“…The proposed method can be viewed as an extension of the HS method using the projection technique. Later, applying the inertial extrapolation method to the CGPMs appeared in References 43 and 33, Ibrahim et al 36,37 proposed inertial accelerated derivative‐free algorithms. To improve the calculation effect of DY method 13 and expand its applications, Liu and Feng, 41 combined with the projection technique, extended and obtained a spectral gradient‐type iterative projection method for the SCMNE.…”

On image restoration problems using new conjugate gradient methods

“…2. In some neighborhood 𝛬 of 𝛺, the gradient of function g is Lipschitz continuous, namely, there exists a constant L > 0 such that: ‖𝑔(𝜊) − 𝑔(𝜏)‖ ≤ 𝐿 ‖𝑜 − 𝜏‖, ∀𝜏, 𝜊 ∈ 𝛬 (26) We demonstrate the Dai et al [17] theorem it is very important for deducing global convergence. Lemma 1.…”

"A descent three-term derivative-free method for signal reconstruction in compressive sensing"

“…Exploiting the simplicity and low storage requirement of the conjugate gradient method [1,2], in recent times, several authors have extended many conjugate gradient algorithms designed to solve unconstrained optimization problems to solve large-scale nonlinear equations (1.6) (see [3][4][5][6][7][8][9][10][11][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]36]). For instance, using the projection scheme of Solodov and Svaiter [35], Xiao and Zhu [38] extended the Hager and Zhang conjugate descent (CG DESCENT) method to solve (1.6).…”