An efficient Dai-Yuan projection-based method with application in signal recovery

Area of Science:

• Numerical Analysis and Optimization
• Applied Mathematics
• Signal Processing

Background:

• The classical Dai-Yuan conjugate gradient (CG) method, while possessing global convergence properties under the Lipschitz condition and satisfying descent conditions with Wolfe line search, suffers from numerical performance issues due to the jamming problem.
• Existing CG algorithms often struggle with efficiency and robustness when applied to complex systems, necessitating advancements for practical applications.

Purpose of the Study:

• To develop an efficient variant of the Dai-Yuan CG algorithm capable of solving nonlinear constrained monotone systems (NCMS).
• To address and resolve the numerical jamming problem inherent in the original Dai-Yuan CG method.
• To demonstrate the numerical robustness and applicability of the proposed variant in compressed sensing (CS) problems.

Main Methods:

• Development of a modified Dai-Yuan conjugate gradient algorithm.
• Theoretical analysis ensuring global convergence under Lipschitz condition and sufficient descent requirements, independent of the line search method.
• Numerical comparisons with existing algorithms from the literature.
• Application of the variant algorithm to sparse signal reconstruction in compressed sensing (CS).

Main Results:

• The proposed variant algorithm maintains global convergence properties, similar to the unmodified Dai-Yuan method, when Lipschitz and sufficient descent conditions are met.
• Numerical computations indicate that the variant algorithm is significantly more robust than existing methods, effectively overcoming the jamming problem.
• The algorithm successfully reconstructs sparse signals in compressed sensing (CS) scenarios, showcasing its practical utility.

Conclusions:

• The efficient variant of the Dai-Yuan CG algorithm provides a robust and numerically stable solution for nonlinear constrained monotone systems.
• This enhanced method overcomes the limitations of the original algorithm, offering improved performance in numerical computations.
• The variant algorithm shows promise for effective application in solving challenging problems such as sparse signal reconstruction in compressed sensing.