An efficient three-term conjugate gradient algorithm with restart strategy and image restoration.

Area of Science:

• Optimization Theory
• Numerical Analysis
• Computational Science

Background:

• Unconstrained optimization is a fundamental problem in applied mathematics and computational science.
• Existing conjugate gradient methods face challenges with large-scale problems and convergence efficiency.
• The development of effective hybrid methods is crucial for advancing optimization techniques.

Purpose of the Study:

• To propose a novel hybrid three-term conjugate gradient method with a restart procedure for unconstrained optimization.
• To introduce a new search direction approximating the memoryless BFGS quasi-Newton direction.
• To analyze the global convergence properties and demonstrate the sufficient descent property of the proposed method.

Main Methods:

• A hybrid structure combining FR, CD, and DY conjugate parameters with a novel search direction.
• Theoretical analysis of the sufficient descent property.
• Global convergence analysis under weak Wolfe line search conditions.
• Numerical experiments on 100 test functions and image restoration applications.

Main Results:

• The proposed method exhibits excellent performance, particularly in large-scale optimization problems.
• Numerical experiments demonstrate that the algorithm outperforms existing conjugate gradient methods.
• The method achieves higher peak signal-to-noise ratio values in image restoration tasks, indicating superior performance.

Conclusions:

• The novel hybrid conjugate gradient method is effective for solving unconstrained optimization problems.
• The algorithm shows significant advantages over other conjugate gradient methods in terms of performance and efficiency.
• The proposed method has practical applications, notably in image restoration, offering improved results.