Two New PRP Conjugate Gradient Algorithms for Minimization Optimization Models.

Area of Science:

• Numerical Analysis
• Optimization Theory
• Computational Mathematics

Background:

• Conjugate gradient methods are fundamental for solving large-scale optimization and nonlinear equation problems.
• Existing methods often rely on line search procedures, which can be computationally intensive.
• There is a continuous need for efficient and robust algorithms with proven convergence properties.

Purpose of the Study:

• To propose two new algorithms utilizing modified Polak–Ribière–Polyak (PRP) conjugate gradient methods.
• To develop algorithms applicable to both unconstrained optimization and solving nonlinear equations.
• To analyze the theoretical properties and numerical performance of the proposed methods.

Main Methods:

• Development of two modified PRP conjugate gradient algorithms.
• Incorporation of function and gradient values in the first algorithm.
• Ensuring trust region and sufficient descent properties without line search.
• Establishing global convergence under specific conditions.

Main Results:

• The first algorithm is effective and competitive for unconstrained optimization problems.
• The second algorithm demonstrates effectiveness for large-scale nonlinear equations.
• Both algorithms exhibit desirable properties like non-negative parameters and descent search directions.
• Numerical experiments validate the practical performance of the proposed methods.

Conclusions:

• The proposed PRP conjugate algorithms offer efficient solutions for unconstrained optimization and nonlinear equations.
• The absence of line search enhances computational efficiency.
• The established global convergence and positive numerical results support their applicability.