A spectral Fletcher-Reeves conjugate gradient method with integrated strategy for unconstrained optimization and...

Area of Science:

• Optimization Theory
• Numerical Analysis
• Computational Finance

Background:

• Large-scale unconstrained optimization problems present significant computational challenges.
• Existing spectral conjugate gradient (SCG) techniques offer efficiency but can be further refined.
• Portfolio selection requires robust methods for optimizing returns and minimizing risk.

Purpose of the Study:

• To introduce a novel structured spectral conjugate gradient (SCG) algorithm.
• To enhance the general structure and performance of conjugate gradient (CG) methods.
• To apply and evaluate the extended SCG method for stock allocation in portfolio selection.

Main Methods:

• Developed a structured SCG approach integrating Quasi-Newton direction and an extended conjugacy condition.
• Incorporated the Fletcher-Reeves conjugate gradient parameter for structural improvement.
• Established global convergence using Wolfe-line search criteria for general functions.

Main Results:

• Numerical experiments demonstrate the superiority of the proposed SCG algorithm over existing CG methods.
• The extended SCG method shows significant improvements in computational efficiency for unconstrained optimization.
• Application to portfolio selection yields optimized stock allocation with minimized risk and enhanced returns.

Conclusions:

• The structured SCG method offers a robust and efficient solution for large-scale unconstrained optimization.
• The algorithm provides a valuable tool for financial applications, particularly in portfolio optimization.
• Empirical evaluations confirm the method's effectiveness in improving computational efficiency and optimization outcomes.