Spectral-like conjugate gradient methods with sufficient descent property for vector optimization.

Area of Science:

• Optimization Theory
• Numerical Analysis
• Computational Mathematics

Background:

• Conjugate Gradient (CG) methods are vital for solving optimization problems.
• Existing CG parameters like PRP, HS, and DL lack guaranteed sufficient descent.
• Ensuring sufficient descent is crucial for the convergence and efficiency of optimization algorithms.

Purpose of the Study:

• Introduce novel spectral-like CG methods with guaranteed sufficient descent.
• Develop methods applicable to arbitrary nonnegative CG parameters.
• Establish global convergence properties without restrictive assumptions.

Main Methods:

• Proposed new spectral-like CG algorithms.
• Theoretically established sufficient descent property independent of line search.
• Proved global convergence using Wolfe line search for four parameters.
• Demonstrated convergence without assuming function convexity or employing restarts.

Main Results:

• The new methods ensure sufficient descent for the search direction.
• Global convergence is established for four specific nonnegative CG parameters (SPRP, SHZ, SDL, SHS).
• The generated sequences satisfy the first-order necessary conditions for Pareto optimality.
• Computational experiments confirm the effectiveness and superior performance of the proposed methods.

Conclusions:

• The developed spectral-like CG methods offer a robust approach to optimization.
• These methods provide guaranteed sufficient descent and global convergence.
• The proposed algorithms outperform existing methods like HZ and SP in efficiency metrics.