An incomplete Hessian Newton minimization method and its application in a chemical database problem

Area of Science:

• Numerical Analysis
• Optimization
• Computational Chemistry

Background:

• Large-scale unconstrained minimization problems with dense Hessian matrices pose significant computational challenges.
• Existing methods may struggle with efficiency and scalability when dealing with large, dense matrices.

Purpose of the Study:

• To propose a novel modified Newton method, the incomplete Hessian Newton method (IHN), for efficient large-scale unconstrained minimization.
• To analyze the theoretical convergence properties of IHN.
• To develop and apply IHN and its variant, T-IHN, to a practical problem in chemical database optimal projection mapping.

Main Methods:

• Development of the incomplete Hessian Newton method (IHN) using an incomplete Hessian matrix.
• Theoretical analysis of IHN's global convergence and linear rate of convergence under specific conditions.
• Construction of the truncated-IHN method (T-IHN) as an effective application.
• Numerical experimentation to validate theoretical results and assess T-IHN's performance.

Main Results:

• IHN demonstrates global convergence and a linear rate of convergence with a suitable incomplete Hessian matrix.
• The Wolfe conditions are satisfied in IHN with a line search step length of one.
• The T-IHN method is effective for large-scale chemical database optimal projection mapping, even with indefinite incomplete Hessian matrices.
• Numerical results confirm IHN's theoretical properties and highlight T-IHN's efficiency.

Conclusions:

• The incomplete Hessian Newton method (IHN) provides an efficient approach for large-scale unconstrained minimization problems.
• The truncated-IHN method (T-IHN) is a promising and efficient algorithm for practical applications, including chemical database mapping.
• IHN and T-IHN offer robust and scalable solutions for complex optimization tasks.