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Principles of Rodent Surgery for the New Surgeon

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Published on: January 6, 2011

Generalizations of the quadratic bound optimization principle.

Xun-Jian Li¹, Guo-Liang Tian², Hua Zhou^1,3

¹Department of Biostatistics, University of California, Los Angeles, CA 90095.

Proceedings of the National Academy of Sciences of the United States of America

|January 23, 2026

Summary

This study introduces a generalized quadratic upper-bound (QUB) principle for optimization, overcoming limitations of the original method. The new approach uses a matrix-valued function for improved convergence in minimization problems.

Keywords:

MM algorithm diagonal matrix-valued function least squares maximum likelihood estimation quadratic bound principle

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Area of Science:

Optimization Algorithms
Numerical Analysis

Background:

The quadratic bound (QB) principle is a specialized form of majorization-minimization/minorization-maximization.
The quadratic upper-bound (QUB) principle aids minimization but faces challenges with constant positive definite matrices.

Purpose of the Study:

To generalize the quadratic bound (QB) principle to overcome existing limitations.
To develop novel QUB algorithms for optimization problems.

Main Methods:

Replacing the constant matrix in QUB with a continuous matrix-valued function.
The function dominates the Hessian and depends on current and potential iterates.
Implementing diagonal matrix-valued functions with separated diagonal entries.

Main Results:

Theoretical analysis confirms global convergence under favorable conditions.
A tangency condition in the scalar case promotes superlinear convergence.
Numerical experiments in Julia demonstrate the effectiveness of the generalized QB principle.

Conclusions:

The generalized QB principle offers a robust alternative to the traditional QUB method.
The proposed algorithms exhibit strong convergence properties.
The approach is validated by practical numerical implementations.