Distance majorization and its applications.

Area of Science:

• Optimization
• Applied Mathematics
• Convex Analysis

Background:

• Minimizing convex functions over intersecting convex sets is a common problem.
• Existing methods like interior point methods struggle with large-scale datasets.
• Projection onto set intersections can be computationally challenging.

Purpose of the Study:

• To develop a novel algorithm for convex programming that scales effectively with high dimensionality.
• To address the limitations of current methods in modern large-scale applications.
• To provide an efficient solution for problems involving projections onto set intersections.

Main Methods:

• The proposed algorithm is an instance of sequential unconstrained minimization.
• It integrates the majorization-minimization principle, classical penalty methods, and quasi-Newton acceleration.
• The approach focuses on accelerating fixed-point algorithms for improved performance.

Main Results:

• The developed algorithm demonstrates good scalability with increasing problem dimensionality.
• Performance is validated through several application examples.
• The method offers an efficient alternative for large-scale convex optimization.

Conclusions:

• The proposed distance majorization algorithm effectively addresses large-scale convex programming problems.
• This approach offers a promising solution for applications in statistics, engineering, and machine learning.
• The integration of multiple optimization techniques leads to a robust and scalable algorithm.