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MM optimization: Proximal distance algorithms, path following, and trust regions.

Alfonso Landeros1, Jason Xu2, Kenneth Lange1,3,4

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Summary
This summary is machine-generated.

The majorization-minimization (MM) principle and proximal distance algorithms offer a unified framework for optimization problems. This review explores their applications and acceleration techniques in statistics, finance, and nonlinear optimization.

Keywords:
computationdata scienceoptimizationstatistics

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Area of Science:

  • Optimization Theory
  • Computational Mathematics

Background:

  • The majorization-minimization (MM) principle is a powerful iterative method for solving optimization problems.
  • Proximal distance algorithms provide a generic approach for constrained optimization using quadratic penalties.

Purpose of the Study:

  • To review the majorization-minimization (MM) principle and proximal distance algorithms.
  • To illustrate their application across statistics, finance, and nonlinear optimization.
  • To explore methods for accelerating MM algorithms.

Main Methods:

  • Review of majorization-minimization (MM) principle and proximal distance algorithms.
  • Application of these principles to diverse problems in statistics, finance, and nonlinear optimization.
  • Exploration of acceleration techniques including matrix decompositions, path following, and cubic majorization.

Main Results:

  • Demonstrated applicability of MM and proximal distance principles to various optimization challenges.
  • Introduced and tested novel acceleration strategies for MM algorithms.
  • Highlighted the MM principle as a versatile framework for algorithm design and reinterpretation.

Conclusions:

  • The majorization-minimization (MM) principle and proximal distance algorithms are effective and broadly applicable optimization frameworks.
  • Acceleration techniques can significantly enhance the performance of MM algorithms.
  • This work unifies and extends understanding of these optimization methodologies.