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Extensions to the Proximal Distance Method of Constrained Optimization.

Alfonso Landeros1, Oscar Hernan Madrid Padilla2, Hua Zhou3

  • 1Department of Computational Medicine, University of California, Los Angeles CA 90095-1596, USA.

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Summary
This summary is machine-generated.

This study introduces a new optimization method combining Beltrami-Courant penalty and proximal distance principles for minimizing functions with complex constraints. The novel steepest descent variant offers superior speed and accuracy for high-dimensional problems.

Keywords:
ADMMMajorization minimizationconvergencesteepest descent

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Area of Science:

  • Optimization Theory
  • Numerical Analysis
  • Applied Mathematics

Background:

  • Minimizing functions with general constraints is a fundamental problem in optimization.
  • Existing methods like ADMM can be computationally expensive for high-dimensional problems.

Purpose of the Study:

  • To develop a novel optimization algorithm for minimizing a loss function subject to general closed set constraints.
  • To analyze the convergence properties and computational efficiency of the proposed method and its variants.

Main Methods:

  • The study combines the Beltrami-Courant penalty method with the proximal distance principle.
  • A majorizing surrogate function is minimized to generate new iterates.
  • A steepest descent variant is developed to avoid costly linear system solves.

Main Results:

  • Convergence to a stationary point is proven for subanalytic loss functions and constraint sets.
  • Convergence rates, including linear local convergence, are established under stronger assumptions.
  • Numerical experiments show the steepest variant is faster and accurate on high-dimensional problems compared to ADMM.

Conclusions:

  • The proposed proximal distance algorithm effectively handles general fusion constraints in optimization.
  • The steepest descent variant provides a computationally efficient alternative for high-dimensional applications.
  • The method demonstrates broad applicability across various domains, including image denoising and matrix projection.