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An accelerated proximal augmented Lagrangian method and its application in compressive sensing.

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We introduce an accelerated Proximal Augmented Lagrangian Method (PALM) with indefinite proximal regularization for convex programming. This enhanced method demonstrates feasibility and efficiency in compressive sensing applications.

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Area of Science:

  • Optimization
  • Numerical Analysis
  • Applied Mathematics

Background:

  • The Augmented Lagrangian Method (ALM) is a standard solver for convex programming with linear constraints.
  • Practical ALM implementations often incorporate semi-definite proximal terms to improve subproblem solvability.
  • Generalizing these proximal terms offers potential for enhanced algorithmic performance.

Purpose of the Study:

  • To propose an accelerated Proximal Augmented Lagrangian Method with indefinite proximal regularization (PALM-IPR).
  • To extend the applicability of proximal terms beyond semi-definite matrices in convex programming.
  • To analyze the convergence properties and demonstrate the practical efficiency of the proposed method.

Main Methods:

  • Development of the PALM-IPR algorithm, generalizing proximal regularization to indefinite matrices.
  • Theoretical analysis establishing the worst-case convergence rate under mild assumptions.
  • Empirical evaluation through numerical experiments on compressive sensing problems.

Main Results:

  • The proposed PALM-IPR method achieves a worst-case convergence rate in a non-ergodic sense.
  • Numerical results confirm the feasibility of PALM-IPR.
  • The method demonstrates significant efficiency in solving compressive sensing problems.

Conclusions:

  • PALM-IPR offers a generalized and effective approach for convex programming with linear constraints.
  • The method shows promise for applications in signal processing and machine learning, particularly compressive sensing.
  • Indefinite proximal regularization is a viable strategy for enhancing first-order optimization methods.