The symmetric ADMM with indefinite proximal regularization and its application.

Area of Science:

• Optimization Theory
• Numerical Analysis
• Convex Programming

Background:

• Symmetric Alternating Direction Method of Multipliers (S-ADMM) is effective but can suffer from small step sizes with large proximal parameters.
• Existing methods typically use positive definite proximal matrices in subproblems.
• The 'too-small-step-size' phenomenon limits practical applications of S-ADMM.

Purpose of the Study:

• To generalize the proximal matrix in S-ADMM from positive definite to indefinite.
• To propose a novel S-ADMM with indefinite proximal regularization (IPS-ADMM).
• To analyze the convergence properties and iteration complexity of the proposed IPS-ADMM.

Main Methods:

• Generalization of proximal matrices to indefinite forms.
• Development of the Indefinite Proximal Regularization for Symmetric Alternating Direction Method of Multipliers (IPS-ADMM).
• Proof of global convergence without additional assumptions.
• Analysis of worst-case convergence rate using iteration complexity in an ergodic sense.

Main Results:

• The proposed IPS-ADMM is proven to converge globally.
• The iteration complexity and worst-case convergence rate in an ergodic sense are analyzed.
• Numerical results demonstrate the efficiency of the IPS-ADMM.

Conclusions:

• IPS-ADMM offers an effective alternative for solving two-block separable convex programming problems with linear constraints.
• The use of indefinite proximal regularization overcomes limitations of traditional S-ADMM.
• The method shows practical efficiency and theoretical convergence guarantees.