A STRICTLY CONTRACTIVE PEACEMAN-RACHFORD SPLITTING METHOD FOR CONVEX PROGRAMMING

Area of Science:

• Optimization Methods
• Numerical Analysis

Background:

• The Douglas-Rachford splitting method (DRSM) is a foundational algorithm in optimization.
• The Peaceman-Rachford splitting method (PRSM) is an alternative splitting method.
• Both methods are related to the alternating direction method of multipliers.

Purpose of the Study:

• To analyze the convergence properties of the Peaceman-Rachford splitting method (PRSM) for convex minimization.
• To compare PRSM with the Douglas-Rachford splitting method (DRSM).
• To propose a modified PRSM with improved convergence guarantees.

Main Methods:

• Analysis of iterative sequence contraction properties.
• Establishing convergence rates in ergodic and non-ergodic senses.
• Introducing a relaxation factor to enhance PRSM's convergence.
• Numerical validation in statistical learning and image processing.

Main Results:

• PRSM converges faster than DRSM when convergent, but requires stricter assumptions.
• DRSM's iterative sequence is strictly contractive, while PRSM's is contractive to the solution set.
• A worst-case O(1/t) ergodic convergence rate for PRSM is established under mild assumptions.
• A strictly contractive PRSM with a relaxation factor achieves a worst-case O(1/t) non-ergodic convergence rate.

Conclusions:

• The modified PRSM demonstrates numerical efficiency for practical applications.
• PRSM can be enhanced to guarantee strict contraction and faster convergence.
• The study provides theoretical convergence rates for PRSM and its variants.