ADRS-M

Accelerated Douglas–Rachford Splitting via Monotone Residual Restart Course research project, Iterative Methods, Courant Institute, NYU (Spring 2026), with Gabriel Abreu.

Inertial acceleration speeds up operator-splitting methods in practice but can destroy the monotonicity guarantees that make them reliable. ADRS-M addresses this with a restart rule that selects between the inertial and standard step based on the fixed-point residual.

Contributions

• Proposed ADRS-M, an inertial Douglas–Rachford method for nonsmooth convex problems that picks the inertial or standard step by fixed-point residual.
• Proved a monitor inequality guaranteeing a nonincreasing residual, via firm nonexpansiveness and Fejér monotonicity.
• Benchmarked on LASSO, elastic net, basis pursuit, and nuclear norm problems (SVD thresholding, projections, least-squares solves).

Results

The method required 30–79% fewer iterations than standard DRS on three of four problem families, and was never slower on the fourth.