Convergence for nonconvex ADMM, with applications to CT imaging.

Area of Science:

• Optimization algorithms
• Computational imaging
• Applied mathematics

Background:

• The alternating direction method of multipliers (ADMM) is widely used for optimization problems.
• Existing convergence theory for ADMM in nonconvex settings typically requires function smoothness.
• Computed tomography (CT) image reconstruction is a key application area for ADMM.

Purpose of the Study:

• To develop new theoretical convergence guarantees for ADMM.
• To extend ADMM applicability to problems with nonsmooth and nonconvex objective functions.
• To provide a theoretical foundation for ADMM in CT image reconstruction without smoothness assumptions.

Main Methods:

• Developed new theoretical convergence results for ADMM.
• Introduced a restricted strong convexity assumption.
• Validated theoretical findings through empirical simulations.

Main Results:

• Established convergence guarantees for ADMM under restricted strong convexity, without requiring function smoothness.
• Demonstrated ADMM's effectiveness on simulated problems with nondifferentiable objective functions.
• Successfully applied the method to a simulated CT image reconstruction task.

Conclusions:

• The new theoretical results expand the applicability of ADMM to a broader class of optimization problems.
• This work provides a rigorous theoretical basis for using ADMM in challenging inverse problems like CT reconstruction.
• The findings pave the way for more robust and versatile optimization solutions in scientific and engineering domains.