Fast Reflected Forward-Backward algorithm: achieving fast convergence rates for convex optimization with linear cone...

Area of Science:

• Optimization Theory
• Convex Analysis
• Numerical Analysis

Background:

• Solving problems involving the sum of maximally monotone and monotone Lipschitz operators is crucial in various scientific fields.
• Existing methods often face limitations in convergence speed and applicability to complex optimization problems.
• The need for efficient algorithms with strong theoretical convergence guarantees is paramount.

Purpose of the Study:

• To introduce a novel Fast Reflected Forward-Backward (Fast RFB) algorithm for solving monotone operator problems.
• To enhance convergence performance by incorporating Nesterov momentum and a correction term.
• To demonstrate the algorithm's efficacy on minimax problems and convex optimization with linear cone constraints.

Main Methods:

• Derivation of the Fast RFB algorithm, extending existing reflected forward-backward methods.
• Theoretical analysis proving weak convergence of the iterative sequence.
• Application to specific problem classes: minimax problems and convex optimization with linear cone constraints.

Main Results:

• The Fast RFB algorithm achieves a last-iterate convergence rate of o(1/k) for discrete velocity and tangent residual.
• Significant improvements in convergence properties for minimax problems compared to state-of-the-art methods.
• A fully splitting primal-dual algorithm for convex optimization yields a last-iterate convergence rate of o(1/k) for objective function, feasibility, and complementarity.

Conclusions:

• The Fast RFB algorithm offers superior theoretical convergence rates for solving monotone operator problems.
• It provides competitive results for challenging optimization tasks, including minimax and primal-dual problems.
• Numerical experiments validate the algorithm's practical performance and convergence behavior.