[Paper Review] Generalized relaxation of string averaging operators based on strictly relaxed cutter operators

Touraj Nikazad, Mahdi Mirzapour|arXiv (Cornell University)|Jun 20, 2017

Optimization and Variational Analysis41 references3 citations

TL;DR

This paper introduces a generalized relaxation scheme for string averaging operators based on strictly relaxed cutter operators in Hilbert spaces, enabling faster convergence and broader relaxation parameter selection (α ∈ (0,2)) compared to prior work. The method improves error reduction and computational efficiency, particularly when combined with multiple strings and optimal parameter tuning, as validated through numerical tests on convex feasibility problems using subgradient projections.

ABSTRACT

Analysis of a generalized relaxation of string averaging operators

Motivation & Objective

• To develop a generalized relaxation framework for string averaging operators based on strictly relaxed cutter operators in Hilbert spaces.
• To extend convergence analysis beyond strictly quasi-nonexpansive operators to include a wider class of operators such as relaxed projections and subgradient projections.
• To improve error reduction and computational efficiency in fixed point iterations for convex feasibility problems.
• To demonstrate the advantage of using relaxation parameters in (0,2) rather than (0,1), enabling faster convergence.

Proposed method

• The string averaging operator is constructed as a convex combination of finitely many operators, each being a composition of strictly relaxed cutter operators.
• Generalized relaxation is applied to the string averaging operator using step size parameters α ∈ (0,2), enhancing convergence speed.
• The projected version of the generalized relaxation is introduced and analyzed for feasibility problems with closed convex sets.
• The convergence analysis relies on the demi-closedness property of individual operators T_i - Id, extending prior results that required demi-closedness of the full string operator or its components.
• The method is applied to subgradient projection methods for solving nonlinear convex feasibility problems.
• Numerical experiments use cyclic subgradient projection schemes with adaptive step sizes and relaxation parameters to evaluate performance.

Experimental results

Research questions

• RQ1 Can generalized relaxation of string averaging operators based on strictly relaxed cutter operators achieve faster convergence than existing methods?
• RQ2 What is the impact of extending relaxation parameters from (0,1) to (0,2) on error reduction and convergence speed?
• RQ3 How does the use of strictly relaxed cutter operators improve convergence compared to strictly quasi-nonexpansive operators?
• RQ4 Does combining generalized relaxation with multiple strings significantly reduce the number of iterations in convex feasibility problems?
• RQ5 Can the proposed method maintain convergence while allowing broader operator classes such as relaxed subgradient projections and resolvents of monotone operators?

Key findings

• Using relaxation parameters α ∈ (0,2) instead of (0,1) leads to faster error reduction and improved convergence speed, as demonstrated in numerical tests.
• The best performance in the quadratic test case was achieved with α = 1.5 and λ = 1.5, reducing both iteration count and computational time compared to α = 1.
• For 10 quadratic examples with E=1, the generalized relaxation technique reduced the average number of iterations from 85 (without grt) to 28 (with grt) when E=10.
• With E=20 strings and generalized relaxation, the average iteration count dropped to 28, compared to 868 without generalized relaxation, showing a dramatic reduction in iterations.
• The projected generalized relaxation scheme significantly outperforms standard schemes, especially when combined with multiple strings and optimal α selection.
• Numerical results confirm that generalized relaxation with α ≠ 1 can yield better performance than the standard α = 1 case, validating the theoretical advantage of extended parameter range.