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Weak convergence on Douglas-Rachford method
B. F. Svaiter
TL;DR
The paper analyzes monotone operators and uses Fitzpatrick-function properties together with auxiliary lemmas and algebraic manipulations. The proof concludes that limi→∞αk,i = 0 for k = 1, . . . , m and characterizes operator membership through equality in the relevant inequality.
Problem
The supplied passages define monotonicity and maximal monotonicity and establish the operator-theoretic setting used in the proof.
Method
The proof combines auxiliary lemmas, direct algebraic manipulations, and the Fitzpatrick function’s lower bound and equality characterization.
Results
limi→∞αk,i = 0 for k = 1, . . . , m, while equality in the relevant inequality implies (x, x∗) ∈T.
Takeaways & Limitations
The argument uses Fitzpatrick-function equality to identify points belonging to the monotone operator T.
Abstract
from arXiv · showhide
We prove that the sequences generate by the Douglas-Rachford method converge weakly to a solution of the inclusion problem
A An auxiliary result
The auxiliary results establish weak×weak* closure properties for maximal monotone operators and use Fitzpatrick’s function to certify graph membership. These tools support the Douglas–Rachford convergence argument by showing subsequential limits lie in the solution set.
- Maximal monotonicity combined with weak×weak* convergence provides an inequality for the limiting pair.
- Equality in that inequality certifies that the limiting pair belongs to the operator graph.
- Fitzpatrick’s function is weak×weak* lower semicontinuous and dominates the duality product, with equality exactly on the operator graph.
- The proof applies these properties to maximal monotone operators and graph-constrained nets.
- Applying the preceding equations and Lemma 4 completes the auxiliary argument.
- The auxiliary estimates imply that α_k,i tends to zero for every k.