Weak Convergence in Operator Splitting

• Weak Convergence Approach is a methodology that formalizes how sequences of random measures, processes, or functions converge in distribution, especially in infinite-dimensional spaces.
• The Douglas–Rachford method utilizes interlaced sequences and summable errors to manage operator inclusion problems, providing a framework for addressing weak convergence where norm convergence fails.
• Convergence proofs leverage quasi-Fejér properties, Opial’s lemma, and decay estimates, ensuring that any weak limit satisfies the original inclusion conditions in practical and theoretical applications.

The weak convergence approach is a foundational methodology in probability theory, stochastic processes, and optimization, formalizing how sequences of random objects (measures, processes, or functions) converge in distribution. In modern mathematical research, it enables rigorous analysis of approximation schemes, operator splitting methods, and asymptotic behavior in infinite-dimensional or non-smooth settings, especially where strong (norm) convergence is unachievable. This entry provides a comprehensive overview of the principles, techniques, and significance of the weak convergence approach, particularly as developed in the context of operator splitting and the Douglas–Rachford method for inclusion problems involving maximal monotone operators (Svaiter, 2010).

1. Problem Formulation and Algorithmic Context

The weak convergence approach systematically addresses the solution of operator inclusion problems in real Hilbert spaces: $0 \in A(x) + B(x)$ where $A,B : H \to 2^H$ are maximal monotone operators and $H$ is a real, infinite-dimensional Hilbert space. The Douglas–Rachford (DR) method is a canonical splitting algorithm for such problems. Given an initial pair $(x_0, b_0)$ with $b_0 \in B(x_0)$, the method generates interlaced sequences $\{(y_k,a_k)\}$ and $\{(x_k,b_k)\}$ through the recursive steps:

• Solve for $(y_k,a_k) \in A$ such that

$y_k + a_k = x_{k-1} - b_{k-1}$

• Solve for $(x_k,b_k) \in B$ such that

$$\left\|x_k + b_k - (y_k + b_{k-1})\right\| \leq \epsilon_k$$

where $\{\epsilon_k\}$ is a summable sequence ($\sum_k \epsilon_k < \infty$), accommodating inexact or perturbed computations.

2. Key Principles of Weak Convergence in Operator Splitting

The DR method, operating under the splitting paradigm, leverages weak topology—a convergence mode where, for any continuous linear functional $\phi$ on $H$, $\phi(x_k) \to \phi(x)$, even if $\|x_k-x\|$ does not vanish. The principal result (Theorem 1 in (Svaiter, 2010)) asserts that if the extended solution set

$$S(A, B) = \{(x, b) \in H \times H : b \in B(x),\; -b \in A(x)\}$$

is nonempty, then

$$(x_k, b_k) \rightharpoonup (x, b), \qquad (y_k, a_k) \rightharpoonup (x, -b)$$

in the weak topology of $H \times H$, with $b \in B(x)$ and $-b \in A(x)$. This property is crucial in infinite-dimensional settings where strong convergence may not be obtainable due to non-compactness or operator degeneracy.

3. Convergence Mechanisms: Analytical Tools

The convergence proof integrates several deep analytical mechanisms:

a) Quasi-Fejér Convergence:

Define $p_k = (x_k, b_k)$. The quasi–Fejér property is manifested as

$$\|p_k - p\| \leq \|p_{k-1} - p\| + \delta_k, \quad \sum_k \delta_k < \infty$$

for any $p \in S(A,B)$. This bounds the deviation of iterates from any solution by a telescoping, summable error—a weak form of monotonicity that implies boundedness and vanishing "error terms" ($\|a_k + b_{k-1}\| \to 0$).

b) Application of Opial’s Lemma:

Opial's lemma is invoked to guarantee uniqueness of weak cluster points. Boundedness and the quasi-Fejér property, together with weak closedness of the solution set (from maximal monotonicity), ensure that the whole sequence converges weakly, not just a subsequence.

c) Key Identities and Limit Estimates:

The convergence analysis relies on identities: $x_{k-1} - y_k = a_k + b_{k-1}$

$\lim_{k\to\infty} (x_{k-1} - y_k) = 0$

and decay inequalities: $$\|p_k - p\|^2 \leq \|p_{k-1} - p\|^2 - \frac{1}{2}\|a_k + b_{k-1}\|^2$$ These provide fine control of the "gap" between DR method substeps and are instrumental in verifying that any limit point satisfies $0 \in A(x) + B(x)$.

d) Auxiliary Lemmas on Weak Closure:

Nets and sequences in Banach spaces that are weakly convergent—critical due to the weak topology—are addressed using auxiliary results, guaranteeing that limit points indeed lie in graphs of maximal monotone operators. This is essential for handling the passage to the limit in the product Hilbert space.

4. Assumptions and Their Operational Implications

The convergence theory is anchored on these specific assumptions:

• Maximal Monotonicity: $A$ and $B$ are maximal monotone on a real Hilbert (and thus reflexive) space, ensuring graph closure and applicability of Minty’s theorem.
• Nonempty Solution Set: $S(A,B)$ is nonempty, precluding pathological divergences.
• Summable Error Tolerance: The sequence $\{\epsilon_k\}$ satisfies $\sum_k \epsilon_k < \infty$, ensuring the cumulative effect of computational inexactness is asymptotically negligible.
• Topological Preliminaries: Reflexivity provides weak sequential compactness—a necessity for extracting weakly convergent subsequences from bounded sequences.

These assumptions enable robust application of the DR method in convex optimization, variational inequalities, and related operator inclusion problems, provided the operator monotonicity and error controls are respected.

5. Significance and Scope of Weak Convergence Results

Weak convergence in Hilbert spaces, as established for the DR method, distinguishes itself from strong convergence:

• It guarantees convergence in the evaluation by continuous linear functionals, not necessarily in the norm.
• In operator splitting and convex feasibility problems, weak convergence is often optimal due to the limitations imposed by infinite-dimensionality or lack of compactness.
• The result ensures that, while $\|x_k - x\|$ may not vanish, every weak limit is a genuine solution to the original inclusion problem. This is the critical convergence guarantee available in many practical applications, particularly in large-scale or function space settings.

6. Summary and Mathematical Formulations

The weak convergence approach in the context of the DR method provides the following essential workflow:

Step	Purpose	Key Formula
Inclusion Problem	Define solution target	$0 \in A(x) + B(x)$
DR Iteration	Generate split sequences	$y_k + a_k = x_{k-1} - b_{k-1}$ and $\|x_k + b_k - (y_k + b_{k-1})\| \leq \epsilon_k$
Fejér Property	Control deviation by telescoping error	$\|p_k - p\| \leq \|p_{k-1} - p\| + \delta_k$
Decay/Gaps	Quantify convergence of sub-terms	$\lim_{k\to\infty} (x_{k-1} - y_k) = 0$
Solution Set	Characterize weak limits	$S(A,B) = \{ (x, b) : b \in B(x),\; -b \in A(x) \}$

The approach demonstrates that DR-generated sequences converge weakly to pairs solving $0 \in A(x) + B(x)$, with the limit components satisfying the requisite operator inclusions. Weak convergence, though strictly less than norm convergence, is frequently the best attainable in high-dimensional variational and monotone inclusion frameworks and suffices for validation of the method’s correctness in convex analysis and related areas.

This synthesis reflects the rigorous structure, operational mechanisms, and foundational role of the weak convergence approach in advanced variational and operator-theoretic analysis (Svaiter, 2010).