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Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification
Weifeng Yang
TL;DR
The paper asks whether Rockafellar’s interior-domain condition ensures maximality of sums in arbitrary Banach spaces. It develops a monotone-polar construction theorem and verifies it on c0, then transfers the resulting counterexample to standard ℓ1. In both spaces, the operators are maximally monotone and satisfy the condition, but their sum is not maximally monotone.
Problem
The paper addresses the unrestricted sum question: whether the interior-domain condition alone guarantees maximality of sums of maximally monotone operators on arbitrary Banach spaces.
Method
The paper computes monotone polars and maximality criteria for a class of graphs, uses a positive rank-one perturbation on c0, and transfers the example through a bounded linear surjection onto c0.
Results
The paper obtains counterexamples on c0 and standard ℓ1 in which both operators are maximally monotone, satisfy the interior-domain condition, and have a nonmaximally monotone sum.
Takeaways & Limitations
The constructions show that Rockafellar’s interior-domain condition is not sufficient for maximality of sums in these nonreflexive Banach spaces.
Takeaways & Limitations
The construction relies on the stated assumptions of the general theorem, including hypotheses such as (H4), rather than applying without conditions to arbitrary graphs.
Abstract
from arXiv · showhide
We construct counterexamples to Rockafellar's sum conjecture in which two maximally monotone operators satisfy the interior-domain condition but their sum is not maximally monotone. We give one counterexample on $c_0$ and another on $\ell^1$ with its usual norm. We establish a general construction theorem that computes the entire monotone polar of a class of graphs, gives a necessary and sufficient condition for their maximal monotonicity, and shows how a positive rank-one perturbation yields a nonmaximal sum under this condition. We verify the theorem's hypotheses and its maximality criterion on $c_0$, thereby obtaining a counterexample to the conjecture. Furthermore, we construct a bounded linear surjection from $\ell^1$ onto $c_0$ and use it to obtain the counterexample on $\ell^1$.
1 Introduction
The paper addresses whether Rockafellar’s interior-domain condition guarantees maximality of operator sums in arbitrary Banach spaces. It develops a general counterexample framework and instantiates it on c0 and standard ℓ1.
- 1 Introduction: The unrestricted sum question asks whether the interior-domain condition alone suffices for maximality on arbitrary real Banach spaces.The question is posed in the full dual pair X × X∗.
- 1 Introduction: The paper constructs counterexamples showing that the interior-domain condition can hold while the sum of two maximally monotone operators is not maximally monotone.The examples are given on c0 and standard ℓ1.
- 1.1 Contributions: The paper’s general construction theorem computes monotone polars, characterizes maximality, and identifies a positive rank-one perturbation producing a nonmaximal sum.Under its assumptions, the constructed graph is maximally monotone, but its sum with an everywhere-defined positive rank-one operator is not.
- 1.1 Contributions: On c0, the authors verify the theorem’s assumptions using coupled triangular maps and establish the original interior-domain condition.The second operator is bounded, positive, rank-one, and everywhere defined, while the first has nonempty domain.
- 1.1 Contributions: A bounded linear surjection from ℓ1 onto c0 transfers the counterexample to standard ℓ1, where both operators remain maximally monotone but their sum is not.The paper uses a pullback construction for this transfer.
2 Preliminaries
The preliminaries fix the Banach-space setting and define monotonicity, monotone polars, maximality, and finite radial bounds. These notions provide the criteria used later to analyze the constructed graphs.
- 2 Preliminaries: The paper works in real Banach spaces with norm-topology interiors and closures, using the continuous dual X∗ and the dual pairing ⟨x,a⟩.These conventions determine the ambient setting for all monotonicity statements.
- 2 Preliminaries: For bounded linear maps, positivity means ⟨x,Px⟩≥0 for every x, while rank one means that the range has dimension one.These properties describe the perturbation operator used in the construction.
- 2 Preliminaries: The monotone polar G^μ consists of points (z,p) satisfying ⟨z−x,p−a⟩≥0 for every (x,a) in G.This definition formalizes being monotonically related to a graph.
- 2 Preliminaries: A graph is maximally monotone exactly when it equals its monotone polar.The monotone polar contains all points monotonically related to every point of the graph.
- 2 Preliminaries: A finite radial bound at the origin for a nonempty graph excluding the origin is equivalent to a uniform lower bound ⟨x,a⟩≥−C∥x∥.The least admissible constant is the radial-bound quantity V(G).
3 Maximal monotonicity and nonmaximal sums
The paper develops a graph construction whose monotone polar and maximality can be characterized explicitly, then uses a positive rank-one perturbation to produce nonmaximal sums satisfying the interior-domain condition. A bounded-surjection pullback transfers the construction from c0 to standard ℓ1.
- Graph construction: The constrained graph is formed by requiring the bounded map M to follow a prescribed Lipschitz curve indexed by the scalar parameter r(a).The method introduces linear maps E and M, a nonzero functional g, and a Lipschitz curve ω to define the graph and its admissible fibres.
- Construction theorem: The construction theorem computes the entire monotone polar and characterizes maximality by excluding dual variables satisfying Eq. (9).Under Assumption 1, the graph is maximally monotone exactly when no (p, τ) with τ in [−1, 1] satisfies r(p) = τ and Mp = (τ, η).
- Polar computation: The proof identifies all polar points by exploiting affine parameter fibres, annihilation of the subspace K, and separate cases for parameters inside and outside [−1, 1].This establishes the explicit polar formula and its maximal monotonicity before deriving the criterion for the original graph.
- Construction theorem: When Eq. (9) holds, adding the everywhere-defined positive rank-one operator P to the maximally monotone operator A yields a nonmaximal sum.Both A and P are maximally monotone and satisfy the interior-domain condition, while (0, 0) gives a proper monotone extension of their sum.
- Pullback lemma: Applying the pullback lemma to a bounded surjection from ℓ1 onto c0 produces the second counterexample in ℓ1.The resulting operators are maximally monotone in ℓ1 × ℓ∞, while their sum remains nonmaximal because (0, 0) is monotonically related to its graph but is not in it.
- Pullback lemma: A bounded linear surjection Q transfers maximal monotonicity from a Banach space V to the pullback graph in U × U∗.The proof shows that the pulled-back dual variable annihilates ker Q, factors through Q, and then invokes maximality of the original operator.
4 Counterexamples on c0 and standard ℓ1
The paper constructs counterexamples on c0 and standard ℓ1 where maximally monotone operators satisfy the interior-domain condition but have a nonmaximally monotone sum.
- A counterexample on c0: The c0 hypotheses are verified using blockwise maps, a Lipschitz curve of block sums, parameter fibres, and the monotone-polar construction theorem.The theorem’s hypotheses and maximality criterion are established through Lemmas 2–4 before applying Theorem 1.
- A counterexample on c0: The c0 construction yields maximally monotone A and P satisfying the interior-domain condition, while A + P is not maximally monotone.The sum admits (0, 0) as a monotone extension point outside its graph.
- A counterexample on c0: The c0 operator A has graph points with ∥x∥∞ > 1/2 and satisfies ⟨x, a⟩ ≥ −2σ∥x∥∞, with σ = π2/96 < 1/8.These bounds imply V(gra A) ≤ 2σ < 1/4.
- A counterexample on c0: Every point of gra(A + P) satisfies ⟨x, b⟩ > 1 −σ > 7/8, making (0, 0) monotonically related to the entire graph.This strict lower bound is the key quantitative obstruction used to prove nonmaximality.
- A counterexample on standard ℓ1: A bounded surjection Q from standard ℓ1 onto c0 transfers the construction, with bounded preimages supplied by an explicit lifting argument.The quotient has ∥Q∥ = 1 and supports the pullback lemma.
- A counterexample on standard ℓ1: On standard ℓ1, T and B are maximally monotone, B is everywhere-defined positive rank-one, and T + B is not maximally monotone.The transferred pair preserves the interior-domain condition and has (0, 0) in the monotone polar but outside the sum graph.
5 Conclusions
The paper concludes that both c0 and standard ℓ1 furnish counterexamples to Rockafellar’s sum conjecture under the original interior-domain condition.
- 5 Conclusions: Both constructed pairs consist of maximally monotone operators satisfying the interior-domain condition whose sum is not maximally monotone.The paper also establishes a finite radial bound for the first operator in each pair.
Use of AI
The supplied passage describes prior materials and iterative discussions that guided further development of the framework for studying failure of maximality under addition.
- Use of AI: The work used prior constructions, obstruction results, and discussions to develop a framework for investigating maximality failure under addition.The passage mentions blockwise triangular operators, positive rank-one perturbations, and finite radial-bound obstructions.