Outer Approximation Methods for Solving Variational Inequalities Defined over the Solution Set of a Split Convex Feasibility Problem

Andrzej Cegielski; Aviv Gibali; Rafa{\l} Zalas; Simeon Reich

Outer Approximation Methods for Solving Variational Inequalities Defined over the Solution Set of a Split Convex Feasibility Problem

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Pith's one-line read This paper proves that a variational inequality over the split feasibility solution set $S=C\cap A^{-1}(Q)$ can be solved by outer approximation steps onto half-spaces, with norm convergence under a closed-range condition on $A$.

desk verdict A competent synthesis of outer-approximation, Landweber, and SQNE tools into one convergence theorem; the closed-range assumption is the real scope limit, but the proof is sound and the paper deserves referee time. read the letter →

arxiv 1908.07398 v1 pith:TIFCLTN3 submitted 2019-08-20 math.OC

Andrzej Cegielski , Aviv Gibali , Simeon Reich , Rafa{l} Zalas This is my paper · ORCID

classification math.OC MSC 47H0947H1047J2047J2565K15

keywords variationalinequalitysplitconvexfeasibilityproblemCQ-methodouterapproximationmethodLandwebertransformstronglyquasi-nonexpansiveoperatorshalf-spaceprojectionnormconvergence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that a variational inequality governed by a strongly monotone, Lipschitz continuous operator $F$ can be solved over the solution set $S=C\cap A^{-1}(Q)$ of a split convex feasibility problem by an outer approximation method in which each metric projection onto $S$ is replaced by a metric projection onto a half-space $H_k$ that contains $S$ and is easy to evaluate. The half-spaces are constructed from the split structure in three variants---product, simultaneous, and alternating combinations of operators acting on $C$ and $Q$---using the Landweber transform to pull the constraint $Az\in Q$ back to $H_1$. The main theorem shows that if the step sizes $\lambda_k$ satisfy $\lambda_k\to 0$ and $\sum\lambda_k=\infty$, if the linear operator $A$ has closed range, and if the relevant families of sets are boundedly regular, then the iterates converge in norm to the unique solution of $\mathrm{VI}(F,S)$. The significance is that projections onto $S$, or onto $A^{-1}(Q)$, are generally expensive, while the half-space projections used here reduce to closed-form formulas involving $A$, $A^*$, and the current iterate.

What carries the argument

The central machinery is the half-space projection step (1.3)--(1.5), where $H_k=\{z\in H_1: \langle u_k-T_k(u_k), z-T_k(u_k)\rangle\le 0\}$ and $T_k$ is a cutter with $S\subseteq \mathrm{Fix}\,T_k$. For the split constraint, the extrapolated Landweber transform $L_\sigma\{V\}(x)=x+\sigma(x)/\|A\|^2\, A^*(V(Ax)-Ax)$ is the transfer device: under $R(A)\cap \mathrm{Fix}\,V\neq\emptyset$ it is $\rho$-strongly quasi-nonexpansive and has fixed point set $A^{-1}(\mathrm{Fix}\,V)$, so applying it to operators on $H_2$ produces operators on $H_1$ that encode the condition $Az\in Q$. Lemma 2.13 then rewrites $H_k$ as $\{z: \langle Au_k-V_k(Au_k), Az-V_k(Au_k)\rangle\le 0\}=A^{-1}(H_2(Au_k,V_k(Au_k)))$, giving a closed-form projection (2.20). The three variants differ only in how $U_k$ and $L_\sigma\{V_k\}$ are combined---as a product, as a convex combination, or in alternation---and the proof's inequalities (3.17) and (3.23) feed the resulting residuals into the regularity conditions (3.8)--(3.9) and $s$-intermittent control sequences, yielding $d(u_k,S)\to 0$.

What would settle it

A concrete check of the boundary: take $H_1=\ell^2$ and $A$ defined by $(Ax)_n=x_n/n$, which is injective with dense, non-closed range and $|A|=0$; set $C=H_1$, $Q=\{0\}$, and $F=\mathrm{Id}$, so $S=\{0\}$ is the unique solution. The theorem's closed-range assumption fails exactly at estimate (3.31), which would divide by $|A|$; showing whether the iteration still converges for this $A$, or where the distance estimate breaks, would decide whether the closed-range hypothesis is intrinsic to the claim or only an artefact of the proof.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the split structure of $S$ can be turned into the driving device of the algorithm rather than an obstacle. Given two sequences of strongly quasi-nonexpansive operators $U_k$ on $H_1$ with $C\subseteq \mathrm{Fix}\,U_k$ and $V_k$ on $H_2$ with $Q\subseteq \mathrm{Fix}\,V_k$, Theorem 3.1 defines algorithmic operators $T_k$ in three ways and proves that the outer approximation recurrence $u_{k+1}=R_k(u_k-\lambda_k F(u_k))$, $R_k=\mathrm{Id}+\alpha_k(P_{H_k}-\mathrm{Id})$, with $H_k=\{z\in H_1: \langle u_k-T_k(u_k), z-T_k(u_k)\rangle\le 0\}$, satisfies $d(u_k,S)\to 0$; with $\sum\lambda_k=\infty$ the sequence converges in norm to the unique solution of $\mathrm{VI}(F,S)$. The proof is carried by the Landweber transform identity $\mathrm{Fix}\,L_\sigma\{V_k\}=A^{-1}(\mathrm{Fix}\,V_k)$ and by the closed-range inequality $d(u,A^{-1}(Q))\le (1/|A|)\,d(Au,R(A)\cap Q)$, which together transfer progress on the $Q$-side to progress on the $S$-side.

Load-bearing premise

The load-bearing assumption is that the linear map $A$ has closed range; without it $|A|$ can be $0$ and the proof's bound $d(u,A^{-1}(Q))\le (1/|A|)\,d(Au,R(A)\cap Q)$ collapses, so the argument cannot transfer convergence from the $Q$-side back to the $S$-side.

Editorial extensions

If this is right

In finite-dimensional spaces, the closed-range and bounded-regularity assumptions hold automatically, so Theorem 3.1 guarantees norm convergence for every bounded linear $A$ and for all three variants.
When the split part is absent, Theorem 3.7 applies the same half-space construction to ordinary convex feasibility and yields the same norm convergence under the regularity condition (3.38).
Each iteration uses only the closed-form projection onto the half-space $H_k$, so the method avoids any projection onto $A^{-1}(Q)$ or onto $S$; when only subgradients are available, Lemma 2.14 provides an explicit formula of the same type.
The step-size requirement is only $\lambda_k\to 0$ with $\sum\lambda_k=\infty$, so the algorithm does not need to know the strong monotonicity or Lipschitz constants of $F$ in order to choose its steps.
For the alternating variant with maximal extrapolation, the half-space is the preimage under $A$ of a half-space in $H_2$, making the method a direct split analogue of the CQ method described in Remark 3.2.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the closed-range condition on $A$ is likely the real boundary of the theory: for a compact injective $A$ with dense non-closed range, $|A|=0$ and inequality (3.31) cannot hold, so a different distance-transfer argument would be needed to cover such operators.
Inference: Lemma 2.14 suggests a nonsmooth variant of the method in which the split constraint is handled by subgradients of $q\circ A$ rather than projections onto $Q$; the paper provides the projection formula but does not develop this as a separate algorithm.
Inference: the paper does not compare the product, simultaneous, and alternating variants quantitatively; a finite-dimensional test on a multiple-set split convex feasibility problem with known solution would be a natural way to see whether the differences in composition and relaxation parameters affect practical speed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a competent synthesis of three known building blocks — outer-approximation half-spaces, the Landweber transform, and SQNE machinery — into a single theorem (3.1) covering product, simultaneous, and alternating variants. The proof is carefully structured and, on my reading, correct. The closed-range assumption on A is the true scope limit, but it is explicit in the statement, so the theorem as stated stands.

The new content is the three half-space constructions and the unified convergence guarantee. The operator theory itself is mostly borrowed from [24], [15], and [8]; that is fine, because the contribution is knowing how to put those pieces together. Lemma 2.13 and 2.14 give explicit projection formulas for the Landweber half-spaces, which is genuinely useful for implementation. Step 3 of the proof handles the product, simultaneous, and alternating cases separately and coherently; Step 4 then makes the right use of Theorem 2.12 to pass from distance to R(A) ∩ Q to distance to A^{-1}(Q).

The soft spots are real but not disabling. The closed-range condition is load-bearing exactly at inequality (3.31): if R(A) is not closed then |A| = 0 and the estimate has no finite right-hand side. The stress-test note is correct about that. Since the theorem states the assumption openly, this is a scope restriction rather than an error, but the paper could be more candid that the advertised infinite-dimensional setting is narrower than it sounds — compact diagonal operators on ℓ2 fail it. Bounded regularity is also assumed on both families; Remark 3.4 notes the finite-dimensional case covers this, but no rates or comparisons are given. There are no numerical experiments, so we learn nothing about whether the three variants behave differently in practice. That is a limitation, not a correctness issue.

The reliance on earlier self-authored work is heavy but not circular: [13], [15], and [24] are parameter-free theorems that do not contain Theorem 3.1. I agree with the reader's low circularity score.

This paper is for specialists in projection and fixed-point algorithms, particularly those working on split feasibility problems with variational inequalities. They will find a clean framework and a reusable template. I would want a referee who knows the regularity machinery to double-check conditions (3.8)–(3.9) and the s-intermittent controls, but I found no algebraic or logical gap. Recommendation: send it out; acceptance is defensible, with minor revision possibly asking for a comment on the closed-range condition and, if available, a small numerical illustration.

Referee Report

0 major / 5 minor

Summary. The paper studies variational inequalities governed by a strongly monotone and Lipschitz continuous operator F over a closed convex set S that is the solution set of a multiple-set split convex feasibility problem, S = C ∩ A^{-1}(Q). It proposes three variants of an outer approximation method — product, simultaneous, and alternating — in which the difficult projection onto S is replaced by a projection onto a half-space built from a Landweber-type operator acting on the split part. Under explicit hypotheses (strong quasi-nonexpansiveness of the constituent operators, s-intermittent control sequences, bounded regularity of the relevant families, closed range of A, and step sizes λ_k satisfying λ_k → 0 and ∑λ_k = ∞), Theorem 3.1 establishes norm convergence of the iterates to the unique solution of the variational inequality. The proof verifies the regularity condition (1.7) of the general convergence result [24, Theorem 3.1] by showing, in Step 3, that small residuals of T_k imply vanishing distances to the individual sets C_i and Q_j, and then, in Step 4, uses bounded regularity and the closed-range estimate (3.31) to conclude d(u_k, S) → 0.

Significance. If the result holds, the paper provides a useful unified framework for solving variational inequalities over split feasibility sets, covering CQ, simultaneous CQ, and alternating variants, with multiple sets and general quasi-nonexpansive operators. The proof is careful and detailed, and the dependence on the authors' earlier results [15, 24] is transparent citation rather than circular reasoning. The assumptions are stated explicitly, and the three cases are genuinely different algorithmic constructions. The main scope limitation is the closed-range assumption on A, which is load-bearing in Step 4 via inequality (3.31): when R(A) is not closed, |A| = 0 and the estimate d(u_k, A^{-1}(Q)) ≤ (1/|A|) d(Au_k, R(A) ∩ Q) collapses. Because the assumption is explicitly part of Theorem 3.1, this is a limitation rather than an error; nevertheless, it should be made more visible to readers interested in infinite-dimensional applications.

minor comments (5)

[Remark 3.2] The first displayed operator should be U_k := P_C, not P_Q; as written, the remark says a projection onto Q acts on H_1, which is undefined.
[Theorem 3.1] The theorem statement should state explicitly that A is nonzero; Definition 2.9 assumes this, and the factor |A| in inequality (3.31) requires it.
[Proof of Theorem 3.1, Step 3] The notation i_k := argmax_{i∈I} d(u_{n_k}, C_i) is ambiguous when the maximum is attained at several indices; the authors should say that i_k is chosen arbitrarily among the maximizers.
[Equations (2.20) and (2.23)] The positive-part notation (·)_+ is used without definition; a brief definition would improve readability.
[Introduction and abstract] The abstract and introduction advertise the general Hilbert-space setting without qualification; since the closed-range assumption on A in Theorem 3.1 is a substantial restriction in infinite dimensions, it would be helpful to flag this limitation in the introduction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the convergence proof is self-contained under the stated hypotheses and the prior citations are load-bearing only as parameter-free external theorems.

full rationale

Theorem 3.1 is a genuine extension, not a restatement of its inputs. The convergence proof invokes Theorem 1.1 from [24] only as a black-box template: it reduces the task to proving the regularity implication (3.10), which the paper proves directly from the construction of T_k and the assumed hypotheses (3.8)–(3.9). The operators U_k, V_k and the Landweber transforms are defined independently of the target set S, and the inclusion S ⊆ Fix T_k is verified from Theorems 2.4–2.6 and 2.12, all of which are stated with explicit assumptions that do not include Theorem 3.1. The only split-specific estimate, inequality (3.31), is quoted from prior work [15, Lemma 4.4] and is exactly the place where the explicit assumption that R(A) is closed is used; it is an assumption, not an output, and the reader's note correctly identifies this as the main scope restriction rather than a circularity. Each step of Step 3 is a direct norm estimate leading to (3.16), and Step 4 applies bounded regularity to reach d(u_{n_k},S) -> 0. No parameter is fitted to a subset of data and later called a prediction, no quantity is defined in terms of the result it is supposed to establish, and no uniqueness theorem is imported to forbid alternatives. The self-citations to [13], [15], and [24] are normal scholarly dependencies on prior published theorems that are stated with assumptions external to the present result. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

All assumptions are listed in Theorem 3.1; none are fitted to data. The main external inputs are prior theorems on SQNE operators, the Landweber transform, and outer approximation convergence, cited rather than reproved. The new content is the construction of half-spaces and the proof that componentwise regularity plus bounded regularity and closed range imply norm convergence. No invented entities or fitted numerical parameters are introduced; algorithm control parameters such as lambda_k, eta_k, rho_k, and sigma_k are design choices with stated ranges, not fitted constants.

assumptions (9)

standard math Closed Range Theorem and the identity |A| = |A*| = sqrt(|AA*|) (Theorem 2.1)
Used to justify |A| > 0 and the distance bound (2.18); proof is cited to [15, Lemma 3.2] and not reproduced here.
standard math Landweber transform properties (Theorem 2.12): for a rho-SQNE V with R(A) intersection Fix V nonempty, L_sigma{V} is rho-SQNE and Fix L_sigma{V} = A^{-1}(Fix V)
Central for the split part; cited to [13, Theorem 4.1] and [15, Lemmata 4.4 and 4.6].
standard math Composition and convex combinations of SQNE operators remain SQNE with controlled constants and fixed point intersections (Theorems 2.5 and 2.6)
Used to prove T_k are cutters and to derive lower bounds (3.17) and (3.23); cited to [8] and [17].
standard math Outer approximation convergence template (Theorem 1.1 from [24, Theorem 3.1])
Assumed as background; the whole proof reduces the new method to verifying condition (1.7).
domain assumption F is L-Lipschitz and alpha-strongly monotone
Guarantees uniqueness of the VI solution and the convergence framework; required by Theorem 1.1.
domain assumption R(A) is closed
Needed for |A| > 0 and for inequality (2.18)/(3.31); automatic in finite dimensions but can fail in infinite-dimensional Hilbert spaces.
domain assumption Bounded regularity of the families {A^{-1}(Q), C_1, ..., C_m} and {R(A), Q_1, ..., Q_n}
Needed in Step 4 to convert max_i d(u,C_i) to 0 and max_j d(Au,Q_j) to 0 into d(u,S) to 0.
domain assumption Sequences U_k and V_k are beta_k- and gamma_k-SQNE with uniform positive lower bounds, C subset Fix U_k, Q subset Fix V_k, and satisfy regularity conditions (3.8) and (3.9)
Defines admissible operators and provides the limit implications used in Step 3.
domain assumption Step sizes lambda_k >= 0 satisfy lambda_k to 0 and the sum of lambda_k is infinite; control index sets are s-intermittent
Standard for gradient-like methods; s-intermittency lets each constraint appear often enough in Step 3.

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Pith. "Pith review of Outer Approximation Methods for Solving Variational Inequalities Defined over the Solution Set of a Split Convex Feasibility Problem." pith.science (2026). https://pith.science/paper/TIFCLTN3

@misc{pith2026190807398,
  author       = {Pith},
  title        = {Pith review of: Outer Approximation Methods for Solving Variational Inequalities Defined over the Solution Set of a Split Convex Feasibility Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TIFCLTN3}},
  note         = {Machine review of arXiv:1908.07398}
}

abstract

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split convex feasibility problem, where $C$ and $Q$ are both closed and convex subsets of two real Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, respectively, and the operator $A$ acting between them is linear. We consider a modification of the gradient projection method the main idea of which is to replace at each step the metric projection onto $S$ by another metric projection onto a half-space which contains $S$. We propose three variants of a method for constructing the above-mentioned half-spaces by employing the multiple-set and the split structure of the set $S$. For the split part we make use of the Landweber transform.

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Optimization 66(3), 417–437 (2017)

Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66(3), 417–437 (2017)

work page 2017

[2] [15]

Optimization (2019)

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work page 2016

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In: Fixed point theory and its applications, pp

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