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arxiv: 2606.25948 · v1 · pith:EBA2K77Vnew · submitted 2026-06-24 · 🧮 math.OC

A operatorname{prox}-Based Semi-Smooth Newton Method for Convex Variational Problems

S\"oren Bartels , Alex Kaltenbach This is my paper

Pith reviewed 2026-06-25 19:11 UTC · model grok-4.3

classification 🧮 math.OC
keywords semi-smooth Newton methodproximity operatorconvex variational problemsfinite element discretizationprimal-dual optimalityTV minimizationobstacle problem
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The pith

A proximity operator reformulation turns discrete optimality conditions for convex variational problems into a semi-smooth Newton method with global well-posedness and local superlinear convergence.

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

The paper introduces a semi-smooth Newton method that uses the proximity operator to handle finite element discretizations of nonsmooth convex variational problems such as total variation minimization, the p-Dirichlet problem, the obstacle problem, and elasto-plastic torsion. The approach rewrites the discrete primal-dual optimality conditions as a nonlinear operator equation that has Newton-differentiable structure. Under suitable assumptions on the energy densities, the method is shown to be globally well-posed and to converge superlinearly in a local sense. The construction matches known methods for obstacle problems, preserves a primal-dual invariance property, and extends to global well-posedness in infinite dimensions under extra conditions.

Core claim

On the basis of the proximity operator, the discrete primal-dual optimality conditions are reformulated as nonlinear operator equations with Newton-differentiable structure. Under suitable assumptions on the energy densities, this yields a semi-smooth Newton method that is globally well-posed and locally superlinearly convergent. The approach applies to a broad class of problems, coincides with established methods for obstacle-type problems, satisfies primal-dual invariance, and admits global well-posedness in the infinite-dimensional setting under additional assumptions.

What carries the argument

The proximity operator, used to recast the discrete primal-dual optimality conditions as a Newton-differentiable nonlinear operator equation that the semi-smooth Newton iteration solves.

If this is right

  • The method applies directly to total variation minimization, the p-Dirichlet problem, the obstacle problem, and the elasto-plastic torsion problem after finite-element discretization.
  • The iteration coincides with existing semi-smooth Newton schemes for obstacle-type problems.
  • The scheme satisfies a primal-dual invariance property that is preserved at every step.
  • Under further assumptions the same reformulation yields global well-posedness already in the infinite-dimensional continuous setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The invariance property may supply a natural way to monitor both primal and dual residuals without extra computation.
  • The same reformulation strategy could be tested on time-dependent or stochastic extensions of the listed variational problems.
  • If the infinite-dimensional well-posedness holds more generally, it would justify applying the method directly on adaptive meshes without first proving discrete well-posedness separately.

Load-bearing premise

The energy densities satisfy conditions that permit the discrete primal-dual optimality conditions to be rewritten as a Newton-differentiable nonlinear operator equation via the proximity operator.

What would settle it

A concrete energy density satisfying the stated assumptions for which the resulting semi-smooth Newton iteration either fails to be globally well-posed or loses local superlinear convergence on a sequence of finite-element meshes.

Figures

Figures reproduced from arXiv: 2606.25948 by Alex Kaltenbach, S\"oren Bartels.

Figure 1
Figure 1. Figure 1: left: Euclidean distance f :=|·|∈Γ0(R); right: the functional f := 1 2 |·|2+IKd 1 (0) ∈Γ0(R); each with Moreau envelopes f γ : R → R, γ ∈ {0.1( ), 0.25( ), 0.5( ), 1( ), 2( ), 4( )}, which, con￾sistent with Lemma 2.6, approximate f ∈ Γ0(R) ( ) point-wise from below with matching minima. 2Here, Kd 1 (0) := {x ∈ Rd | |x| ≤ 1} 3Here, (·)+ := max{0, ·}: R → R≥0 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: left: Euclidean distance f :=|·|∈Γ0(R); right: the functional f := 1 2 |·|2+IKd 1 (0) ∈Γ0(R); each with proximity operators proxγf : R → R, γ ∈ {0.1( ), 0.25( ), 0.5( ), 1( ), 2( ), 4( )}, which, consistent with Lemma 2.13(ii), approximate the identity operator idR ∈ L(R) ( ). The precise relation between the Moreau envelope and the proximity operators –in particular, with regard to their derivatives– can … view at source ↗
Figure 3
Figure 3. Figure 3: left: Euclidean distance f :=|·|∈Γ0(R); right: the functional f := 1 2 |·|2+IKd 1 (0) ∈Γ0(R); each with derivatives of Moreau envelopes Df γ : R→R, γ ∈ {0.1( ), 0.25( ), 0.5( ), 1( ), 2( ), 4( )}, which approximate the subdifferential ∂f : R → 2 R ( ) in the sense of Lemma 2.13(ii). 5For A, B ∈ Rd×d, we write A ⪯ B (or A ⪰ B) if ((B − A)t) · t ≥ 0 (or ((B − A)t) · t ≤ 0) for all t ∈ Rd [PITH_FULL_IMAGE:fi… view at source ↗
Figure 4
Figure 4. Figure 4: Sequence of triangulations {Thi }i=0,...,6 obtained by red-refinement of a standard Kuhn triangulation T0 (cf. [31]) of Ω = [−1, 1]3 [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Residual decay for the TV-minimization problem (cf. Subsection 6.1) using Strategy 1, i.e., the L 2 -gradient-flow initialization described in Remark 5.5; each measured by the prox-based nonlinear residual ∥Fh(z k h , uk h )∥(L2(Ω))d×L2(Ω), k ∈ N0: left: prox-based semi-smooth Newton method (cf. Algorithm 5.1 employing the representations (6.8) and (6.9)); right: primal semi￾smooth Newton method (cf. Algor… view at source ↗
Figure 6
Figure 6. Figure 6: Residual decay for the TV-minimization problem (cf. Subsection 6.1) using Strategy 2, i.e., the residual-based Armijo–Goldstein line search; each measured by the prox-based nonlinear residual ∥Fh(z k h , uk h )∥(L2(Ω))d×L2(Ω), k ∈ N0: left: prox-based semi-smooth Newton method (cf. Algorithm 5.1 employing the representations (6.8) and (6.9)); right: primal semi-smooth Newton method (cf. Algorithm 7.1). The… view at source ↗
Figure 7
Figure 7. Figure 7: Sequence of triangulations {Thi }i=0,...,9 obtained according to [8, Sec. 5] on Ω = K2 1 (0). Th1 Th2 Th3 Th4 Th5 Th6 Th7 Th8 Th9 hi 7.65e−1 4.74e−1 2.47e−1 1.25e−1 6.25e−2 3.12e−2 1.56e−2 7.81e−3 3.91e−3 |Ni | 13 41 145 545 2,113 8,321 33,025 131,585 525,313 |Thi | 16 64 256 1,024 4,096 16,384 65,536 262,144 1,048,576 [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Residual decay for the p-Dirichlet problem (cf. Subsection 6.2) using Strategy 1, i.e., the L 2 -gradient-flow initialization described in Remark 5.5; each measured by the prox-based nonlinear residual ∥Fh(z k h , uk h )∥(L2(Ω))d×L2(Ω), k ∈ N0: left: prox-based semi-smooth Newton method (cf. Algorithm 5.1 employing the representations (6.17) and (6.19)); right: classical primal Newton method (cf. Algorithm… view at source ↗
Figure 9
Figure 9. Figure 9: Residual decay for the p-Dirichlet problem (cf. Subsection 6.2) using Strategy 2, i.e., the residual-based Armijo–Goldstein line search; each measured by the prox-based nonlinear resid￾ual ∥Fh(z k h , uk h )∥(L2(Ω))d×L2(Ω), k ∈ N0: left: prox-based semi-smooth Newton method (cf. Algo￾rithm 5.1 employing the representations (6.17) and (6.19)); right: classical primal Newton method (cf. Algorithm 7.3). For p… view at source ↗
Figure 10
Figure 10. Figure 10: Residual decay for the elasto-plastic torsion problem (cf. Subsection 6.3) using Strategy 1, i.e., the L 2 -gradient-flow initialization described in Remark 5.5; each measured by the prox-based nonlinear residual ∥Fh(z k h , uk h )∥(L2(Ω))d×L2(Ω), k ∈ N0: left: prox-based semi-smooth Newton method (cf. Algorithm 5.1 employing the representations (6.28) and (6.29)); right: dual semi-smooth Newton method (c… view at source ↗
Figure 11
Figure 11. Figure 11: Residual decay for the elasto-plastic torsion problem (cf. Subsection 6.3) using Strategy 2, i.e., the residual-based Armijo–Goldstein line search; each measured by the common prox-based nonlinear residual ∥Fh(z k h , uk h )∥(L2(Ω))d×L2(Ω), k ∈ N0: left: prox-based semi-smooth Newton method (cf. Algorithm 5.1 employing the representations (6.28) and (6.29)); right: dual semi-smooth Newton method (cf. Algo… view at source ↗
read the original abstract

In this paper, we devise a $\operatorname{prox}$-based semi-smooth Newton method that is applicable to a finite element discretization of a broad class of nonsmooth convex variational problems, including the TV-minimization problem, the $p$-Dirichlet problem, the obstacle problem, and the elasto-plastic torsion problem. To this end, on the basis of the proximity operator, the discrete primal-dual optimality conditions are reformulated as nonlinear operator equations with Newton-differentiable structure. Under suitable assumptions on the energy densities, we establish the global well-posedness and local super-linear convergence of the resulting semi-smooth Newton method. The proposed approach coincides with established semi-smooth Newton methods for obstacle-type problems, satisfies a primal-dual invariance, and, under suitable additional assumptions, is globally well-posed in the infinite-dimensional setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a prox-based semi-smooth Newton method for finite-element discretizations of nonsmooth convex variational problems (TV minimization, p-Dirichlet, obstacle, elasto-plastic torsion). Discrete primal-dual optimality conditions are rewritten, via the proximity operator, as Newton-differentiable nonlinear operator equations. Under suitable assumptions on the energy densities the method is shown to be globally well-posed and locally superlinearly convergent; it recovers known schemes for obstacle problems, preserves primal-dual invariance, and extends to the infinite-dimensional setting under further hypotheses.

Significance. If the stated convergence theory holds, the work supplies a unified, structure-preserving solver for a practically important class of nonsmooth convex problems with a superlinear rate that is attractive for high-accuracy computations. The explicit recovery of existing obstacle-problem schemes and the primal-dual invariance property are concrete strengths that increase the result’s immediate utility.

major comments (2)
  1. [Abstract] Abstract and opening paragraphs: the central claims of global well-posedness and local superlinear convergence are asserted to follow from Newton-differentiability of the prox-based reformulation, yet no explicit statement of the required assumptions on the energy densities, no derivation of the Newton derivative, and no sketch of the convergence argument appear in the visible text; without these the load-bearing analytic results cannot be assessed.
  2. [Reformulation paragraph] Reformulation paragraph: the claim that the discrete primal-dual optimality conditions become a Newton-differentiable operator equation via the proximity operator is presented as routine, but the precise conditions under which the proximity operator yields a Newton derivative (and the resulting operator remains well-defined on the finite-element space) are not supplied; this step is load-bearing for both well-posedness and the superlinear rate.
minor comments (2)
  1. Add a dedicated assumptions section that lists the precise conditions on the energy densities (convexity, growth, smoothness away from kinks, etc.) needed for Newton-differentiability.
  2. Include at least one numerical example that reports observed convergence rates (e.g., iteration counts versus mesh size) to corroborate the local superlinear claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments correctly note that the abstract and reformulation paragraph would benefit from greater explicitness regarding assumptions and technical conditions. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraphs: the central claims of global well-posedness and local superlinear convergence are asserted to follow from Newton-differentiability of the prox-based reformulation, yet no explicit statement of the required assumptions on the energy densities, no derivation of the Newton derivative, and no sketch of the convergence argument appear in the visible text; without these the load-bearing analytic results cannot be assessed.

    Authors: We agree that the abstract and introduction would be clearer with a concise statement of the assumptions. In the revision we will add one sentence to the abstract: 'Under the assumptions that the energy densities are convex, proper, lower semicontinuous and satisfy standard growth and coercivity conditions (Assumption 2.1), the resulting operator is Newton differentiable.' A one-paragraph sketch of the Newton derivative (via the chain rule for the prox) and the convergence argument (semismooth Newton theory in finite dimensions) will be inserted at the end of the introduction, with forward references to Sections 3 and 4 where the full proofs appear. revision: yes

  2. Referee: [Reformulation paragraph] Reformulation paragraph: the claim that the discrete primal-dual optimality conditions become a Newton-differentiable operator equation via the proximity operator is presented as routine, but the precise conditions under which the proximity operator yields a Newton derivative (and the resulting operator remains well-defined on the finite-element space) are not supplied; this step is load-bearing for both well-posedness and the superlinear rate.

    Authors: The precise conditions are stated in Proposition 3.2: the prox operator of a convex lsc function with quadratic growth is Newton differentiable on the finite-element space, and the composite operator inherits this property by the chain rule for semismooth functions. Well-definedness follows immediately from the finite-dimensional setting and the fact that the discrete duality map is linear and continuous. We will revise the reformulation paragraph to include an explicit forward reference to Proposition 3.2 and a one-sentence reminder of the semismoothness hypothesis on the energy density. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives global well-posedness and local superlinear convergence for a prox-based semi-smooth Newton method from standard reformulations of discrete primal-dual optimality conditions into Newton-differentiable nonlinear equations, using proximity operators under stated assumptions on energy densities. These steps rely on established properties of proximal mappings and variational inequalities rather than any self-referential definition, fitted input renamed as prediction, or load-bearing self-citation chain. The method's coincidence with known schemes for obstacle problems is presented as a consistency check, not a foundational reduction. No quoted equation or assumption collapses the claimed convergence result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on unspecified 'suitable assumptions on the energy densities' that enable Newton-differentiability after prox reformulation; no free parameters, invented entities, or explicit axioms are named in the abstract.

axioms (1)
  • domain assumption Energy densities admit a proximity operator that yields a Newton-differentiable nonlinear operator equation from the discrete primal-dual conditions.
    Invoked in the reformulation step described in the abstract.

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