Hybrid subgradient and simulated annealing method for hemivariational inequalities

Adil M. Bagirov; Anna Ochal; Piotr Bartman-Szwarc

REVIEW 3 major objections 4 minor 16 references

Hybrid subgradient and simulated annealing method for hemivariational inequalities

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that hemivariational inequalities can be solved numerically as nonsmooth nonconvex energy minimization, combining a finite-terminating aggregate subgradient descent with simulated annealing restarts.

desk verdict The hybrid method is a reasonable engineering heuristic, but its convergence theory rests on an assumption that, taken at face value, forces convexity—so the main theorems don't hold for the nonconvex problems they claim to solve. read the letter →

arxiv 2505.23676 v1 pith:7TCZROTP submitted 2025-05-29 math.OC cs.CE

Piotr Bartman-Szwarc , Adil M. Bagirov , Anna Ochal This is my paper

classification math.OCcs.CE MSC 49J4049J5265K0590C2674M15

keywords HemivariationalinequalitiesNonsmoothoptimizationAggregatesubgradientmethodSimulatedannealingContactmechanicsNonconvexenergyFiniteelementGlobal

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

Contact problems with layered materials have energy landscapes that are both nonsmooth and nonconvex: a cracked composite layer's reaction force drops abruptly, creating many local minima. The paper's aim is to solve the resulting hemivariational inequality by minimizing its energy function directly, using a local aggregate subgradient method to find stationary points and simulated annealing to escape them by proposing new starting points. The authors prove that their local algorithm terminates in finitely many iterations at an approximately stationary point, provided a one-sided subgradient inequality holds. They further assert that the full hybrid converges to a global minimizer with probability one, because the annealing is used only to seed the local search. On a clamped beam resting on a composite foundation with 2, 3, 7, and 10 layers, the hybrid 'global subgradient' method finds lower energy than BFGS, conjugate gradient, and Powell methods, and does so faster than Powell.

What carries the argument

The object that carries the argument is the aggregate subgradient $\bar v_k = \lambda_k v_k + (1-\lambda_k)\tilde v_k$, where $\lambda_k$ minimizes $\phi_k(\lambda)=\|\lambda v_k + (1-\lambda)\tilde v_k\|^2$ over $[0,1]$, combining the freshly computed subgradient $v_k \in \partial L(u+\eta d_{k+1})$ with the previous aggregate $\tilde v_k$. Its norm decides the next move: if $\|\bar v_k\|\le\delta$, the step is null and $\eta$ is multiplied by $\gamma$; otherwise the direction $d_{k+1}=-\|\bar v_k\|^{-1}\bar v_k$ is tested against the sufficient-decrease condition (5), and a failed test forces a new subgradient outside the convex hull of the previous two, making $\|\bar v_k\|^2$ shrink by a factor $C_2<1$ each inner iteration. This gives finite termination for fixed $\eta$, and the bound $M\le (L(u_0)-L_*)/(c_2\eta\delta)+1$ for the outer loop. The global wrapper uses the Metropolis function $R(w,u,T)=\min\{1,\exp((L(u)-L(w))/T)\}$ to decide whether to accept a random coordinate perturbation as the next starting point.

What would settle it

Compute both sides of inequality (3) for the layered potentials $j_7$ and $j_{10}$ at a nondifferentiability point, with $d$ pointing into a decreasing branch of the contact law and with a subgradient $v\in\partial L(u+\tau d)$ chosen at the kink. A single violation would invalidate the descent argument in Proposition 1. A cheaper experiment is to run Algorithm 1 on a simple locally Lipschitz function that violates (3) and observe whether it still terminates at an $(\eta,\delta)$-stationary point.

Watch

Extended reading notes

Core claim

The paper's central claim is that the discrete hemivariational inequality can be replaced by the unconstrained minimization of $L(u) = \frac12\langle Au,u\rangle + \langle b,u\rangle + J(u)$, where $J$ is locally Lipschitz but generally nonsmooth and nonconvex, and that this minimization can be solved reliably by a two-phase method. Algorithm 1 forms an aggregate subgradient $\bar v_k$ from the two most recent subgradients; if its norm is small the step is declared null and the step-size parameter $\eta$ shrinks, otherwise a descent direction is tested and a serious step is taken. The proof shows that the inner loop terminates in finitely many iterations because each failed descent forces the aggregated subgradient norm to shrink geometrically, and the outer loop terminates at an $(\eta,\delta)$-stationary point. Algorithm 2 wraps this local search in a simulated-annealing loop whose Metropolis acceptance rule proposes new starting points; Remark 1 asserts that the resulting hybrid converges to a global minimizer of $L$ with probability one. The numerical experiments on a two-dimensional clamped beam report that the 'global subgradient' variant attains lower values of the energy functional than BFGS, conjugate-gradient, and Powell solvers across 2, 3, 7, and 10 composite layers, with growing advantage as layers increase.

Load-bearing premise

The descent and finite-termination proofs rely on the one-sided subgradient inequality $L(u+\tau d)-L(u)\le\tau\langle v,d\rangle$ for all $v\in\partial L(u+\tau d)$; the paper notes this holds for difference-of-convex $J$ but does not verify it for the multi-layer potentials $j_n$ used in the experiments.

Editorial extensions

If this is right

Algorithm 1 is guaranteed to reach an $(\eta,\delta)$-stationary point of the energy in finitely many iterations whenever the subgradients along the unit sphere are bounded and $L$ is bounded below.
For fixed tolerances, the inner-loop iteration count is logarithmic in $\delta/C_1$, so the local phase is practical on finite-element discretizations.
If Remark 1's global-convergence assertion holds, the hybrid is a global optimization method for this class of nonconvex nonsmooth energies, not merely a local descent.
On the layered-beam benchmark the hybrid attains lower energy values than BFGS, CG, Powell, and gradient-estimated variants, with the largest gaps for 7 and 10 layers.
Because null steps reuse only two subgradients, the memory cost per iteration is constant, making the method suitable for large-scale contact simulations.

Reading between the lines

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

The probability-one global convergence statement is stronger than the finite-termination theorem for the local phase; it inherits whatever hypotheses the cited simulated-annealing convergence result requires, so a reader should check whether those hypotheses actually hold for the energy $L$ and the random coordinate-step proposal.
A natural benchmark extension is to run Algorithm 2 on three-dimensional contact problems or on quasistatic and dynamic formulations, where the number of local minima grows and the annealing component may matter even more.
If inequality (3) is ever violated for a real contact potential, one could restore the descent proof by replacing the Clarke subdifferential with a larger set or by adding a line-search safeguard; the numerical section does not currently establish (3) for the layered potentials $j_n$.
The constant-two-subgradient aggregate makes the method nearly memory-free, so it could serve as a cheap warm-start strategy for bundle or quasi-Newton methods on nonsmooth nonconvex problems.

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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

The paper proposes a pragmatic hybrid: run the aggregate subgradient method of Bagirov et al. as a local solver, then use simulated annealing to pick restart points. The Algorithm 2 wrapper is the genuinely new bit, and the numerical results on a layered composite beam suggest it often finds lower-energy states than BFGS, CG, or Powell. They also ship the conmech package, so the experiments are reproducible in principle. I give credit for that.

The problem is the theory. The whole convergence proof of Algorithm 1 hangs on assumption (3), but that inequality is not a mild regularity condition. If you require it for every v in the Clarke subdifferential, it forces L to be convex: plugging u=b and τd=a-b gives L(b) ≥ L(a)+⟨v,b-a⟩ for all b, which means every subgradient is a global supporting subgradient. The paper says (3) holds when J is difference-of-convex; take J(x)=-x² and it fails. If instead (3) is intended only for some v, the proof never shows that the v_{k+1} selected in Step 7 satisfies it. Either reading, Propositions 1 and 2 don't apply to the nonconvex j_n potentials they use in the numerics. Remark 1's "converges to the global minimizer with probability one" is then just an assertion; the standard SA theory in [9] doesn't cover the hybrid chain, which interleaves deterministic local minimizations.

So the theoretical core is not just missing a detail—it's built on a false assumption. The numerical section is suggestive but weak on its own: no error bars, no tuning parameters for the SA schedule or the subgradient tolerances, no baseline tuning for the numpy solvers. Ten runs per load case can't reliably separate methods.

That said, this deserves a serious referee, not a desk rejection. The hybrid idea is sensible and the contact problem is hard enough that a robust heuristic is worth publishing. The fix is honest: either prove (3) under assumptions that cover the actual superpotentials, or drop the convergence theorems and present the method as a heuristic with good numerical behavior. A referee should demand this before acceptance.

Me, I'd use it in a reading group to show how a plausible regularity assumption can silently imply convexity. But I wouldn't cite it in its current form.

Referee Report

3 major / 4 minor

Summary. The paper proposes a hybrid method for minimizing nonsmooth, nonconvex energy functions that arise from hemivariational inequalities in contact mechanics. Algorithm 1 is an aggregate subgradient local search with an inner loop and an outer loop; Algorithm 2 combines this local search with a simulated annealing restart mechanism. The authors prove finite termination of the inner loop and an iteration bound for reaching an (η,δ)-stationary point, claim in Remark 1 that the hybrid method converges to a global minimizer with probability one, and report numerical comparisons on a layered beam contact problem against BFGS, CG, Powell, and two gradient-aware variants.

Significance. If the theoretical claims were valid, the paper would offer a practically useful global optimization method for a genuinely nonsmooth, nonconvex problem class in contact mechanics, with the welcome feature of explicit finite-termination bounds and an open-source implementation in the conmech package. The numerical study also has a sensible structure: it compares six solvers over multiple load levels and layer counts. However, the central convergence theory rests on assumption (3), which is not satisfied by the nonconvex problems in Section 4, and the global-convergence claim in Remark 1 is asserted rather than derived. The numerical evidence, while indicative, lacks parameter values and statistical detail, so the headline comparison is not fully substantiated.

major comments (3)

[Section 2, Eq. (3)] Assumption (3) is not a mild regularity condition and is not satisfied by the nonconvex energies considered in the paper. If (3) is required for every v in the Clarke subdifferential ∂L(u+τd), then taking u=b, d=(a-b)/||a-b||, τ=||a-b|| gives L(a)-L(b) ≤ ⟨v,a-b⟩ for every v∈∂L(a), i.e. L(b) ≥ L(a)+⟨v,b-a⟩ for all b. Thus every Clarke subgradient is a global supporting subgradient, which forces L to be convex. The energy (2) with a genuinely nonconvex superpotential J cannot satisfy this, and the paper's statement that (3) holds when J is difference-of-convex is false for nonconvex DC functions. If (3) is instead intended only for a particular v, then the proof must show that the subgradient v_{k+1} selected in Step 7 always satisfies the inequality, but no selection rule is given. Since Proposition 1 uses (3) to pass from failure of the descent test (5) to the inner-product bound (7), and Proposition 2 relies on the resulting finite termination, the convergence theory does not cover the j_n potentials used in Section 4.
[Section 2, Remark 1] The claim that the hybrid method 'converges to the global minimizer of the function L with probability one' is not proved. The cited convergence theory for simulated annealing (reference [9]) applies to the annealing process itself under specific conditions on the cooling schedule, neighborhood structure, and the objective; none of these conditions are verified for Algorithm 2. Moreover, using simulated annealing only to generate starting points for a local method that terminates at stationary points does not, by itself, imply convergence to a global minimizer, because the local method may converge to a nonglobal stationary point from every generated restart. The remark should either be replaced by a precise theorem with assumptions, or weakened to a heuristic statement.
[Section 4, numerical results] The numerical comparison is not reproducible as reported. The values of ε, δ, γ, c1, c2, T0, Tmin, α, and the number of restarts are not stated; the mesh size is only bounded above by 1.75mm, not specified; and the 10 simulations per case are summarized without error bars or any measure of dispersion. The text says the 'global subgradient' method used five starting points per run, but Algorithm 2 as written does not contain such a parameter. These omissions make it impossible to assess whether the reported advantage over Powell's method is robust.

minor comments (4)

[Section 2, Proposition 1 proof, Eq. (9)] Inequality (9) appears to have a typographical error: the second term should read (1 + 2λ(c1−1))∥¯vk∥², not (1 + 2λ(c1−1)∥¯vk∥²)∥¯vk∥². The stated form contains an extra factor ∥¯vk∥² inside the parentheses.
[Section 4, Figure 4] The left-column y-axis label is ambiguous; the text explains that the plot shows the negative of the cost function on a logarithmic scale, but the axis label should state this explicitly, e.g. '−L(u)'.
[Section 4, paragraph after Figure 3] The sentence 'The function j2 represents two composite layers represents a soft base covered by a thin (3mm) protective layer' contains a grammatical error and should be rewritten.
[Algorithm 2 and Section 4] Step 3 of Algorithm 2 generates a trial point by perturbing one coordinate by µ∈[0,1], but the relationship between this procedure and the 'five starting points' mentioned in Section 4 is unclear; the paper should specify how the number of restarts enters Algorithm 2.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the convergence proofs are self-contained and the numerical comparisons are direct; only minor self-citations appear.

full rationale

Propositions 1 and 2 derive finite termination and (eta,delta)-stationarity from explicit assumptions (local Lipschitzness, bounded subgradients, boundedness below, and inequality (3)) using self-contained estimates; the proof of Proposition 1 states the use of (3) explicitly when passing from failure of the descent test to the inner-product bound (7), and Proposition 2 supplies an explicit iteration bound. Inequality (3) is introduced as an assumption about L rather than being deduced from the results of the paper; its disputed justification for difference-of-convex J is a verification/correctness issue, not a circular dependency. Algorithm 2's global-convergence remark is an inference from the cited simulated annealing convergence theory and not a re-labelling of a fitted quantity. The numerical section compares the proposed methods directly with standard numpy solvers (BFGS, CG, Powell) on cost-function values for a specific contact problem; no parameter is fitted to a subset and then renamed as a prediction, and the 'global minimum' terminology refers to the lowest value found among the tested methods. The self-citations ([2,3,7,12]) point to the authors' earlier subgradient methods and the conmech package, but the paper's convergence argument does not reduce to those references: the key steps are proved in the text and the assumptions are stated independently. Therefore no central claim is equivalent to its inputs by construction.

Assumptions & free parameters 7 free parameters · 5 assumptions · 1 invented entities

The central convergence results rest on a regularity inequality (3), boundedness of the subdifferential (6), boundedness below (10), and the reduction from the hemivariational inequality to the discrete optimization problem taken from prior work [7]. None of these are derived in this paper.

free parameters (7)

delta (Algorithm 1 inner-loop tolerance) = not reported
Controls the stopping test in Algorithm 1 Step 4 and defines an (eta,delta)-stationary point; all numerical runs depend on it.
epsilon (Algorithm 1 outer-loop tolerance) = not reported
Determines when eta is small enough to stop Algorithm 1; not specified in the numerical section.
gamma (outer-loop reduction factor) = not reported
Scales eta in Algorithm 1 Step 9; its value affects the number of outer iterations and the quality of the stationary point.
c1 and c2 (descent control constants) = not reported
Appear in the acceptance tests (5) and step-size rule in Algorithm 1; values are not stated.
T0, Tmin, alpha (simulated annealing schedule) = not reported
Initial temperature, minimum temperature, and reduction factor in Algorithm 2; these determine how much exploration is done and are not specified.
number of restart points for global search = 5
Section 4 says a search algorithm identified five starting points for each optimization run; this count is chosen by hand and affects the results.
finite element mesh size = <= 1.75 mm
Discretization parameter h in Section 4; affects both accuracy and wall time.

assumptions (5)

domain assumption L is bounded below, L^* = inf L > -infinity (equation (10))
Used in Proposition 2 to guarantee finite outer iterations; not proved for the contact energy, though it is plausible.
domain assumption One-sided subgradient inequality (3) holds for the energy L
Assumed for all u,d,v; guarantees inner-loop finite termination and descent. The paper says it holds for DC functions but does not verify it for the contact potentials used in Section 4.
domain assumption Boundedness of subgradients over a sphere, C1 < +infinity (equation (6))
Assumed in Proposition 1; plausible for a locally Lipschitz function on a compact set but not stated for the discrete contact problem.
domain assumption Existence and equivalence of the discrete optimization reformulation P^h_opt
Taken from reference [7]; the paper does not reprove that solving P^h_opt solves the discretized hemivariational inequality.
standard math Clarke subdifferential calculus and standard finite-dimensional normed space facts
Background used in Propositions 1 and 2.

invented entities (1)

none
purpose: No new particles, forces, dimensions, or conserved quantities are introduced.
The multi-layer contact laws j_n are modeling choices within an existing problem class, not new physical entities.

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Cite this review

Pith. "Pith review of Hybrid subgradient and simulated annealing method for hemivariational inequalities." pith.science (2026). https://pith.science/paper/7TCZROTP

@misc{pith2026250523676,
  author       = {Pith},
  title        = {Pith review of: Hybrid subgradient and simulated annealing method for hemivariational inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7TCZROTP}},
  note         = {Machine review of arXiv:2505.23676}
}

read the original abstract

In this paper, we employ a global aggregate subgradient method for the numerical solution of hemivariational inequality problems arising in contact mechanics. The method integrates a global search procedure to identify starting points for a local minimization algorithm. The algorithm consists of two types of steps: null steps and serious steps. In each null step, only two subgradients are utilized: the aggregate subgradient and the subgradient computed at the current iteration point, which together determine the search direction. Furthermore, we compare the performance of the proposed method with selected solvers using a representative contact mechanics problem as a case study.

Figures

Figures reproduced from arXiv: 2505.23676 by the authors.

**Figure 1.** Modeled beam under loading. As the beam deflects under the applied force and penetrates the obstacle. The composite nature of the foundation is characterized by the subgradient of the functional jn. Examples of the functional jn for n = 2 (black) and n = 7 (gray) are illustrated in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** A nonmonotone contact law and the corresponding functional j(u). For a sake of simplicity, we neglect the body internal forces, i.e., f0 = 0. Elasticity tensors are given by (Aτ )ij = Eκ (1 + κ)(1 − 2κ) (τ11 + τ22)δij + E 1 + κ τij ∀ τ = (τij ) ∈ S 2 , i, j = 1, 2. Here and below δij is the Kronecker delta, and E and κ are Young’s modulus and Poisson’s ratio of the body material, respectively. We chose E = 9.646 · 1… view at source ↗

**Figure 3.** Deflection of the beam under loading. We present the results of numerical experiments conducted for the contact law based on the functions j2, j3, j7, and j10. For each case, we performed 10 simulations under varying load conditions. The optimization problem in each simulation was solved using six different methods which was: four optimization techniques from the numpy package [6] and two proposed in the following p… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Energy function values for selected simulations (left column) and the corresponding computation time (right column) [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [9]

Locatelli, M.: Simulated annealing algorithms for continuous global optimization: convergence conditions. J. Optim. Theory and Appl.104, 121–133 (2000). https: //doi.org/10.1023/A:1004680806815

work page doi:10.1023/a:1004680806815 2000

[2] [1]

Berlin/New York: Springer Verlag (2014)

Bagirov, A., Karmitsa, N., Mäkelä, M.: Introduction to Nonsmooth Opti- mization. Berlin/New York: Springer Verlag (2014). https://doi.org/10.1007/ 978-3-319-08114-4

work page 2014

[3] [2]

Bagirov, A., Jin, L., Karmitsa, N., Al Nuaimat, A., Sultanova, N.: Subgradient method for nonconvex nonsmooth optimization. J. Optim. Theory and Appl.157, 416–435 (2013). https://doi.org/10.1007/s10957-012-0167-6

work page doi:10.1007/s10957-012-0167-6 2013

[4] [3]

Optimization Letters15, 83–96 (2021)

Bagirov, A., Taheri, S., Joki, K., Karmitsa, N., Mäkelä, M.: Aggregate subgradient method for nonsmooth DC optimization. Optimization Letters15, 83–96 (2021). https://doi.org/10.1007/s11590-020-01586-z

work page doi:10.1007/s11590-020-01586-z 2021

[5] [4]

Acta Numerica 28, 175–286 (2019)

Han, W., Sofonea, M.: Numerical analysis of hemivariational inequalities in con- tact mechanics. Acta Numerica 28, 175–286 (2019). https://doi.org/10.1017/ S0962492919000023

work page 2019

[6] [5]

Journal of Nonlinear and Variational Analysis6, 517– 534 (2022)

Han, W., Sofonea, M.: Numerical analysis of a general elliptic variational- hemivariational inequality. Journal of Nonlinear and Variational Analysis6, 517– 534 (2022). https://doi.org/10.23952/jnva.6.2022.5.06

work page doi:10.23952/jnva.6.2022.5.06 2022

[7] [6]

Nature 585(7825), 357–362 (Sep 2020)

Harris, C.R., Millman, K.J., van der Walt, S.J., Gommers, R., Virtanen, P., Cour- napeau,D.,Wieser,E.,Taylor,J.,Berg,S.,Smith,N.J.,Kern,R.,Picus,M.,Hoyer, S., van Kerkwijk, M.H., Brett, M., Haldane, A., Fern’andez del R’io, J., Wiebe, M., Peterson, P., G’erard-Marchant, P., Sheppard, K., Reddy, T., Weckesser, W., Abbasi, H., Gohlke, C., Oliphant, T.E.: Ar...

doi:10.1038/s41586-020-2649-2 2020

[8] [7]

Applied Mathematics & Op- timization 83, 1465–1485 (2021)

Jureczka, M., Ochal, A.: A nonsmooth optimization approach for hemivariational inequalities with applications to contact mechanics. Applied Mathematics & Op- timization 83, 1465–1485 (2021). https://doi.org/10.1007/s00245-019-09593-y

work page doi:10.1007/s00245-019-09593-y 2021