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Subspace Newton's Method for $\ell_0$-Regularized Optimization Problems with Box Constraint

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Pith's one-line read The paper establishes a subspace Newton method for box-constrained ℓ0-regularized optimization that converges globally and, under a Hessian Lipschitz condition, quadratically to a strict local minimizer.

desk verdict Useful box-constrained extension of NL0R, but the main theorem analyzes a different algorithm than the one stated unless the missing τ is a typo. read the letter →

arxiv 2505.17382 v1 pith:L7QESDED submitted 2025-05-23 math.OC

Yuge Ye , Qingna Li This is my paper · ORCID
classification math.OC MSC 90C5390C3090C2665K10
keywords box-constrainedℓ0regularizationsubspaceNewtonmethodτ-stationarypointactivesetidentificationglobalconvergencequadraticcompressedsensingproximalgradient
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

The paper's goal is to give the box-constrained version of ℓ0-regularized sparse optimization the same second-order guarantee that, until now, only the unconstrained case had: an algorithm that provably converges from any start and speeds up to quadratic convergence near the answer. It defines a τ-stationary point through a proximal operator with a hard-thresholding closed form, then partitions the support into active variables (at their box bounds) and inactive variables (candidates for Newton refinement). The proposed BNL0R method updates inactive variables by a subspace Newton step and active variables by a proximal gradient step, falling back to the proximal gradient direction whenever the Newton step is not a valid descent direction. Under strong smoothness, strict complementarity, and a constraint qualification at the limit, the authors prove the whole sequence converges to a strict local minimizer and that Newton steps are eventually always accepted, with a quadratic rate when the Hessian is Lipschitz near the solution. The numerical experiments on compressed-sensing and image-recovery instances show the method reaching machine-precision recovery in four to six iterations and outperforming an ℓ1 projected-gradient solver and a proximal iterative hard-thresholding solver.

What carries the argument

The load-bearing object is the τ-stationary point. With τ>0, x is τ-stationary when it solves x = Prox_{τλ(‖·‖0+δΩ)}(x−τ∇f(x)); the paper shows the proximal operator has a closed form: coordinates stay themselves when inside the box and above √(2τλ), clip to u_i or −l_i when pushed past the boundary, and zero out otherwise. That closed form yields the index sets Θ (inactive, sign-uncertain, gradient-free target), Γ (active, at a box bound), and I (support complement, forced to zero), and the stationarity equation Fτ(x;Θ,Γ,I)=0. The algorithm applies a Newton step to the Θ block of that equation, sets Γ coordinates by projection and I coordinates to zero; a feasibility/descent criterion checks that the direction is acceptable, and when it fails a proximal gradient step (projection of x−τ∇f(x) onto the box) is used instead. The convergence proof leans on the projection inequality, the Armijo line search with a step-size lower bound, eventual stability of the index sets, and a final appeal to a standard result on unit stepsizes to convert linear contraction into the quadratic estimate.

What would settle it

A one-dimensional check settles the parameter condition: with box [−0.5, 0.5], λ=1, τ=1, and z=1, the minimizer of ½(y−1)²+‖y‖0+δΩ(y) is 0, not 0.5 as the claimed case split would return, so any proof relying on that formula would not hold for such τ.

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Extended reading notes

Core claim

The central claim is that the box-constrained ℓ0 problem min f(x)+λ‖x‖0 subject to −l≤x≤u can be solved by a hybrid subspace Newton/proximal-gradient scheme with guarantees that first-order methods for the same model lack. The authors introduce τ-stationary points, characterized by x = Prox_{τλ(‖·‖0+δΩ)}(x−τ∇f(x)), prove that every global minimizer is τ-stationary for τ<1/L and that τ-stationary points are local (or, for strongly convex f with τ>1/ℓ, unique global) minimizers. Around such a point the algorithm declares a support set, splits it into an active set at the box boundaries and an inactive set where the gradient should vanish, and solves the stationarity equation by Newton's method on the inactive block while projecting the active block onto the box and zeroing the complement. Theorem 7 asserts global convergence to a strict local minimizer, eventual unconditional acceptance of the Newton direction, and, if the Hessian is locally Lipschitz, the quadratic bound ||$x^{{k+1}}$−x*|| ≤ (M*/(2ℓ̃*))||x^k−x*||²—making this, in the authors' account, the first box-constrained ℓ0 method with a proven quadratic rate.

Load-bearing premise

The load-bearing premise is that the step size τ is small enough that √(2τλ) is smaller than every box bound, equivalently τ < min_i min(l_i²,u_i²)/(2λ); without that inequality the closed-form proximal threshold that defines the supports is not exact.

Editorial extensions

If this is right

  • For the first time, box-constrained ℓ0 problems inherit a second-order convergence theory: global convergence from any start plus local quadratic rate under Hessian Lipschitzness.
  • Support identification becomes algorithmic: after finitely many iterations the active/inactive partition is fixed, so Newton updates act on small reduced Hessians rather than the full n-dimensional system.
  • The proximal-gradient fallback is not just a safety net: it keeps iterates feasible, decreases the objective by a fixed margin, and is eventually replaced by Newton steps, so the method automatically transitions from first-order to second-order behavior.
  • On compressed-sensing problems, the method recovers the ground truth to machine precision in a handful of iterations and, in the reported experiments, dominates both ℓ1 projected gradient and proximal iterative hard thresholding in accuracy and runtime.
  • For noisy recovery and image reconstruction, the same method yields substantially higher PSNR with far fewer nonzeros, indicating the box constraints plus ℓ0 model are practically useful, not only theoretically clean.

Reading between the lines

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

  • If the τ-stationary condition is reused as a stopping certificate, the residual ‖x−Prox_{τλp}(x−τ∇f(x))‖ gives a computable, thresholded check of optimality; nothing in the paper develops it as such, but the equivalence results make it natural.
  • The independence of the proof from the specific structure of f suggests the same active-set split could be applied to ℓ0 group sparsity or ℓp penalties with box constraints; the proximal formula would need re-deriving, but the Newton-on-inactive/zero-on-rest machinery is generic.
  • Because the theorem assumes strict complementarity and a constraint qualification at the limit, degenerate limits—where a coordinate is optimal simultaneously at zero and at a bound that coincides with zero—are the likely trouble zone; a test with l_i=0 or u_i=0 would probe whether the method still converges and how the rate degrades.
  • The reported machine-precision accuracy in noise-free recovery suggests BNL0R could serve as a refinement oracle inside larger sparse-modeling pipelines, for example after a first-order method has identified a support, rather than as a replacement for inexpensive warm starts.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes BNL0R, a subspace Newton method for box-constrained ℓ0-regularized minimization. It introduces a τ-stationary point via the proximal operator of the ℓ0-plus-indicator objective, partitions the support into active and inactive sets, applies Newton's method to the inactive variables, and uses a projected gradient step as a fallback when the Newton direction is not acceptable. Under assumptions of strong smoothness, local strong convexity, strict complementarity, and Abadie's constraint qualification, the authors claim global convergence of the full sequence to a strict local minimizer, eventual acceptance of the Newton step, and local quadratic convergence when the Hessian is locally Lipschitz. The numerical section reports compressed sensing and image recovery experiments comparing BNL0R with PGA and PIHT.

Significance. If the convergence theory is made correct, the contribution is significant: it would provide a box-constrained ℓ0 method with a proven global convergence result and a local quadratic rate, extending the unconstrained subspace Newton method of [25] to the box-constrained setting and going beyond the first-order methods previously available for problem (1.1). The τ-stationarity characterization and the support-identification results are also useful. The numerical experiments are reasonably extensive, covering noise-free and noisy recovery, varying box constraints, and image recovery, and the reported gains in iteration count, time, and accuracy over PGA and PIHT support the practical value of the approach. However, as written the central theorems are not established for the algorithm that is actually stated: the standing parameter condition in (2.3) is inconsistent with the proof of Lemma 2.1 and with the later parameter choice (4.1), and the Newton direction in Algorithm 1 omits the factor τ that is used throughout the convergence analysis. These are load-bearing issues that must be repaired before the advertised results are valid.

major comments (3)
  1. [§2.1, Lemma 2.1 and Eq. (2.3)] The standing assumption (2.3), τ < 1/(2λa) with a = min_i min(l_i²,u_i²), is inconsistent with the proof of Lemma 2.1. In Cases 2 and 3 of the proof, the boundary comparison h(u_i) < h(0) requires τ < u_i²/(2λ), and similarly τ < l_i²/(2λ) for every i, i.e. τ < a/(2λ). For a < 1 the stated bound 1/(2λa) is larger than a/(2λ), so (2.3) can hold while the required inequality fails; for a > 1 it is stronger than needed. The later parameter choice (4.1) correctly uses a/(2λ), showing that the manuscript itself switches conventions. This should be repaired by replacing (2.3) with the correct condition τ < a/(2λ) and propagating it consistently through Definition 1, Lemma 2.2, and Proposition 2.
  2. [§3, Eqs. (3.2) and (3.4); §4, Eqs. (4.43)–(4.45)] The algorithm as stated defines the Newton direction on Γ_k by d^k_{Γ_k} = [Π_Ω(x^k − ∇f(x^k))]_{Γ_k} − x^k_{Γ_k} and evaluates the trial point with Π_Ω(x^k − g^k). The entire convergence analysis, however, is for the τ-projection: Proposition 4 gives (3.10) only for d_F = [Π_Ω(x−τg)]_F − x_F, Lemma 3.1 applies this inequality to d^k_{Γ_k}, and the quadratic-rate proof in (4.43)–(4.45) replaces d^k_{Γ_k} with [Π_Ω(z^k)]_{Γ_k} − x^k_{Γ_k} for z^k = x^k − τ∇f(x^k). Unless τ = 1, which is not assumed, the descent, acceptance, and quadratic-convergence proofs concern a different method from Algorithm 1. The displayed definitions in (3.2) and (3.4) should be changed to use x−τ∇f(x), or the theorem statements must be restricted to τ = 1.
  3. [§4, Lemmas 4.1 and 4.2] The descent property (4.2) and the Armijo step-size lower bound (4.5) are load-bearing: they enter (4.12)–(4.13), Lemma 4.3, and Theorem 7. Yet Lemma 4.1's proof is deferred by saying that replacing T_k and S_k with I_k and S̃_k in [25, Lemma 3] makes the proof analogous, and Lemma 4.2 similarly defers to [25, Lemma 4] after introducing ψ. This is not a self-contained proof in a journal submission, especially because the box constraint changes the projection and the inequalities with τ must be re-verified. Please supply the full arguments or quote the exact results from [25] and verify all hypotheses, including the corrected τ-condition from (2.3).
minor comments (5)
  1. [Throughout] The text refers to 'Algorithm 3' in several places (e.g., Lemmas 4.3 and 4.4, Theorem 6) while the displayed algorithm is Algorithm 1; please harmonize the numbering.
  2. [§4, Theorem 7(ii)] The sentence 'we ready to prove that condition (4.12) always holds' appears to refer to the acceptance criterion (3.3), not to the descent inequality (4.12); please correct the reference.
  3. [§5.3, Table 5] The caption of Table 5 says 'Results on E3 with m = n' but the text in §5.3 specifies m = 0.25n; the two statements should be made consistent.
  4. [§4, Eq. (4.21)] In Theorem 6(ii), the formula x^*_{Γ_l∞} = −l_{Γ_u∞} should presumably be x^*_{Γ_l∞} = −l_{Γ_l∞}; please fix this typo.
  5. [Throughout] There are numerous typographical slips such as 'marix', 'Amijio', 'T able', 'Barzilar-Borwein', and duplicated I_2 in §5.3; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence and quadratic-rate claims are derived from explicit assumptions and external benchmarks, not from fitted inputs or self-referential definitions.

full rationale

The derivation chain is not circular. The τ-stationary condition (Definition 1, Eq. (2.2)) is a fixed-point characterization, and Proposition 3 independently links it to local and global minimizers; Theorem 7 then proves global convergence and local quadratic convergence under explicit strong-smoothness, local strong-convexity, strict-complementarity, and Abadie CQ assumptions, with constants in (4.1) chosen from problem data rather than fitted to output. The repeated use of [25] (e.g., Lemmas 4.1–4.2, where the proof is said to follow by replacing T_k and S_k with I_k and S̃_k in [25, Lemma 3]) is dependence on an external unconstrained result, not a self-citation, and the box-constrained proximal operator (Lemma 2.1), index-set identification (Lemmas 4.4–4.5), and feasibility arguments are new work. No experimental quantity is produced by tuning parameters to reproduce the reported residuals; the reported res values are measured against ground truth. Two internal-consistency gaps are present but are not circularity: (a) Lemma 2.1's proof asserts 'zi ≥ ui ⇒ −uizi ≤ −u2i < −2τλ' under condition (2.3), but that strict inequality requires τλ < u_i^2/2, which (2.3) does not supply and which the later parameter choice (4.1) repairs by using a/(2λ); (b) Algorithm 1's displayed direction dΓ = [ΠΩ(x − ∇f(x))]Γ − xΓ in (3.2)/(3.4) is not the Newton direction for Fτ, whose inactive equation uses z = x − τ∇f(x) in (2.10) and (4.43). Both are correctness concerns, not reductions of the theorem to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard smoothness and convexity assumptions, plus active-set regularity conditions and algorithm parameter choices. No data-fitted free parameters or invented entities are introduced. The main issue is that condition (2.3) as written is weaker than what Lemma 2.1 requires, and the paper's own parameter selection in (4.1) uses a different bound.

assumptions (5)
  • domain assumption f is twice continuously differentiable and bounded from below
    Assumed in problem (1.1) and used throughout; standard for sparse optimization.
  • domain assumption f is strongly smooth with constant L and locally strongly convex with constant ℓ* near x*
    Definitions 2 and Theorem 7; required for descent and for the quadratic rate.
  • domain assumption The limit point satisfies strict complementarity and Abadie's CQ
    Assumed in Theorems 6 and 7 to guarantee finite identification of the active index sets.
  • ad hoc to paper ∇f(0) ≠ 0 and λ ∈ (0, λ̄)
    Remark 1 and S0 exclude the trivial case where 0 is already stationary and are used to ensure convergence to a nonzero limit.
  • ad hoc to paper Algorithm parameters satisfy (4.1)
    The conditions on τ, δ, σ, β are chosen to make the descent and line-search proofs work; they are not fitted to data.

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Pith. "Pith review of Subspace Newton's Method for $\ell_0$-Regularized Optimization Problems with Box Constraint." pith.science (2026). https://pith.science/paper/L7QESDED

@misc{pith2026250517382,
  author       = {Pith},
  title        = {Pith review of: Subspace Newton's Method for $\ell_0$-Regularized Optimization Problems with Box Constraint},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L7QESDED}},
  note         = {Machine review of arXiv:2505.17382}
}
abstract

This paper investigates the box-constrained $\ell_0$-regularized sparse optimization problem. We introduce the concept of a $\tau$-stationary point and establish its connection to the local and global minima of the box-constrained $\ell_0$-regularized sparse optimization problem. We utilize the $\tau$-stationary points to define the support set, which we divide into active and inactive components. Subsequently, the Newton's method is employed to update the non-active variables, while the proximal gradient method is utilized to update the active variables. If the Newton's method fails, we use the proximal gradient step to update all variables. Under some mild conditions, we prove the global convergence and the local quadratic convergence rate. Finally, experimental results demonstrate the efficiency of our method.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 36 canonical work pages

  1. [25]

    Numerical Algorithms, 1–30 (2021)

    Zhou, S., Pan, L., Xiu, N.: Newton method for ℓ0-regularized optimization. Numerical Algorithms, 1–30 (2021)

    work page 2021

  2. [1]

    IEEE Transactions on information theory 52(2), 489–509 (2006)

    Cand` es, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on information theory 52(2), 489–509 (2006)

    2006

  3. [2]

    IEEE Transactions on Information Theory 51(12), 4203–4215 (2005) https://doi.org/10.1109/TIT.2005

    Candes, E.J., Tao, T.: Decoding by linear programming. IEEE Transactions on Information Theory 51(12), 4203–4215 (2005) https://doi.org/10.1109/TIT.2005. 858979

    doi:10.1109/tit.2005 2005

  4. [3]

    IEEE Transactions on information theory 52(4), 1289–1306 (2006) 35

    Donoho, D.L.: Compressed sensing. IEEE Transactions on information theory 52(4), 1289–1306 (2006) 35

    work page 2006

  5. [4]

    Proceedings of the IEEE 98(6), 1031–1044 (2010)

    Wright, J., Ma, Y., Mairal, J., Sapiro, G., Huang, T.S., Yan, S.: Sparse repre- sentation for computer vision and pattern recognition. Proceedings of the IEEE 98(6), 1031–1044 (2010)

    work page 2010

  6. [5]

    IEEE Transactions on Image Processing 21(10), 4349–4360 (2012)

    Yuan, X.-T., Liu, X., Yan, S.: Visual classification with multitask joint sparse representation. IEEE Transactions on Image Processing 21(10), 4349–4360 (2012)

    work page 2012

  7. [6]

    IEEE transactions on neural networks and learning systems 23(3), 399–411 (2012)

    Bian, W., Chen, X.: Smoothing neural network for constrained non-lipschitz opti- mization with applications. IEEE transactions on neural networks and learning systems 23(3), 399–411 (2012)

    work page 2012

  8. [7]

    In: Machine Learning, Optimization, and Data Science: 6th International Conference, LOD 2020, Siena, Italy, July 19–23, 2020, Revised Selected Papers, Part II 6, pp

    Dinh, T., Wang, B., Bertozzi, A., Osher, S., Xin, J.: Sparsity meets robustness: Channel pruning for the feynman-kac formalism principled robust deep neural nets. In: Machine Learning, Optimization, and Data Science: 6th International Conference, LOD 2020, Siena, Italy, July 19–23, 2020, Revised Selected Papers, Part II 6, pp. 362–381 (2020). Springer

    work page 2020

Show all 40 references
  1. [8]

    IEEE transactions on neural networks and learning systems 31(2), 574–588 (2019)

    Lin, S., Ji, R., Li, Y., Deng, C., Li, X.: Toward compact convnets via structure- sparsity regularized filter pruning. IEEE transactions on neural networks and learning systems 31(2), 574–588 (2019)

    2019

  2. [9]

    SIAM Journal on Imaging Sciences 8(4), 2294–2322 (2015)

    Bian, W., Chen, X.: Linearly constrained non-lipschitz optimization for image restoration. SIAM Journal on Imaging Sciences 8(4), 2294–2322 (2015)

    2015

  3. [10]

    IEEE Transactions on Image Processing 21(12), 4709–4721 (2012)

    Chen, X., Ng, M.K., Zhang, C.: Non-lipschitz ℓp-regularization and box con- strained model for image restoration. IEEE Transactions on Image Processing 21(12), 4709–4721 (2012)

    2012

  4. [11]

    Springer (2010)

    Elad, M.: Sparse and Redundant Representations: from Theory to Applications in Signal and Image Processing. Springer (2010)

    2010

  5. [12]

    Proceedings of the IEEE 98(6), 972–982 (2010)

    Elad, M., Figueiredo, M.A., Ma, Y.: On the role of sparse and redundant representations in image processing. Proceedings of the IEEE 98(6), 972–982 (2010)

    2010

  6. [13]

    Com- mun

    Cand` es, E., Recht, B.: Exact matrix completion via convex optimization. Com- mun. ACM 55(6), 111–119 (2012) https://doi.org/10.1145/2184319.2184343

    2012

  7. [14]

    Journal of Computational and Graphical Statistics 16(4), 782–798 (2007)

    Liu, Y., Wu, Y.: Variable selection via a combination of the l0 and l1 penalties. Journal of Computational and Graphical Statistics 16(4), 782–798 (2007)

    2007

  8. [15]

    IEEE Transactions on Audio, Speech, and Language Processing 14(1), 50–57 (2006) https://doi.org/10.1109/TSA.2005.860346

    Blumensath, T., Davies, M.: Sparse and shift-invariant representations of music. IEEE Transactions on Audio, Speech, and Language Processing 14(1), 50–57 (2006) https://doi.org/10.1109/TSA.2005.860346

    2006

  9. [16]

    Journal of the 36 Royal Statistical Society: Series B (Methodological) 58(1), 267–288 (2018) https: //doi.org/10.1111/j.2517-6161.1996.tb02080.x

    Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the 36 Royal Statistical Society: Series B (Methodological) 58(1), 267–288 (2018) https: //doi.org/10.1111/j.2517-6161.1996.tb02080.x

    2018

  10. [17]

    Journal of Fourier analysis and Applications 14, 629–654 (2008)

    Blumensath, T., Davies, M.E.: Iterative thresholding for sparse approximations. Journal of Fourier analysis and Applications 14, 629–654 (2008)

    2008

  11. [18]

    Mathematical Programming 137(1), 91– 129 (2013)

    Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi- algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized gauss–seidel methods. Mathematical Programming 137(1), 91– 129 (2013)

    2013

  12. [19]

    Bertsimas, D., King, A., Mazumder, R.: Best subset selection via a modern optimization lens (2016)

    2016

  13. [20]

    Journal of Global Optimization 76(4), 769–791 (2020)

    Cheng, W., Chen, Z., Hu, Q.: An active set barzilar–borwein algorithm for ℓ0 regularized optimization. Journal of Global Optimization 76(4), 769–791 (2020)

    2020

  14. [21]

    Inverse Problems 30(1), 015001 (2013) https://doi

    Ito, K., Kunisch, K.: A variational approach to sparsity optimization based on lagrange multiplier theory. Inverse Problems 30(1), 015001 (2013) https://doi. org/10.1088/0266-5611/30/1/015001

    2013 doi

  15. [22]

    Journal of Machine Learning Research 19(10), 1–37 (2018)

    Huang, J., Jiao, Y., Liu, Y., Lu, X.: A constructive approach to l0 penalized regression. Journal of Machine Learning Research 19(10), 1–37 (2018)

    2018

  16. [23]

    Journal of Machine Learning Research 22(12), 1–45 (2021)

    Zhou, S., Xiu, N., Qi, H.-D.: Global and quadratic convergence of newton hard- thresholding pursuit. Journal of Machine Learning Research 22(12), 1–45 (2021)

    2021

  17. [24]

    SIAM Journal on Optimization 31(4), 3184–3211 (2021)

    Zhou, S., Pan, L., Xiu, N., Qi, H.-D.: Quadratic convergence of smoothing new- ton’s method for 0/1 loss optimization. SIAM Journal on Optimization 31(4), 3184–3211 (2021)

    2021

  18. [26]

    Optimization Letters 9(1), 57–74 (2015)

    Liuzzi, G., Rinaldi, F.: Solving ℓ0-penalized problems with simple constraints via the frank–wolfe reduced dimension method. Optimization Letters 9(1), 57–74 (2015)

    2015

  19. [27]

    arXiv preprint arXiv:2104.13107 (2021)

    Wu, F., Bian, W., Xue, X.: Smoothing fast iterative hard thresholding algo- rithm for ℓ0 regularized nonsmooth convex regression problem. arXiv preprint arXiv:2104.13107 (2021)

    2021 arXiv

  20. [28]

    Science China Mathematics 58, 1–20 (2015)

    Zhao, Z., Xu, F., Li, X.: Adaptive projected gradient thresholding methods for constrained ℓ0 problems. Science China Mathematics 58, 1–20 (2015)

    2015

  21. [29]

    SIAM Journal on Numer- ical Analysis 58(1), 858–883 (2020) https://doi.org/10.1137/18M1186009 37 https://doi.org/10.1137/18M1186009

    Bian, W., Chen, X.: A smoothing proximal gradient algorithm for nons- mooth convex regression with cardinality penalty. SIAM Journal on Numer- ical Analysis 58(1), 858–883 (2020) https://doi.org/10.1137/18M1186009 37 https://doi.org/10.1137/18M1186009

    2020 doi

  22. [30]

    Mathematical Programming 147(1), 125–154 (2014)

    Lu, Z.: Iterative hard thresholding methods for ℓ0 regularized convex cone programming. Mathematical Programming 147(1), 125–154 (2014)

    2014

  23. [31]

    Neural Networks 143, 678–689 (2021) https://doi.org/10.1016/j.neunet.2021.08.001

    Li, W., Bian, W.: Smoothing neural network for l0 regularized optimization problem with general convex constraints. Neural Networks 143, 678–689 (2021) https://doi.org/10.1016/j.neunet.2021.08.001

    2021 doi

  24. [32]

    Journal of Scientific Computing 79(2), 809–826 (2019)

    Zhang, X., Zhang, X.: A new proximal iterative hard thresholding method with extrapolation for ℓ0minimization. Journal of Scientific Computing 79(2), 809–826 (2019)

    2019

  25. [33]

    Journal of Global Optimization 76(4), 819–840 (2020)

    Wu, F., Bian, W.: Accelerated iterative hard thresholding algorithm for ℓ0 reg- ularized regression problem. Journal of Global Optimization 76(4), 819–840 (2020)

    2020

  26. [34]

    Mathematical Programming 39(1), 93–116 (1987)

    Calamai, P.H., Mor´ e, J.J.: Projected gradient methods for linearly constrained problems. Mathematical Programming 39(1), 93–116 (1987)

    1987

  27. [35]

    Mathematical Programming 48(1-3), 161–220 (1990)

    Harker, P.T., Pang, J.-S.: Finite-dimensional variational inequality and nonlin- ear complementarity problems: a survey of theory, algorithms and applications. Mathematical Programming 48(1-3), 161–220 (1990)

    1990

  28. [36]

    Springer (2003)

    Facchinei, F., Pang, J.-S.: Finite-dimensional Variational Inequalities and Com- plementarity Problems. Springer (2003)

    2003

  29. [37]

    SIAM Journal on scientific and statistical computing 4(3), 553–572 (1983)

    Mor´ e, J.J., Sorensen, D.C.: Computing a trust region step. SIAM Journal on scientific and statistical computing 4(3), 553–572 (1983)

    1983

  30. [38]

    Operations Research Letters 17(3), 131–137 (1995)

    Facchinei, F.: Minimization of sc1 functions and the maratos effect. Operations Research Letters 17(3), 131–137 (1995)

    1995

  31. [39]

    Optimization Letters 11(1), 55–70 (2017)

    Zhang, H., Cheng, L.: Projected shrinkage algorithm for box-constrained ℓ1- minimization. Optimization Letters 11(1), 55–70 (2017)

    2017

  32. [40]

    Applied Numerical Mathematics 193, 179–195 (2023) https://doi.org/10.1016/j.apnum.2023.07.004 38

    Cheng, W., LinPeng, Z., Li, D.: An inexact quasi-newton algorithm for large- scale ℓ1 optimization with box constraints. Applied Numerical Mathematics 193, 179–195 (2023) https://doi.org/10.1016/j.apnum.2023.07.004 38

    2023 doi

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