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Inertial dynamics with vanishing Tikhonov regularization for multiobjective optimization

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Pith's one-line read For convex multiobjective problems, a damped inertial flow with vanishing Tikhonov regularization achieves $O(t^{-p})$ merit decay and, in the main regime, strong convergence to the minimum-norm weak Pareto point.

desk verdict A technically careful multiobjective analogue of TRIGS with a genuinely new regularization path, but the headline strong-convergence result is conditional on a restrictive continuity assumption (A3) that can fail for simple convex smooth problems. read the letter →

arxiv 2411.18422 v2 pith:56X5AQQ3 submitted 2024-11-27 math.OC

Radu Ioan Bot , Konstantin Sonntag This is my paper · ORCID
classification math.OC MSC 90C2990C3090C2591A1291B5534G2034E1037L05
keywords multiobjectiveoptimizationParetooptimalityTikhonovregularizationinertialdynamicsasymptoticvanishingdampingstrongconvergencemeritfunctionLyapunovanalysis
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 studies a second-order inertial dynamics for convex differentiable multiobjective problems, in which both the damping $\alpha/t^q$ and the Tikhonov regularization $\beta/t^p$ vanish as $t\to+\infty$. Its main assertion is that, for $q\in(0,1)$ and $p

What carries the argument

The machinery is a Lyapunov analysis built around the generalized Tikhonov regularization path. For $\lambda>0$ and $r\in[q,1]$, the paper defines energies $$E^r_{\$\lambda$}(t)=$t^{{2r}}$\varphi_t(x(t))+\tfrac12\|\$\lambda$(x(t)-z(t))+t^r\dot x(t)\|^2+\tfrac{\$\lambda$}{2}\bigl(r $t^{{r-1}}$+\$\alpha$ $t^{{r-q}}$-2\$\lambda$\bigr)\|x(t)-z(t)\|^2,$$ where $\varphi_t$ is the merit function of the regularized problem and $z(t)$ is the generalized path. Estimates on the derivative of this energy, combined with an integrating factor and an auxiliary integrability lemma, yield the decay rates and the key estimate $\|x(t)-z(t)\|\to0$. Strong convergence follows once the continuity assumption (A3) guarantees that $z(t)$ itself converges to the minimum-norm element of the limiting lower-level set.

What would settle it

Take the two convex functions defined in Example 2.3 with the explicit regularization path (2.12); as $t\to+\infty$ the second coordinate of $z(t)$ oscillates between $2.25$ and $2.75$, so $z(t)$ does not converge to the minimum-norm weak Pareto point $(0,2)$, demonstrating that without the continuity assumption (A3) the path-convergence mechanism of Theorem 2.1 — and therefore the strong convergence in Theorem 4.8 — fails.

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Extended reading notes

Core claim

The central claim is Theorem 4.8: under Assumptions (A1)-(A3), if $q\in(0,1)$ and $p<q+1$, every bounded trajectory solution $x(\cdot)$ of (MTRIGS), $$\ddot x(t)+\frac{\$\alpha$}{t^q}\dot x(t)+\operatorname{proj}_{C(x(t))+\frac{\$\beta$}{t^p}x(t)+\ddot x(t)}(0)=0,\qquad C(x)=\operatorname{conv}\{\nabla f_i(x):i=1,\dots,m\},$$ converges strongly to a weak Pareto optimal point $x^*$ that minimizes the norm on $\bigcap_{i=1}^m L(f_i,f_i(x^*))$, and the merit function satisfies $\varphi(x(t))=O(t^{-p})$ (Theorem 4.7). The proof proceeds by attaching to each trajectory a generalized regularization path $z(t)=\arg\min_z \max_i(f_i(z)-f_i(x(t)))+\frac{\beta}{2t^p}\|z\|^2$, showing $z(t)\to x^*$ and then $\|x(t)-z(t)\|\(\to0$). The paper also establishes weak convergence in the regimes $q+1<p$ and the boundary case $p=2$, $\beta\geq q(1-q)$, and the rate $\varphi(x(t))=O(t^{-2q})$ when $2q<p$.

Load-bearing premise

The load-bearing premise is assumption (A3): the map that sends a reference vector $q$ to the minimum-norm solution $z_0(q)$ of $\arg\min_z\max_i(f_i(z)-q_i)$ must be continuous, and the paper's Example 2.3 shows this continuity can fail for perfectly convex functions with Lipschitz gradients; if it fails, the regularization path can oscillate and the strong-convergence conclusion collapses.

Editorial extensions

If this is right

  • In the regime $q\in(0,1)$ with $p<q+1$, any bounded trajectory of (MTRIGS) converges strongly to a weak Pareto optimal point that is the minimum-norm element of the intersection of lower level sets at the limit function values.
  • The merit function values decay as $O(t^{-p})$; since $p$ can be chosen below but arbitrarily close to $q+1<2$, the value convergence can be made arbitrarily close to the $O(t^{-2})$ rate typical of Nesterov-type accelerated dynamics.
  • In the regime $q+1<p$ and in the boundary case $p=2$, $\beta\geq q(1-q)$, bounded trajectories converge weakly to a weak Pareto optimal point, with the velocity satisfying $\int^{\infty} t\|\dot x(t)\|^2\,dt<+\infty$.
  • These results lay the groundwork for fast gradient and proximal point methods in multiobjective optimization that carry strong convergence guarantees for the iterates.
  • Numerical experiments indicate that the Tikhonov term actively steers the trajectory to the minimal-norm Pareto point, in contrast to the unregularized inertial multiobjective system.

Reading between the lines

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

  • A practical consequence is that the strong-convergence guarantee should only be claimed for problems where the auxiliary map $z_0(\cdot)$ is continuous; verifying this condition, or using regularization schemes that recover it, becomes part of algorithm design, and the paper's Example 2.3 is a concrete test case.
  • The excluded boundary $p=q+1$ is the natural place to look next: the single-objective theory exhibits a threshold phenomenon in this regime, and one would expect a similar phase transition for the multiobjective system.
  • The same Lyapunov construction should translate to discrete-time multiobjective proximal or fast gradient algorithms, predicting $O(1/k^2)$-type decreases of the merit function and strong convergence of the iterates to the minimum-norm weak Pareto point.
  • Because the continuity assumption (A3) is about the geometry of the weak Pareto set in value space, an empirical check is possible: compute $z_0(q)$ on a grid of $q$ vectors and look for jumps; problems with kinked Pareto fronts are the likely failure cases.
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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

2 major / 4 minor

Summary. The paper introduces a second-order inertial multiobjective gradient-like dynamical system with asymptotically vanishing damping and vanishing Tikhonov regularization, (MTRIGS), defined in a Hilbert space for convex differentiable objectives. It proves existence of trajectory solutions in finite dimensions, derives Lyapunov-type energy estimates, and obtains merit-function decay rates O(t^{-2q}) or O(t^{-p}) depending on the parameter regime. In the regime q in (0,1), p < q+1, it establishes strong convergence of bounded trajectories to a weak Pareto optimal point that is the minimum-norm element of the intersection of the lower level sets of the objectives (Theorem 4.8). In other regimes it proves weak convergence via Opial's argument, and it supports the theory with numerical experiments. The standing assumptions (A1)-(A3) include the new regularity condition (A3), which requires continuity of the projection onto the argmin sets S(q).

Significance. The results, if correct, provide a multiobjective analogue of the single-objective Tikhonov-regularized inertial gradient theory, combining fast merit-function decay with a minimum-norm selection principle. The proofs are detailed and carefully parameterized, with a clear summary table of rates and a useful new tool in the generalized regularization path z(t). The paper is transparent about the restrictive standing assumption (A3), and Example 2.3 convincingly shows that (A3) is not a consequence of convexity and Lipschitz smoothness. This limits the scope of the advertised strong-convergence result but does not invalidate the conditional theorems. The numerical experiments illustrate the predicted behavior, although they are not a substitute for the analytical proofs.

major comments (2)
  1. [§1.2.2 and Theorem 4.8] Assumption (A3) is load-bearing for the paper's central strong-convergence claim. In the proof of Theorem 2.1, continuity of z0 is used both to bound the sequence {z_k} and to force every weak cluster point to equal z0(q*); without it, a cluster point can lie in S(q*) without being the minimum-norm element. Example 2.3 demonstrates that for convex, differentiable functions with Lipschitz gradients in R^2, z0 can be discontinuous and the regularization path z(t) can oscillate without converging to z0(0). Consequently, Theorem 4.8's strong convergence and minimum-norm selection are genuinely conditional on (A3), which is a restrictive condition beyond the standard convexity and smoothness hypotheses. The manuscript should state this limitation prominently in the abstract and in the statement of Theorem 4.8, and it should point the reader to the sufficient condition in Definition 1.8/Theorem 1.10 for verifying (A3).
  2. [§4.2, proof of Theorem 4.7] In inequality (4.23), the notation μ_q(t) appears, but the subsequent integrating factor M_r(t) is defined with μ_r(t) = λ/t^r - 2r/t. This is presumably a typographical error: for a fixed r in the theorem, the differential inequality should use μ_r(t), not μ_q(t). The mistake does not affect the argument, but it should be corrected for readability.
minor comments (4)
  1. [Table 1] The row for the regime p < q+1 reports exponents involving max(q, p-q), but Theorem 4.7 is stated for arbitrary r in [q,1) ∩ [p-q,1). The table should explicitly state that the displayed exponents correspond to the choice r = max(q, p-q), since this is the sharpest choice among the admissible values.
  2. [Definition 3.1 and Proposition 3.4] The notation "projC(x(t))+ β/tp x(t)+¨x(t)(0)" should read "proj_{C(x(t)) + β/t^p x(t) + ddot{x}(t)}(0)" to make clear that the projection is onto the set C(x(t)) + β/t^p x(t) + ddot{x}(t). The current typesetting obscures the intended expression.
  3. [Proposition 2.5] In the continuity proof, several displayed formulas have missing or misplaced parentheses, e.g., "max_i (fi(z(t) - qi(t))" and the subsequent term "max_i (fi(z(t) - qi(t)) - β/(2t^p) ∥z(t)∥²". The intended expressions are clear from context, but the formulas should be typeset correctly.
  4. [Theorem 4.12 and Remark 4.13] The notation "r := q+1/2" in the proof is ambiguous; it should be "r := (q+1)/2" to match the endpoint used in the argument leading to ∫ s ∥dot{x}(s)∥² ds < ∞.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the strong-convergence/minimum-norm result is conditional on the disclosed regularity assumption A3, and the proofs are self-contained Lyapunov and Tikhonov-path arguments rather than reductions to fitted inputs or self-citations.

full rationale

The central derivation chain is not circular. Theorem 4.8 combines two in-paper results: Theorem 2.1, which proves convergence of the generalized regularization path z(·) to z0(f*) = proj_{S(f*)}(0) under the explicit continuity assumption (A3), and Theorem 4.7, which proves ||x(t)-z(t)|| -> 0 via Lyapunov energy estimates and integration lemmas. The minimum-norm characterization of the limit is a consequence of these arguments, not an assumed input; the target point is identified after the limit is obtained, and the equality S(f*) = ∩_{i=1}^m L(f_i, f_i(x*)) is derived from weak Pareto optimality and the definition of S(·). Assumption (A3) is disclosed as a standing hypothesis, and Example 2.3 explicitly demonstrates that it is not implied by convexity and Lipschitz gradients; this is a genuine regularity condition, not a hidden restatement of the conclusion. The paper does reuse prior work [33,34] for existence results and for the (MAVD) system, but those citations are supporting rather than load-bearing for the new rates and the strong-convergence selection principle, and the cited results are themselves external published results. There is no fitted parameter renamed as a prediction, no imported uniqueness theorem, and no ansatz smuggled in through self-citation. The unusual label 'Sonntag-Attouch Theorem' attached to Beer's Mosco-convergence equivalence is a naming oddity, but it does not make the argument circular because the theorem's content is standard and its use is not the source of the paper's central claims. The restrictive character of (A3) is best regarded as a correctness/robustness concern, not as circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No free parameters are fitted to data. The constants alpha, beta, p, and q are design parameters of the dynamical system; the theorems state convergence for ranges of them, and the numerical experiments fix them for illustration. The Lyapunov parameter lambda is chosen inside an interval during the proof and is not an empirical fit. The axioms listed are the paper's own standing assumptions plus the finite-dimensionality restriction for existence.

assumptions (4)
  • domain assumption Each component function f_i is convex and continuously differentiable with Lipschitz continuous gradients.
    Stated as Assumption (A1) in Section 1.2 and used throughout the paper.
  • domain assumption For the initial data, the expanded level set contains weak Pareto points and the uniform bound R in (1.9) is finite.
    Stated as Assumption (A2) in Section 1.2. It is used to bound the generalized regularization path z(t) in Proposition 2.4 and to relate phi and phi_t in Proposition 2.6.
  • domain assumption For all q in R^m, S(q) is nonempty and the projection z0(q) = proj_{S(q)}(0) is continuous.
    Stated as Assumption (A3) in Section 1.2. It is essential for Theorem 2.1, which gives convergence of the regularization path, and hence for the strong convergence result Theorem 4.8.
  • domain assumption The Hilbert space H is finite dimensional for the existence proof.
    Theorem 3.2 and Appendix B establish existence of trajectory solutions only for finite-dimensional H, although the system is introduced in a general Hilbert space.
invented entities (1)
  • Generalized regularization path z(t), defined as argmin_z max_i(f_i(z)-f_i(x(t))) + (beta/(2t^p))||z||^2.
    purpose: Used to measure the trajectory's distance to the minimum-norm weak Pareto point and to enable the strong convergence proof.
    This is a defined mathematical construction, not an empirically observed entity. Its convergence is derived from Assumption (A3), and it has no independent observable evidence outside the paper.

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Pith. "Pith review of Inertial dynamics with vanishing Tikhonov regularization for multiobjective optimization." pith.science (2026). https://pith.science/paper/56X5AQQ3

@misc{pith2026241118422,
  author       = {Pith},
  title        = {Pith review of: Inertial dynamics with vanishing Tikhonov regularization for multiobjective optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/56X5AQQ3}},
  note         = {Machine review of arXiv:2411.18422}
}
read the original abstract

In this paper, we introduce, in a Hilbert space setting, a second order dynamical system with asymptotically vanishing damping and vanishing Tikhonov regularization that approaches a multiobjective optimization problem with convex and differentiable components of the objective function. Trajectory solutions are shown to exist in finite dimensions. We prove fast convergence of the function values, quantified in terms of a merit function. Based on the regime considered, we establish both weak and, in some cases, strong convergence of trajectory solutions towards a weak Pareto optimal point. To achieve this, we apply Tikhonov regularization individually to each component of the objective function. Furthermore, we conduct numerical experiments to validate the theoretical results and investigate the qualitative behavior of the dynamical system. This work extends results from convex single objective optimization into the multiobjective setting. The results presented in this paper lay the groundwork for the development of fast gradient and proximal point methods in multiobjective optimization, offering strong convergence guarantees.

Figures

Figures reproduced from arXiv: 2411.18422 by the authors.

Figure 1
Figure 1. Visualization of (A2) with a trajectory x(t) ∈ LPw(F, F(x0) + a). 1.2.2 Discussion of assumption (A3) We need assumption (A3) to show the strong convergence of the generalized regularization path for multiob￾jective optimization problems. We illustrate the necessity of this assumption with an example in Section 2. In the following we show that the continuity of the projection q 7→ z0(q) := projS(q) (0) is closely co… view at source ↗
Figure 2
Figure 2. Contour plots of the functions f1 and f2 defined in (2.9): (a) The weak Pareto sets of (2.10) and (2.11) for ε ∈ {10−1 , 10−1.5 , 10−2 , 10−2.5 , 10−3}. (b) The weak Pareto set of (2.10) and the regularization path z(·) defined in (2.12) with parameters p = 1, β = 1 2 , η = 1 50 . and the Tikhonov regularized problem min x∈H  f1(x) + ε 2 ∥x∥ 2 f2(x) + ε 2 ∥x∥ 2  . (2.11) Figure 2a illustrates the weak Pareto set P… view at source ↗
Figure 3
Figure 3. Contour plots of f1 and f2 defined in (5.1), the weak Pareto set Pw of the problem (MOP-Ex1) and the trajectory solutions x(·) of (MTRIGS) and (MAVD) with identical initial conditions, respectively. 5.1 Comparison of (MTRIGS) with (MAVD) In the first example, we consider the following instance of (MOP). Define the sets S1 := {−1} × [1, 2] ⊆ R 2 and S2 := {1} × [1, 2] ⊆ R 2 , and the functions fi : R 2 → R, x 7→ fi(x… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The merit function values φ(x(t)) and the distance ∥x(t) − z(t)∥ of the trajectory solutions to the generalized regularization path for (MTRIGS) and (MAVD) for the problem (MOP-Ex1). • For both systems, we approximate the first and second derivatives by ˙x(tk) = x(tk+1…
Figure 5
Figure 5. Figure 5: The merit function values φ(x(t)) and the distance ∥x(t) − z(t)∥ of the trajectory to the generalized regularization path for q = 0.8 and p ∈ {0.25, 0.75, 1.25, 1.75} [PITH_FULL_IMAGE:figures/full_fig_p032_5.png]
Figure 6
Figure 6. Figure 6: The merit function values φ(x(t)) and the distance ∥x(t) − z(t)∥ of the trajectory to the generalized regularization path for p = 1.1 and q ∈ {0.3, 0.6, 0.8, 0.99}. particularly for q = 0.99, where convergence is significantly faster. For the smallest value q = 0.3, th…
Figure 7
Figure 7. Figure 7: The sets Mi ⊆ R 2 for i = 1, 2, 3. Denoting M1 :=  x ∈ R 2 : |x1| ≤ 1, x2 + 1 ≤ q 1 − x 2 1  , M2 :=  x ∈ R 2 : |x1| > 1, x2 + 1 ≤ 0 [PITH_FULL_IMAGE:figures/full_fig_p039_7.png]

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Works this paper leans on

43 extracted references · 33 canonical work pages

  1. [1]

    Alvarez, On the Minimizing Property of a Second Order Dissipative System in Hilbert Spaces , SIAM Journal on Control and Optimization, 38 (2000), pp

    F. Alvarez, On the Minimizing Property of a Second Order Dissipative System in Hilbert Spaces , SIAM Journal on Control and Optimization, 38 (2000), pp. 1102–1119, https://doi.org/10.1137/S0363012998335802

    work page doi:10.1137/s0363012998335802 2000

  2. [2]

    Attouch, A

    H. Attouch, A. Balhag, Z. Chbani, and H. Riahi , Damped inertial dynamics with vanishing tikhonov regularization: Strong asymptotic convergence towards the minimum norm solution , Journal of Differential Equations, 311 (2022), pp. 29–58, https://doi.org/10.1016/j.jde.2021.12.005

    work page doi:10.1016/j.jde.2021.12.005 2022

  3. [3]

    Attouch, Z

    H. Attouch, Z. Chbani, J. Peypouquet, and P. Redont , Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity , Mathematical Programming, 168 (2018), pp. 123–175, https://doi.org/10.1007/ s10107-016-0992-8

    work page 2018

  4. [4]

    Multiibjective optimization : an inertial dynamical approach to Pareto optima

    H. Attouch and G. Garrigos, Multiobjective optimization: an inertial dynamical approach to Pareto optima , arXiv preprint arXiv:1506.02823, (2015)

    work page Pith review arXiv 2015

  5. [5]

    Attouch, X

    H. Attouch, X. Goudou, and P. Redont , The Heavy Ball with Friction Method, I. The Continuous Dynamical System: Global Exploration of the Local Minima of a Real-Valued Function by Asymptotic Analysis of a Dissipative Dynamical System, Communications in Contemporary Mathematics, 2 (2000), pp. 1–34, https://doi.org/10.1142/S0219199700000025

    doi:10.1142/s0219199700000025 2000

  6. [6]

    Attouch and S

    H. Attouch and S. L ´aszl´o, Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution , Mathematical Methods of Operations Research, 99 (2024), pp. 307–347, https://doi.org/10.1007/s00186-024-00867-y

    work page doi:10.1007/s00186-024-00867-y 2024

  7. [7]

    Aubin and A

    J.-P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory , vol. 264, Springer, Berlin, 2012, https://doi.org/10.1007/978-3-642-69512-4

    doi:10.1007/978-3-642-69512-4 2012

  8. [8]

    Aubin and H

    J.-P. Aubin and H. Frankowska , Set-Valued Analysis , Springer, Boston, 2009, https://doi.org/10.1007/ 978-0-8176-4848-0

    work page 2009

Show all 43 references
  1. [9]

    Bauschke and P

    H. Bauschke and P. Combettes , Convex Analysis and Monotone Operator Theory in Hilbert Spaces , Springer, 01 2017, https://doi.org/10.1007/978-3-319-48311-5

    2017 doi

  2. [10]

    Beer , On Mosco convergence of convex sets , Bulletin of the Australian Mathematical Society, 38 (1988), pp

    G. Beer , On Mosco convergence of convex sets , Bulletin of the Australian Mathematical Society, 38 (1988), pp. 239–253, https://doi.org/10.1017/S0004972700027519

    1988 doi

  3. [11]

    G. Beer, R. Rockafellar, and R. J.-B. Wets , A characterization of epi-convergence in terms of convergence of level sets , Proceedings of the American Mathematical Society, 116 (1992), pp. 753–761, https://doi.org/10.2307/2159443

    1992 doi

  4. [12]

    J. M. Borwein and S. Fitzpatrick, Mosco convergence and the Kadec property, Proceedings of the American Mathematical Society, 106 (1989), pp. 843–851, https://doi.org/10.1090/S0002-9939-1989-0969313-4

    1989 doi

  5. [13]

    R. I. Bot ¸, E. R. Csetnek, and S. C. L´aszl´o, On the strong convergence of continuous newton-like inertial dynamics with tikhonov regularization for monotone inclusions , Journal of Mathematical Analysis and Applications, 530 (2024), p. 127689

    2024

  6. [14]

    Brezis, Operateurs Maximaux Monotones Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert, North Holland, Amsterdam, 1973, https://doi.org/10.1016/S0304-0208(08)72386-7

    H. Brezis, Operateurs Maximaux Monotones Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert, North Holland, Amsterdam, 1973, https://doi.org/10.1016/S0304-0208(08)72386-7

    1973 doi

  7. [15]

    Cabot, H

    A. Cabot, H. Engler, and S. Gadat , On the long time behavior of second order differential equations with asymptotically small dissipation, Transactions of the American Mathematical Society, 361 (2009), pp. 5983–6017, https://doi.org/10.1090/ S0002-9947-09-04785-0

    2009

  8. [16]

    Cabot, H

    A. Cabot, H. Engler, and S. Gadat, Second-order differential equations with asymptotically small dissipation and piecewise flat potentials, Electronic Journal of Differential Equations, 2009 (2009), pp. 33–38

    2009

  9. [17]

    Chen, Generalized viscosity approximation methods in multiobjective optimization problems , Computational Optimization and Applications, 49 (2011), pp

    Z. Chen, Generalized viscosity approximation methods in multiobjective optimization problems , Computational Optimization and Applications, 49 (2011), pp. 179–192, https://doi.org/10.1007/s10589-009-9282-1

    2011 doi

  10. [18]

    Z. Chen, C. Xiang, K. Zhao, and X. Liu , Convergence analysis of tikhonov-type regularization algorithms for multiobjective optimization problems, Applied Mathematics and Computation, 211 (2009), pp. 167–172, https://doi.org/10.1016/j.amc. 2009.01.037

    2009 doi

  11. [19]

    T. D. Chuong , Tikhonov-type regularization method for efficient solutions in vector optimization , Journal of Computational and Applied Mathematics, 234 (2010), pp. 761–766, https://doi.org/10.1016/j.cam.2010.01.040

    2010 doi

  12. [20]

    T. D. Chuong and J.-C. Yao, Viscosity-type approximation method for efficient solutions in vector optimization , Taiwanese Journal of Mathematics, 14 (2010), pp. 2329 – 2342, https://doi.org/10.11650/twjm/1500406078

    2010

  13. [21]

    Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization (Vector Optimization), Springer, Berlin, 2008, https://doi.org/10.1007/978-3-540-79159-1

    G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization (Vector Optimization), Springer, Berlin, 2008, https://doi.org/10.1007/978-3-540-79159-1

    2008 doi

  14. [22]

    Karapetyants and S

    M. Karapetyants and S. C. L ´aszl´o, A nesterov type algorithm with double tikhonov regularization: fast convergence of the function values and strong convergence to the minimal norm solution , Applied Mathematics & Optimization, 90 (2024), p. 17

    2024

  15. [23]

    S. C. L ´aszl´o, A proximal-gradient inertial algorithm with tikhonov regularization: strong convergence to the minimal norm solution, arXiv preprint arXiv:2407.10350, (2024)

    2024 arXiv

  16. [24]

    S. C. L ´aszl´o, On the convergence of an inertial proximal algorithm with a tikhonov regularization term , Communications in Nonlinear Science and Numerical Simulation, (2025), p. 108924

    2025

  17. [25]

    S. C. L ´aszl´o, Solving convex optimization problems via a second order dynamical system with implicit hessian damping and tikhonov regularization, Computational Optimization and Applications, 90 (2025), pp. 113–149

    2025

  18. [26]

    C. G. Liu, K. F. Ng, and W. H. Yang , Merit functions in vector optimization , Mathematical Programming, 119 (2009), pp. 215–237, https://doi.org/10.1007/s10107-008-0208-y . 40

    2009 doi

  19. [27]

    S. C. L ´aszl´o, On the strong convergence of the trajectories of a Tikhonov regularized second order dynamical system with asymptotically vanishing damping , Journal of Differential Equations, 362 (2023), pp. 355–381, https://doi.org/10.1016/j. jde.2023.03.014

    2023 doi

  20. [28]

    May, Asymptotic for a second-order evolution equation with convex potential andvanishing damping term , Turkish Journal of Mathematics, 41 (2017), pp

    R. May, Asymptotic for a second-order evolution equation with convex potential andvanishing damping term , Turkish Journal of Mathematics, 41 (2017), pp. 681–685, https://doi.org/10.3906/mat-1512-28

    2017 doi

  21. [29]

    Mosco , Convergence of convex sets and of solutions of variational inequalities , Advances in Mathematics, 3 (1969), pp

    U. Mosco , Convergence of convex sets and of solutions of variational inequalities , Advances in Mathematics, 3 (1969), pp. 510–585, https://doi.org/10.1016/0001-8708(69)90009-7

    1969 doi

  22. [30]

    Y. E. Nesterov , A method for solving the convex programming problem with convergence rate O(1/k2), Doklady Akademii Nauk SSSR, 269 (1983), pp. 543–547

    1983

  23. [31]

    B. T. Polyak, Some methods of speeding up the convergence of iteration methods , USSR Computational Mathematics and Mathematical Physics, 4 (1964), pp. 1–17, https://doi.org/10.1016/0041-5553(64)90137-5

    1964 doi

  24. [32]

    R. T. Rockafellar and R. J.-B. Wets , Variational Analysis , Springer, Berlin, 1998, https://doi.org/10.1007/ 978-3-642-02431-3

    1998

  25. [33]

    Sonntag and S

    K. Sonntag and S. Peitz, Fast convergence of inertial multiobjective gradient-like systems with asymptotic vanishing damp- ing, SIAM Journal on Optimization, 34 (2024), pp. 2259–2286, https://doi.org/10.1137/23M1588512

    2024 doi

  26. [34]

    Sonntag and S

    K. Sonntag and S. Peitz, Fast multiobjective gradient methods with Nesterov acceleration via inertial gradient-like systems , Journal of Optimization Theory and Applications, (2024), https://doi.org/10.1007/s10957-024-02389-3

    2024 doi

  27. [35]

    W. Su, S. Boyd, and E. J. Cand `es, A differential equation for modeling Nesterov’s accelerated gradient method: Theory and insights, Journal of Machine Learning Research, 17 (2016), pp. 1–43, http://jmlr.org/papers/v17/15-084.html

    2016

  28. [36]

    Tanabe, E

    H. Tanabe, E. H. Fukuda, and N. Yamashita , A globally convergent fast iterative shrinkage-thresholding algorithm with a new momentum factor for single and multi-objective convex optimization , arXiv preprint arXiv:2205.05262, (2022), https: //doi.org/10.48550/arXiv.2205.05262

    2022 doi arXiv

  29. [37]

    Tanabe, E

    H. Tanabe, E. H. Fukuda, and N. Yamashita , Convergence rates analysis of a multiobjective proximal gradient method , Optimization Letters, (2022), pp. 1–18, https://doi.org/10.1007/s11590-022-01877-7

    2022 doi

  30. [38]

    Tanabe, E

    H. Tanabe, E. H. Fukuda, and N. Yamashita , An accelerated proximal gradient method for multiobjective optimization , Computational Optimization and Applications, (2023), pp. 1–35, https://doi.org/10.1007/s10589-023-00497-w

    2023 doi

  31. [39]

    Tanabe, E

    H. Tanabe, E. H. Fukuda, and N. Yamashita , New merit functions for multiobjective optimization and their properties , Optimization, (2023), pp. 1–38, https://doi.org/10.1080/02331934.2023.2232794

    2023

  32. [40]

    Terazono and A

    Y. Terazono and A. Matani , Continuity of optimal solution functions and their conditions on objective functions , SIAM Journal on Optimization, 25 (2015), pp. 2050–2060, https://doi.org/10.1137/110850189

    2015 doi

  33. [41]

    A. N. Tikhonov , Regularization of incorrectly posed problems, Soviet Mathematics Doklady, 4 (1963), pp. 1624–1627

    1963

  34. [42]

    A. N. Tikhonov , Solution of incorrectly formulated problems and the regularization method , Soviet Mathematics Doklady, 4 (1963), pp. 1035–1038

    1963

  35. [43]

    X. Q. Yang and J. C. Yao , Gap functions and existence of solutions to set-valued vector variational inequalities , Journal of Optimization Theory and Applications, 115 (2002), pp. 407–417, https://doi.org/10.1023/A:1020844423345. 41

    2002 doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.