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Projected proximal gradient trust-region algorithm for nonsmooth optimization

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A trust-region method for nonsmooth composite problems finds epsilon-critical points at the same worst-case rate as the smooth case, even when model Hessians grow unboundedly.

desk verdict A credible complexity extension with a promising subproblem solver, but the PPG backtracking as implemented is not proven to satisfy the sufficient-decrease condition behind the theory. read the letter →

arxiv 2501.04889 v1 pith:LCBD3LX4 submitted 2025-01-09 math.OC

Minh N. Dao , Hung M. Phan , Lindon Roberts This is my paper · ORCID
classification math.OC MSC 47H0549M3765K0565K1090C30
keywords trust-regionmethodsnonsmoothoptimizationproximalgradientdescentweakconvexityworst-casecomplexityunboundedHessiangrowthprojectedcomposite
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

Trust-region methods are extended to composite objectives $F = f + h$, where $f$ is smooth but nonconvex and $h$ is convex and nonsmooth, and their worst-case iteration complexity is derived for the case where model Hessians can grow as $O(k^t)$ with $t \in [0,1]$. The result matches the smooth case exactly: $O(\epsilon^{-2/(1-t)})$ iterations to reach an $\epsilon$-approximate first-order critical point when $0 \le t < 1$, and $e^{O(\epsilon^{-2})}$ iterations when $t = 1$. A new subproblem solver, the projected proximal gradient (PPG) method, performs several proximal-gradient steps on the local model and then projects the final step into the trust region; the paper proves this solver satisfies the sufficient-decrease condition required by the main algorithm. Numerical experiments on 154 $\ell_1$-regularized CUTEst problems show PPG is competitive with the spectral proximal gradient solver and outperforms it when high-accuracy solutions are requested.

What carries the argument

The stationarity measure $\pi(x,\gamma) = \frac{1}{\gamma}\|\operatorname{Prox}_{\gamma h}(x - \gamma \nabla f(x)) - x\|$ (Equation 6), which vanishes exactly at first-order critical points of $F$ and is continuous in $x$ and $\gamma$, is what the algorithm drives below $\epsilon$. The sufficient-decrease condition (Assumption 3.3) links the model decrease to this measure and to the trust-region radius, and the quantity $a_k$ (Equation 24) tracks the interplay between radius, Hessian growth, and achieved decrease, providing the ratio-like bound used in the complexity proof. The PPG subproblem solver (Algorithm 2) is the central new object: starting from the current iterate, it runs up to $N$ proximal-gradient steps on the model $m(p)$, then projects the final point into the trust region via $p^* = \frac{\Delta}{\max\{\Delta, \|u_i - x\|\}}(u_i - x)$; its decrease guarantee comes from $\lambda$-convexity of the model and nonexpansiveness of the proximity operator.

What would settle it

Construct a model with $\lambda = -L$ (the worst weak-convexity case) and a large $L$, run Algorithm 2 with the Remark 4.5 backtracking, and record the accepted $\gamma$; if it systematically lands outside $[c_*/(10L), c_*/L]$ while the output $p^*$ still satisfies (83), the practical solver is not certified by the paper's Lemma 4.3. A direct counterexample would be a pair $(\gamma, L, \text{model})$ where all model decreases are positive but $m(0) - m(p^*) < \kappa_s \pi(x,\gamma) \min\{\Delta, \pi(x,\gamma)/L\}$.

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Extended reading notes

Core claim

The paper's central claim is that the trust-region algorithm for (1) retains the same worst-case complexity as smooth trust-region methods under Assumption 3.2: if model Hessians grow like $\|H_k\| = O(k^t)$, then for $t \in [0,1)$ an $\epsilon$-approximate first-order critical point is found in $O(\epsilon^{-2/(1-t)})$ iterations, and for $t = 1$ in $\widetilde{O}(e^{c \epsilon^{-2}})$ iterations (Theorem 3.14, Corollary 3.15). This extends the global-convergence result of the proximal trust-region framework to quantitative rates. The second claim is that the new PPG subproblem solver (Algorithm 2) meets the sufficient-decrease requirement (Assumption 3.3) that drives these bounds, with the model decrease bounded below by $\theta \|u_1 - x\| \min\{\Delta, \|u_1 - x\|\}$ (Theorem 4.2) and a backtracking argument (Corollary 4.4) that guarantees the needed decrease for stepsizes in $[c_*/(10L), c_*/L]$. Numerical results indicate PPG is a practical and often superior alternative to SPG, particularly for high-accuracy solutions.

Load-bearing premise

The paper's convergence guarantee for the PPG subproblem solver assumes the backtracking rule chooses a stepsize in the interval $[c_*/(10L), c_*/L]$, but the rule actually implemented only checks that model decreases are positive, which is not shown to imply that interval.

Editorial extensions

If this is right

  • When model Hessians are uniformly bounded ($t = 0$), the bound recovers the standard $O(\epsilon^{-2})$ iteration count for smooth trust-region methods.
  • Quasi-Newton models whose Hessians grow linearly ($t = 1$, e.g. SR1) do not alter the asymptotic rate relative to the smooth case.
  • Any subproblem solver satisfying Assumption 3.3 inherits the same worst-case complexity, making the theory solver-agnostic.
  • The PPG solver's simple structure—proximal-gradient steps plus one final projection—makes it a drop-in replacement for existing subproblem solvers in nonsmooth trust-region implementations.
  • Numerical evidence shows PPG scales with subproblem iteration budget $N$ and beats SPG at high accuracy ($\tau = 10^{-6}$), suggesting it is preferable when tight tolerances are needed.

Reading between the lines

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

  • If the backtracking rule actually forces $\gamma$ into the certified interval $[c_*/(10L), c_*/L]$, then the implemented PPG solver is fully covered by Corollary 4.4; if not, the practical success is not explained by the paper's sufficient-decrease theorem, and a separate analysis of the positive-decrease test would close the gap.
  • Because only the final projection depends on the trust-region geometry, the same projected proximal gradient idea should extend to other convex trust regions (e.g. $\ell_\infty$ or simplex constraints) by replacing the Euclidean projection, provided the projected step remains in the region and the decrease bound adapts.
  • A natural stress test is to compare PPG and SPG on problems where the model Hessian is actually unbounded in practice (e.g. SR1 updates), since the theory allows $t = 1$ but the experiments here use exact Hessians.
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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 studies trust-region methods for minimizing F = f + h, where f is smooth (possibly nonconvex) and h is convex and nonsmooth. The first contribution extends worst-case iteration complexity results for trust-region methods from the smooth setting to this composite setting when model Hessian norms may grow at rate O(k^t), t in [0,1]. The second contribution proposes a projected proximal gradient (PPG) subproblem solver that performs several proximal gradient steps and one final projection into the trust region, and claims that this solver satisfies the sufficient decrease condition required by the main trust-region algorithm. The paper proves an O(epsilon^{-2/(1-t)}) bound for t in [0,1) and a doubly exponential-type bound for t = 1, matching the smooth-case rates of Diouane et al. (2024). Numerical experiments on L1-regularized CUTEst problems compare PPG with the spectral proximal gradient (SPG) solver.

Significance. The paper's central complexity analysis appears internally consistent and provides a valuable extension of recent smooth-case results to composite nonsmooth optimization, with a clearly stated set of assumptions and a careful adaptation of the techniques of [DHO24]. The proposed PPG solver is simple and the numerical results are promising, especially at high accuracy. However, the theoretical guarantee that the subproblem solver enforces the required sufficient decrease (Assumption 3.3) is not fully linked to the implemented backtracking rule, and the statement of Corollary 4.4 contains a missing condition; these issues must be addressed before the results can be accepted as stated.

major comments (2)
  1. [Section 4, Corollary 4.4] The proof of Corollary 4.4 asserts that for the geometric stepsize schedule gamma_j = alpha^j gamma_0 with alpha in (0.1,1), 'there must be at least one j for which gamma_j in [c*/(10L), c*/L]'. This is not true for arbitrary gamma_0 > 0, because the sequence is strictly decreasing and if gamma_0 < c*/(10L), all subsequent terms are below the interval. The corollary as stated is therefore false unless an additional lower bound such as gamma_0 >= c*/(10L) is imposed, or the backtracking procedure is allowed to increase the stepsize. Please add the missing condition or correct the claim.
  2. [Section 4, Remark 4.5 and Lemma 4.3] The backtracking rule actually implemented (Remark 4.5) accepts gamma_j whenever all computed model decreases m(0)-m(u_i) and m(0)-m(p*) are positive. This test is strictly weaker than the interval condition gamma_j in [c*/(10L), c*/L] that underlies the sufficient-decrease inequality (88). For arbitrarily small gamma, the proximal gradient steps become tiny and the model decreases can remain positive, while the right-hand side of (83) or (88) can remain bounded away from zero at nonstationary points. Consequently, the paper does not prove that the implemented rule produces steps satisfying Assumption 3.3, so the complexity certificate of Theorem 3.14 does not formally apply to Algorithm 1 as implemented and tested in Section 5. To close this gap, the authors should either modify the acceptance test to check the interval condition (which is implementable since L is known) or prove that positive model decrease implies a quantitative decrease of the form (83).
minor comments (4)
  1. [Section 5] The statement that the modified trust-region updating rule (89) 'does not affect the complexity theory' is not substantiated. The theory relies on the specific parameters gamma_dec, gamma_inc, eta and on the acceptance condition rho_k >= eta, whereas (89) uses different thresholds and an additional boundary condition on ||p_k||. Please provide a direct argument or soften the claim.
  2. [Section 3, Lemma 3.5 and Lemma 3.7] The phrase 'for all in N' should read 'for all k in N' in both lemmas.
  3. [Section 3, Lemma 3.9] The displayed inequality in the proof of Lemma 3.9 contains typographical errors ('a2 + sqrt(a2^2 + 4a1)/2' is typeset incorrectly), and the statement 'k* = Theta(a1)' is not derived explicitly. Please clean up the proof.
  4. [References] The reference [Toi88] has a duplicated article in its title: 'a a of trust-region methods' should be 'a class of trust-region methods'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the complexity result is conditional on an independent sufficient-decrease assumption, and the PPG solver's decrease guarantee is proved from first principles; the noted implementation gap is a correctness risk, not a circularity.

full rationale

After walking the derivation chain, I find no circular step. The worst-case complexity result (Theorem 3.14) is explicitly conditional on Assumption 3.3, a sufficient-decrease condition stated independently of the algorithm's implementation; the proof follows the external smooth-case argument of [DHO24] and the nonsmooth trust-region framework of [BK23a, BK23b], which are not authored by the present authors. The second contribution, the PPG subproblem solver, is proved to satisfy the required decrease from first principles: Lemma 4.1 derives a per-iteration model decrease from lambda-convexity and the proximal-gradient optimality condition, Lemma 4.3 converts this into a bound involving the stationarity measure pi(x,gamma), and Corollary 4.4 lifts the bound to pi(x,gamma_max) using the monotonicity of pi (Proposition 2.3). The cited self-papers [DP19, DPT24] are used only for standard lambda-convexity facts (equivalent convexity of f - (lambda/2)||.||^2 and the quadratic lower bound), not for the paper's novel claims. The numerical benchmarks use the external CUTEst test set and compare against the external SPG solver. I note, as a correctness concern rather than circularity, that the practical backtracking rule in Remark 4.5 (accept gamma_j when computed model decreases are positive) is weaker than the interval condition gamma_j in [c*/(10L), c*/L] used in Corollary 4.4, and Corollary 4.4 also implicitly needs gamma_0 >= c*/(10L) to guarantee the geometric sequence enters the required interval; positive model decrease can occur for arbitrarily small gamma while the right-hand side of (88) stays bounded away from zero. This is a gap between implementation and theory, not a reduction of a prediction to an input, so it does not affect the circularity score.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central complexity theorem rests on the three stated assumptions (smoothness and boundedness of F, Hessian growth, sufficient decrease) and on cited technical lemmas from [DHO24]. The sufficient-decrease assumption is not free for PPG: it is proved in Corollary 4.4 under an idealized stepsize selection, though the practical acceptance rule is weaker. No free parameters are fitted to data; the listed parameters are implementation choices for numerical experiments.

free parameters (2)
  • trust-region parameters (gamma_dec, gamma_inc, eta, Delta0) = gamma_dec=0.5, gamma_inc=2, eta=10^{-3}, Delta0=1
    Standard trust-region hyperparameters chosen for the numerical experiments; they do not enter the complexity constants.
  • PPG parameters (N, mu_u, alpha, gamma0 heuristic) = N in {15,30,50}, mu_u=2, alpha=0.9, gamma0=2||g0||/(3||H0g0||)
    Chosen by hand for the numerical comparison; the practical acceptance rule for backtracking is not part of the theoretical guarantee.
assumptions (6)
  • domain assumption Assumption 3.1: f is differentiable with L-Lipschitz gradient; h is proper, lower semicontinuous and convex; F is bounded below.
    Stated as the problem class for the trust-region algorithm and used in Lemma 3.5 and Theorem 3.14.
  • domain assumption Access to f, gradient of f, evaluations of h, and the proximity operator of h.
    Stated as informal access assumptions in the introduction and used by Algorithm 2.
  • domain assumption Assumption 3.2: model Hessians satisfy max_{j<=k} ||H_j|| <= mu(1+k^t) for some mu>0 and t in [0,1].
    Unbounded Hessian growth assumption taken from [DHO24]; the basis for the t-dependent complexity bounds.
  • domain assumption Assumption 3.3: the step p_k satisfies the sufficient decrease condition m_k(0)-m_k(p_k) >= kappa_p pi_k(gamma_max) min{Delta_k, pi_k(gamma_max)/(1+||H_k||)}.
    The key condition for Theorem 3.4 and Theorem 3.14; proved for PPG under idealized stepsize selection in Corollary 4.4.
  • standard math Standard convex and nonsmooth analysis facts: subdifferential sum rule, proximity operator properties, nonexpansiveness, and lambda-convexity inequalities.
    Used in Section 2 and Lemma 4.1 with citations to [Mor06, Bec17, DP19, DPT24].
  • standard math Technical lemmas from [DHO24]: Lemma 3.8 (integral bound for the Hessian growth series) and Lemma 3.12 (counting successful iterations).
    Cited as [DHO24, Lemma 5] and [DHO24, Lemma 7] and used directly in the proof of Theorem 3.14.

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Pith. "Pith review of Projected proximal gradient trust-region algorithm for nonsmooth optimization." pith.science (2026). https://pith.science/paper/LCBD3LX4

@misc{pith2026250104889,
  author       = {Pith},
  title        = {Pith review of: Projected proximal gradient trust-region algorithm for nonsmooth optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LCBD3LX4}},
  note         = {Machine review of arXiv:2501.04889}
}
read the original abstract

We consider trust-region methods for solving optimization problems where the objective is the sum of a smooth, nonconvex function and a nonsmooth, convex regularizer. We extend the global convergence theory of such methods to include worst-case complexity bounds in the case of unbounded model Hessian growth, and introduce a new, simple nonsmooth trust-region subproblem solver based on combining several iterations of proximal gradient descent with a single projection into the trust region, which meets the sufficient descent requirements for algorithm convergence and has promising numerical results.

Figures

Figures reproduced from arXiv: 2501.04889 by the authors.

Figure 1
Figure 1. Comparison of PPG (Algorithm 2) and SPG [BK23a] subproblem solvers with increasing maximum iterations N ∈ {15, 30, 50} in the trust-region method Algorithm 1. performance profiles [DM02]. For solvers S ∈ S and problems P ∈ P, let K(S, P, τ ) be the first iteration with πk(1) ≤ τ (with K = +∞ if this never occurs). For a given solver S and accuracy level τ , data profiles measure the proportion of problems solved aft… view at source ↗

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Works this paper leans on

44 extracted references · 41 canonical work pages

  1. [1]

    A Levenberg-Marquardt Method for Nonsmooth Regularized Least Squares

    Aleksandr Y. Aravkin, Robert Baraldi, and Dominique Orban. A L evenberg- M arquardt method for nonsmooth regularized least squares. arXiv preprint arXiv:2301.02347, 2022

    work page Pith review arXiv 2022

  2. [2]

    Aravkin, Robert Baraldi, and Dominique Orban

    Aleksandr Y. Aravkin, Robert Baraldi, and Dominique Orban. A proximal quasi- N ewton trust-region method for nonsmooth regularized optimization. SIAM Journal on Optimization , 32(2):900--929, 2022

    work page 2022

  3. [3]

    Line Search and Trust-Region Methods for Convex-Composite Optimization

    James V. Burke and Abraham Engle. Line search and trust-region methods for convex-composite optimization. arXiv preprint arXiv:1806.05218, 2019

    work page Pith review arXiv 2019

  4. [4]

    Burke and Abraham Engle

    James V. Burke and Abraham Engle. Strong metric (sub)regularity of K arush-- K uhn-- T ucker mappings for piecewise linear-quadratic convex-composite optimization and the quadratic convergence of N ewton's method. Mathematics of Operations Research , 45(3):1164--1192, 2020

    work page 2020

  5. [5]

    First-order methods in optimization

    Amir Beck. First-order methods in optimization . SIAM, 2017

    2017

  6. [6]

    Efficient proximal subproblem solvers for a nonsmooth trust-region method

    Robert J Baraldi and Drew P Kouri. Efficient proximal subproblem solvers for a nonsmooth trust-region method. Optimization Online, https://optimization-online.org/?p=24879, 2023

    work page 2023

  7. [7]

    Baraldi and Drew P

    Robert J. Baraldi and Drew P. Kouri. A proximal trust-region method for nonsmooth optimization with inexact function and gradient evaluations. Mathematical Programming , 201(1-2):559--598, 2023

    work page 2023

  8. [8]

    Baraldi and Drew P

    Robert J. Baraldi and Drew P. Kouri. Local convergence analysis of an inexact trust-region method for nonsmooth optimization. Optimization Letters , 18(3):663--680, 2024

    work page 2024

Show all 44 references
  1. [9]

    A. R. Conn, Nick Gould, A. Sartenaer, and Ph . L. Toint. Global convergence of a class of trust region algorithms for optimization using inexact projections on convex constraints. SIAM Journal on Optimization , 3(1):164--221, 1993

    1993

  2. [10]

    A. R. Conn, N. I. M. Gould, and Ph . L. Toint. Global convergence of a class of trust region algorithms for optimization with simple bounds. SIAM Journal on Numerical Analysis , 25(2):433--460, 1988

    1988

  3. [11]

    Conn, Nicholas I

    Andrew R. Conn, Nicholas I. M. Gould, and Philippe L. Toint. Trust-Region Methods , volume 1 of MPS-SIAM Series on Optimization . MPS/SIAM, Philadelphia, 2000

    2000

  4. [12]

    Cartis, N

    C. Cartis, N. I. M. Gould, and P. L. Toint. An adaptive cubic regularization algorithm for nonconvex optimization with convex constraints and its function-evaluation complexity. IMA Journal of Numerical Analysis , 32(4):1662--1695, 2012

    2012

  5. [13]

    Coralia Cartis, Nicholas I. M. Gould, and Philippe L. Toint. Evaluation Complexity of Algorithms for Nonconvex Optimization: Theory, Computation and Perspectives . Number 30 in MOS-SIAM Series on Optimization. MOS/SIAM, Philadelphia, 2022

    2022

  6. [14]

    A trust-region method for nonsmooth nonconvex optimization

    Ziang Chen, Andre Milzarek, and Zaiwen Wen. A trust-region method for nonsmooth nonconvex optimization. Journal of Computational Mathematics , 41(4):639--672, 2023

    2023

  7. [15]

    Optimality conditions and a smoothing trust region N ewton method for nonlipschitz optimization

    Xiaojun Chen, Lingfeng Niu, and Yaxiang Yuan. Optimality conditions and a smoothing trust region N ewton method for nonlipschitz optimization. SIAM Journal on Optimization , 23(3):1528--1552, 2013

    2013

  8. [16]

    Complexity of trust-region methods in the presence of unbounded Hessian approximations

    Youssef Diouane, Mohamed Laghdaf Habiboullah, and Dominique Orban. Complexity of trust-region methods in the presence of unbounded Hessian approximations. arXiv preprint arXiv:2408.06243, 2024

    2024

  9. [17]

    J. E. Dennis, Jr., Shou-Bai B Li, and Richard A Tapia. A unified approach to global convergence of trust region methods for nonsmooth optimization. Mathematical Programming , 68(1-3):319--346, 1995

    1995

  10. [18]

    Dolan and Jorge J

    Elizabeth D. Dolan and Jorge J. Mor \'e . Benchmarking optimization software with performance profiles. Mathematical Programming , 91(2):201--213, 2002

    2002

  11. [19]

    M. N. Dao and H. M. Phan. Adaptive D ouglas-- R achford splitting algorithm for the sum of two operators. SIAM Journal on Optimization , 29(4):2697--2724, 2019

    2019

  12. [20]

    M. N. Dao, T. N. Pham, and P. T. Tung. Doubly relaxed forward- D ouglas-- R achford splitting for the sum of two nonconvex and a DC function. arXiv preprint arXiv:2405.08485, 2024

    2024 arXiv

  13. [21]

    Inexact derivative-free optimization for bilevel learning

    Matthias J Ehrhardt and Lindon Roberts. Inexact derivative-free optimization for bilevel learning. Journal of Mathematical Imaging and Vision , 63(5):580--600, 2021

    2021

  14. [22]

    PyCUTEst : An open source P ython package of optimization test problems

    Jaroslav Fowkes, Lindon Roberts, and \'A rp \'a d B u rmen. PyCUTEst : An open source P ython package of optimization test problems. Journal of Open Source Software , 7(78):4377, 2022

    2022

  15. [23]

    Garmanjani, D

    R. Garmanjani, D. J \'u dice, and L. N. Vicente. Trust-region methods without using derivatives: Worst case complexity and the nonsmooth case. SIAM Journal on Optimization , 26(4):1987--2011, 2016

    1987

  16. [24]

    Nicholas I. M. Gould, Dominique Orban, and Philippe L. Toint. CUTEst : A constrained and unconstrained testing environment with safe threads for mathematical optimization. Computational Optimization and Applications , 60(3):545--557, 2015

    2015

  17. [25]

    A derivative-free trust-region algorithm for composite nonsmooth optimization

    Geovani Nunes Grapiglia, Jinyun Yuan, and Ya-xiang Yuan. A derivative-free trust-region algorithm for composite nonsmooth optimization. Computational and Applied Mathematics , 35(2):475--499, 2016

    2016

  18. [26]

    Model-based derivative-free methods for convex-constrained optimization

    Matthew Hough and Lindon Roberts. Model-based derivative-free methods for convex-constrained optimization. SIAM Journal on Optimization , 32(4):2552--2579, 2022

    2022

  19. [27]

    Directional sparsity in optimal control of partial differential equations

    Roland Herzog, Georg Stadler, and Gerd Wachsmuth. Directional sparsity in optimal control of partial differential equations. SIAM Journal on Control and Optimization , 50(2):943--963, 2012

    2012

  20. [28]

    Globalized inexact proximal N ewton-type methods for nonconvex composite functions

    Christian Kanzow and Theresa Lechner. Globalized inexact proximal N ewton-type methods for nonconvex composite functions. Computational Optimization and Applications , 78(2):377--410, 2021

    2021

  21. [29]

    A scalable trust-region algorithm with application to mixed-norm regression

    Dongmin Kim, Suvrit Sra, and Inderjit Dhillon. A scalable trust-region algorithm with application to mixed-norm regression. In Proceedings of the 27th International Conference on Machine Learning , Haifa, Israel, 2010

    2010

  22. [30]

    Lam, and Lindon Roberts

    Yanjun Liu, Kevin H. Lam, and Lindon Roberts. Black-box optimization algorithms for regularized least-squares problems. arXiv preprint arXiv:2407.14915, 2024

    2024 arXiv

  23. [31]

    The indefinite proximal gradient method

    Geoffroy Leconte and Dominique Orban. The indefinite proximal gradient method. arXiv preprint arXiv:2309.08433, 2023

    2023 arXiv

  24. [32]

    Lee, Yuekai Sun, and Michael A

    Jason D. Lee, Yuekai Sun, and Michael A. Saunders. Proximal N ewton-type methods for minimizing composite functions. SIAM Journal on Optimization , 24(3):1420--1443, 2014

    2014

  25. [33]

    Ching-pei Lee and Stephen J. Wright. Inexact successive quadratic approximation for regularized optimization. Computational Optimization and Applications , 72(3):641--674, 2019

    2019

  26. [34]

    D. S. Mitrinovi\' c . Analytic Inequalities . Springer-Verlag, 1970

    1970

  27. [35]

    B. S. Mordukhovich. Variational Analysis and Generalized Differentiation I. Basic Theory . Springer, 2006

    2006

  28. [36]

    Mor \'e and Stefan M

    Jorge J. Mor \'e and Stefan M. Wild. Benchmarking derivative-free optimization algorithms. SIAM Journal on Optimization , 20(1):172--191, 2009

    2009

  29. [37]

    Numerical optimization

    Jorge Nocedal and Stephen Wright. Numerical optimization . Springer Science & Business Media, 2006

    2006

  30. [38]

    A trust region-type normal map-based semismooth N ewton method for nonsmooth nonconvex composite optimization

    Wenqing Ouyang and Andre Milzarek. A trust region-type normal map-based semismooth N ewton method for nonsmooth nonconvex composite optimization. Mathematical Programming , 2024

    2024

  31. [39]

    M. J. D. Powell. On the global convergence of trust region algorithms for unconstrained minimization. Mathematical Programming , 29:297--303, 1984

    1984

  32. [40]

    A trust region algorithm for minimization of locally L ipschitzian functions

    Liqun Qi and Jie Sun. A trust region algorithm for minimization of locally L ipschitzian functions. Mathematical Programming , 66(1-3):25--43, 1994

    1994

  33. [41]

    Model-Based Derivative-Free Optimization Methods and Software

    Tom Ma \"e l Ragonneau. Model-Based Derivative-Free Optimization Methods and Software . PhD thesis, Hong Kong Polytechnic University, 2022

    2022

  34. [42]

    Philippe L. Toint. Global convergence of a a of trust-region methods for nonconvex minimization in H ilbert space. IMA Journal of Numerical Analysis , 8(2):231--252, 1988

    1988

  35. [43]

    Wright and Benjamin Recht

    Stephen J. Wright and Benjamin Recht. Optimization for Data Analysis . Cambridge University Press, 2022

    2022

  36. [44]

    Proximal gradient algorithm with trust region scheme on R iemannian manifold

    Shimin Zhao, Tao Yan, and Yuanguo Zhu. Proximal gradient algorithm with trust region scheme on R iemannian manifold. Journal of Global Optimization , 88(4):1051--1076, 2024

    2024

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