Scalable Derivative-Free Optimization Algorithms with Low-Dimensional Subspace Techniques

Zaikun Zhang

REVIEW 4 major objections 5 minor 2 cited by

Scalable Derivative-Free Optimization Algorithms with Low-Dimensional Subspace Techniques

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A low-dimensional subspace method scales derivative-free optimization to about 10,000 variables with linear-in-n evaluation cost.

desk verdict A honest draft that re-introduces a known subspace DFO framework; the linear-in-n complexity claim rests on an assumption that fails for the noisy function values advertised. read the letter →

arxiv 2501.04536 v1 pith:XEF2SIIN submitted 2025-01-08 math.OC

Zaikun Zhang This is my paper · ORCID

classification math.OC MSC 90C5690C3065K05

keywords derivative-freeoptimizationsubspacemethodsapproximategradientsworst-caseevaluationcomplexityfinitedifferencesnonconvexhigh-dimensionalOptimIST

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 claims that derivative-free optimization can be made scalable by never searching the full space. Its OptimIST algorithm builds, at each iteration, a low-dimensional subspace that contains an approximate gradient, then solves a cheap subproblem inside that subspace and takes a safeguarded step along the approximate-gradient direction. The analysis gives a worst-case function-evaluation complexity of $O(n\epsilon^{-2})$ for smooth nonconvex objectives when the finite-difference step is chosen as $O(n^{-1/2})$, linear in the number of variables instead of the quadratic dependence of full-space derivative-free methods. Numerical experiments on problems with $10^4$ variables, using only three significant digits of the function values, support the claim that this scaling is real. A reader should care because the result would make derivative-free methods plausible for high-dimensional black-box optimization.

What carries the argument

The load-bearing object is the pair $(\hat g_k, \delta_k)$: an approximate gradient whose error is controlled by the same step size that controls the decrease. Step size update doubles $\delta_k$ whenever $\|\hat g_k\|\ge \eta\delta_k$ and the decrease $f_{k+1}\le f_k-\eta\delta_k^2$ occurs, otherwise it halves $\delta_k$. The implication (3.4) shows that whenever $\delta_k\le \mu\|g_k\|$, the safeguarded step $x_k-\delta_k\hat g_k/\|\hat g_k\|$ is guaranteed to achieve the required decrease, so $\delta_k$ keeps growing relative to $\|g_k\|$; this yields the lower bound $f_k-f_{k+1}\ge \eta\nu^2\|\tilde g_k\|^2$ and hence the complexity counts. The stencil $Y_k = \{x_k\}\cup\{x_k+\tau\delta_k e_i\}$ is the concrete derivative-free construction of $\hat g_k$, and the subspace $S_k$ built around it is what keeps each subproblem cheap.

What would settle it

Run Algorithm 3.1 with $\tau = n^{-1/2}$ on a family of smooth nonconvex problems of increasing dimension, and record the number of function evaluations needed to reach $\|\nabla f(x_k)\|\le\epsilon$. If that count grows faster than $n\epsilon^{-2}$ in either $n$ or $\epsilon^{-1}$, the claimed worst-case bound is not realized; the same run should also track $\sup_k \|\hat g_k-g_k\|/\delta_k$ to check whether Assumption 3.1 is actually satisfied.

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Extended reading notes

Core claim

The paper argues that Algorithm 3.1, derivative-free OptimIST, is globally convergent and has worst-case iteration complexity $K_\epsilon = O(\nu^{-2}\epsilon^{-2})$ for nonconvex smooth objectives under Assumption 3.1, with $K_\epsilon = O(\nu^{-2}\epsilon^{-1})$ for convex objectives and $O(\nu^{-2}|\log\epsilon|)$ for strongly convex ones. The constant is $\nu = \min\{\delta_0/\|g_0\|, \mu/2\}$ with $\mu = 2/(L+2\eta+4\zeta)$, so the complexity depends on the dimension only through the approximate-gradient error constant $\zeta$. When $\hat g_k$ comes from coordinate finite differences on the stencil $\{x_k\}\cup\{x_k+\tau\delta_k e_i\}$, one has $\zeta = \tau\sqrt n L/2$, and choosing $\tau = O(n^{-1/2})$ balances the stencil cost against the error term, yielding $K_f^\epsilon = O(n\epsilon^{-2})$ function evaluations. The convergence argument does not require any particular choice of subspace: any $S_k$ containing $\hat g_k$ works, so the subspace is a mechanism for cheap exploration rather than a condition for convergence.

Load-bearing premise

All convergence and complexity conclusions rely on Assumption 3.1, that the approximate gradient error is bounded by a constant times the current step size $\delta_k$ at every iteration; if noisy or truncated function values break that bound, the guaranteed decrease and the $O(n\epsilon^{-2})$ count no longer follow.

Editorial extensions

If this is right

In the smooth nonconvex case, the guarantee is a worst-case budget of $O(n\epsilon^{-2})$ function evaluations to reach $\|\nabla f\|\le\epsilon$, so dimensions of order $10^3$ to $10^4$ move into the range of derivative-free methods.
For convex objectives the same search needs $O(\nu^{-2}\epsilon^{-1})$ evaluations, and for strongly convex ones $O(\nu^{-2}|\log\epsilon|)$, so the subspace framework inherits the usual acceleration for easier problems.
Because the convergence condition is only $\hat g_k\in S_k$, the subspace can be chosen freely to include conjugate-gradient directions, limited-memory quasi-Newton directions, or approximate-Newton and negative-curvature directions without changing the theory.
The subspace subproblem can be solved inexactly with only $O(1)$ function evaluations per outer iteration, so the per-iteration cost is dominated by forming the finite-difference stencil, which can be evaluated in parallel.

Reading between the lines

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

If the linear-in-$n$ bound survives a polished proof, derivative-free optimization becomes a candidate for hyperparameter tuning and simulation-based design at scales where only noisy function values are available; the paper's own experiments stop at three significant digits and smooth test functions, so this is a forward inference.
The theory's indifference to the subspace suggests a testable extension: replacing the coordinate stencil by random subspace projections or sketched gradients should preserve the $O(n\epsilon^{-2})$ rate whenever the error condition (3.2) holds with high probability; the probabilistic trust-region techniques cited by the paper are the natural tool for that analysis.
A practical diagnostic follows directly: record $\|\hat g_k-g_k\|/\delta_k$ during runs. If this ratio frequently exceeds the theoretical $\zeta$, the algorithm has left the regime where the complexity guarantee applies, which would explain and predict stagnation on ill-conditioned 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

Zaikun's draft is exactly what it says on the tin: a quick set of notes on a subspace DFO framework that goes back to his 2012 thesis and to Conn et al.'s iterated-subspace work. The new material is the numerical study with NEWUOAs, not the algorithm or the convergence skeleton. That is fine if the paper is read as a research announcement, but the current form overclaims what is actually proved.

The framework itself is sensible and the paper explains it clearly. The observation that convergence is guaranteed by (3.1) and (3.2) regardless of the subspace choice is a useful unifying perspective. The numerical results at n=10^4 are encouraging, and the comparison with fminunc is fair enough as a first look. I also appreciate that the author flags this as a draft and says the final paper will be very different.

The soft spots are mostly in the complexity analysis. The stress test is right: Assumption 3.1, equation (3.2), is not justified when function values are truncated to 3 significant digits. With an absolute noise level ε_f, a forward-difference gradient has error O(ε_f/(τδ_k)) in addition to the truncation term O(Lτδ_k), so it cannot satisfy ||ĝ_k − g_k|| ≤ ζδ_k with a constant ζ as δ_k → 0. Since (3.2) is what converts sufficient decrease into the lower bound (3.5), the claimed O(n ε^{-2}) evaluation complexity is not established for the noisy setting emphasized in the abstract. This does not mean the bound is false; it means the draft has not supplied the machinery to handle noise. The paper also leaves the derivation of (3.4) to the reader, and the numerical experiments lack error bars, incomplete settings, and no reproducible code or data. Those are exactly the things a full version must fix.

Novelty is genuinely limited: the algorithm is from the author's own thesis, and the complexity analysis is sketched as an application of prior techniques in [8,9]. The self-citation is not a problem per se; it is the absence of new proof content that matters.

Who should read this? DFO researchers who want to know where subspace methods are headed, and anyone about to start high-dimensional DFO. It is a useful pointer to NEWUOAs and to the upcoming SPRIMA package. But it is not yet a self-contained paper.

Recommendation: send it to peer review, but only after the author revises it into a full paper with detailed proofs, a rigorous treatment of noisy evaluations, and reproducible experiments. As it stands, it deserves a serious referee because the question matters and the author is credible, but the referee should insist on the missing analysis.

Referee Report

4 major / 5 minor

Summary. The paper reintroduces a derivative-free subspace optimization framework originally developed in the author's 2012 PhD thesis. Algorithm 3.1 constructs an approximate gradient, builds a low-dimensional subspace containing it, performs a safeguarded subspace minimization, and adapts a step-size/trust parameter delta_k. The paper sketches global convergence and worst-case evaluation complexity results, claiming O(n epsilon^{-2}) evaluation complexity in the smooth nonconvex case when tau = O(n^{-1/2}), and presents numerical experiments on CUTEst problems with n = 200 and n = 10^4 using function values truncated to 3 significant digits. The manuscript is explicitly labeled as a draft with sketched proofs, and the final paper is promised to be substantially different.

Significance. If the claimed linear-in-n evaluation complexity were rigorously established, this would be a valuable contribution to derivative-free optimization at high dimensions, complementing recent scalable subspace DFO methods such as RSDFO. The framework is simple, flexible, and usefully separates the subspace choice from the convergence mechanism. The paper is honest about provenance, and it provides links to open-source implementations and a reproducible CUTEst benchmark, which are strengths. However, the central theoretical claim is not established in the current manuscript: the key Assumption 3.1 is incompatible with the inaccurate function evaluations used in the experiments, and the proof of the central implication (3.4) is left as a sketch. The numerical results are encouraging but preliminary and do not directly test the theory. The contribution is potentially significant, but the submission is not in publishable form.

major comments (4)

[§3.2, Eq. (3.6) and §3.4] Assumption 3.1 (Eq. (3.2)) cannot hold for the inaccurate function evaluations used in the experiments. For exact evaluations, the forward-difference model (3.6) gives ||g_k - g_k|| <= L*tau*delta_k*sqrt(n)/2, which is O(delta_k). If function values are truncated to 3 significant digits, the absolute error is bounded below by roughly 10^{a-3} when f is of order 10^a, and the coordinate difference quotient carries a noise term of order epsilon_f/(tau*delta_k), which grows as delta_k goes to zero. Thus no constant zeta independent of k can satisfy (3.2) uniformly as delta_k -> 0. With tau = O(n^{-1/2}) and n ~ 10^4, the sampling radius tau*delta_k is small enough that this noise likely dominates at late iterations. Since (3.2) is used to pass from (3.1) to (3.5) and hence to the claimed K_f^epsilon = O(n epsilon^{-2}) bound, the advertised worst-case evaluation complexity is not established for the inaccurate-evaluation setting highlighted in the abstract and used in Section 4.2.
[§3.1, Eq. (3.4)] The key implication (3.4) is asserted with the phrase "we can establish" but no proof is provided, and it is not a one-line consequence of the triangle inequality and Taylor expansion. One needs explicit conditions relating eta, zeta, L, and mu, and a verification that delta_k <= mu ||g_k|| implies both ||g_k|| >= eta*delta_k and the decrease inequality f(x_k - delta_k*g_k/||g_k||) <= f_k - eta*delta_k^2. This implication is the only bridge from Assumption 3.1 to the lower bound (3.5) and to the complexity statements in Section 3.4; leaving it as an exercise is not acceptable in a submission whose central contribution is the complexity result.
[§3.3–3.4] The claim that Step 3 of Algorithm 3.1 costs O(1) function evaluations is not justified. Section 3.3 imposes no quality requirement on x_s^k, but it does not specify a fixed budget for the subspace subproblem solver; if (3.9) is solved by a DFO solver run to convergence, the per-iteration cost is not automatically O(1). The evaluation complexity K_f^epsilon = O(n^{2-2p} epsilon^{-2}) in Section 3.4 therefore requires an explicit rule (for example, at most one trial point in S_k besides the safeguarded point x_g^k) before the linear-in-n conclusion K_f^epsilon = O(n epsilon^{-2}) follows.
[§4.2] The numerical study is not sufficient to support the theoretical claims. The experiments use only 3 significant digits, which, as discussed above, violates the model underlying Assumption 3.1, and no diagnostic is reported for whether ||g_k - g_k|| <= zeta*delta_k holds during the runs. Table 2 reports no accuracy measure such as final gradient norm or distance to a known solution, gives no stopping criteria for fminunc, and reports no repeated runs or variability information. The results are encouraging, but they cannot substitute for a direct test of the convergence theory.

minor comments (5)

[§1] Assumption 1.1 states "The function f : R -> R" but the problem is on R^n; this should read f : R^n -> R.
[§3.1] The definition nu = min{delta_0/||g_0||, mu/2} is undefined when g_0 = 0; the degenerate case should be handled explicitly.
[§3.4] The statement K_f^epsilon = O(n epsilon^{-2}) should specify the dependence of the constant on L, eta, zeta, delta_0, and f(x_0) - f_*; hiding all of these in the O notation makes the scalability claim difficult to compare with existing bounds.
[Figure 1] The performance profiles do not state the evaluation budget, the convergence test details, or clearly identify which curve corresponds to NEWUOAs and which to NEWUOA in the printed caption; please add this information.
[References] Reference [9] is incomplete: it lacks a publication venue and year; if it is a preprint, the arXiv identifier should be given.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the O(n epsilon^{-2}) complexity is a conditional consequence of Assumption 3.1, and the numerical evidence uses external benchmarks; the self-citations are present but not load-bearing.

full rationale

The paper's main theoretical chain is conditional rather than circular. Algorithm 3.1's convergence and the claimed O(n epsilon^{-2}) evaluation complexity follow from the sufficient-decrease condition (3.1) and Assumption 3.1, Eq. (3.2), via the lower bound (3.5) and the standard trust-region complexity argument sketched in Section 3.4. Section 3.2 gives an explicit finite-difference construction, the interpolation set Y_k in Eq. (3.6), that would satisfy (3.2) with zeta = tau*sqrt(n)*L/2 when function values are exact; the complexity calculation in Section 3.4 then reduces to algebra. No fitted parameter is relabeled as a prediction, and no quantity used in the proof is defined in terms of the conclusion it is supposed to establish. The citations to the author's Ph.D. thesis and to references [8,9] are historical or technical: the algorithm is fully specified in the paper, and the cited works are standard trust-region complexity tools that do not assume the present paper's target result. The numerical experiments are run against external benchmarks (98 CUTEst problems and 12 ten-thousand-dimensional problems) and external solvers (NEWUOA and fminunc), so the empirical claim is not a self-referential loop. The one substantive concern is not circularity: Assumption 3.1 is justified in Section 3.2 only for exact function evaluations, whereas the experiments truncate function values to 3 significant digits; under noise the finite-difference error may scale as epsilon_f/(tau*delta_k), which can violate (3.2) as delta_k decreases. That is an assumption-to-experiment robustness gap, consistent with the draft's own caveat that the theory is only sketched, and it does not make the derivation circular. Overall, the central derivation is self-contained; the paper earns a low score for minor, non-load-bearing self-citation rather than for any reduction of the proof to its own inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. Its central claims rest on smoothness and gradient-accuracy assumptions, a finite-difference model quality assumption, an O(1) subproblem-cost assumption, and the author's prior analysis techniques.

free parameters (4)

eta = not reported
Step acceptance threshold in Algorithm 3.1 Steps 3 and 4; the convergence analysis holds for any positive value, but the numerical experiments do not state the value used.
delta_0 = not reported
Initial step radius in Algorithm 3.1; value used in experiments not specified.
tau = not reported (analysis uses tau = O(n^{-p}), p = 1/2)
Finite-difference step factor in the interpolation set (3.6). The choice p = 1/2 yields O(n epsilon^{-2}) evaluation complexity; this is a design choice, not fitted to data.
m = not reported
Number of memory pairs in the limited-memory quasi-Newton subspace (3.8); affects subspace dimension and is not specified in the experiments.

assumptions (5)

domain assumption Assumption 1.1: f is bounded below, differentiable, and nabla f is Lipschitz continuous with constant L.
Standard smoothness assumption for DFO convergence analysis; stated at the start of Section 1.
domain assumption Assumption 3.1: ||g_hat_k - g_k|| <= zeta delta_k for all k.
Approximate gradient error must scale with the step radius. The entire convergence and complexity sketch in Section 3.1 rests on this coupling.
domain assumption The finite-difference interpolation set (3.6) produces a fully linear model with zeta = tau sqrt(n) L / 2.
Stated in Section 3.2; requires f smooth enough and the step tau delta_k small. With function values truncated to 3 significant digits, this may fail in practice.
ad hoc to paper Solving the subspace subproblem (3.9) can be done with O(1) function evaluations independent of n.
Section 3.4 asserts this because the subproblem is low-dimensional and no quality requirement is imposed on x_k^s, but no algorithm or proof of the constant is given.
standard math The worst-case iteration complexity bounds follow from the techniques in [8,9].
The paper does not reproduce these proofs; it cites the author's prior work as the source of the complexity arguments.

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Pith. "Pith review of Scalable Derivative-Free Optimization Algorithms with Low-Dimensional Subspace Techniques." pith.science (2026). https://pith.science/paper/XEF2SIIN

@misc{pith2026250104536,
  author       = {Pith},
  title        = {Pith review of: Scalable Derivative-Free Optimization Algorithms with Low-Dimensional Subspace Techniques},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XEF2SIIN}},
  note         = {Machine review of arXiv:2501.04536}
}

read the original abstract

We re-introduce a derivative-free subspace optimization framework originating from Chapter 5 of the Ph.D. thesis [Z. Zhang, On Derivative-Free Optimization Methods, Ph.D. thesis, Chinese Academy of Sciences, Beijing, 2012] of the author under the supervision of Ya-xiang Yuan. At each iteration, the framework defines a (low-dimensional) subspace based on an approximate gradient, and then solves a subproblem in this subspace to generate a new iterate. We sketch the global convergence and worst-case complexity analysis of the framework, elaborate on its implementation, and present some numerical results on solving problems with dimensions as high as 10^4 using only inaccurate function values.

Figures

Figures reproduced from arXiv: 2501.04536 by the authors.

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Forward citations

Cited by 2 Pith papers

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A Model-Based Derivative-Free Optimization Algorithm for Partially Separable Problems
math.OC 2025-06 conditional novelty 6.0 of 10

UPOQA exploits partial separability by building individual quadratic models with per-element trust regions and an approximate 'Steinmetz projection', cutting function evaluations versus baselines in numerical tests.

Reference graph

Works this paper leans on

21 extracted references · 20 canonical work pages · cited by 2 Pith papers

[1]

A. S. Bandeira, K. Scheinberg, and L. N. Vicente. Convergence of trust-region methods based on probabilistic models. SIAM J. Optim., 24:1238–1264, 2014

work page 2014
[2]

Cartis and L

C. Cartis and L. Roberts. Scalable subspace methods for derivative-free nonlinear least- squares optimization. Math. Program., 199:461–524, 2023

work page 2023
[3]

A. R. Conn, K. Scheinberg, and L. N. Vicente. Introduction to Derivative-Free Optimization, volume 8 of MOS-SIAM Ser. Optim.SIAM, Philadelphia, 2009

work page 2009
[4]

A. R. Conn, Ph. L. Toint, A. Sartenaer, and N. I. M. Gould. On iterated-subspace minimization methods for nonlinear optimization. In Adams L. and Nazareth J. L., editors, Linear and Nonlinear Conjugate Gradient-Related Methods, pages 50–78. SIAM, Philadelphia, 1996

work page 1996
[5]

E. D. Dolan and J. J. Mor´ e. Benchmarking optimization software with performance profiles. Math. Program., 91:201–213, 2002. 10

work page 2002
[6]

Fasano and S

G. Fasano and S. Lucidi. A nonmonotone truncated Newton-Krylov method exploiting negative curvature directions, for large scale unconstrained optimization. Optim. Lett., 3:521–535, 2009

work page 2009
[7]

N. I. M. Gould, D. Orban, and Ph. L. Toint. CUTEst: a constrained and unconstrained testing environment with safe threads for mathematical optimization. Comput. Optim. Appl., 60:545–557, 2015

work page 2015
[8]

Gratton, C

S. Gratton, C. W. Royer, L. N. Vicente, and Z. Zhang. Complexity and global rates of trust-region methods based on probabilistic models. IMA J. Numer. Anal., 38:1579–1597, 2018

work page 2018

Show all 21 references

[9]

Gratton, L

S. Gratton, L. N. Vicente, and Z. Zhang. Optimization by space transformation and decomposition, 2021

2021
[10]

Gu et al

T. Gu et al. BBGP-sDFO: Batch Bayesian and Gaussian process enhanced subspace derivative free optimization for high-dimensional analog circuit synthesis. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 43:417–430, 2024

2024
[11]

W. Hare, G. Jarry-Bolduc, and C. Planiden. Hessian approximations. arXiv:2011.0 2584, 2020

2011
[12]

Hare and C

W. Hare and C. W. Royer. Detecting negative eigenvalues of exact and approximate Hessian matrices in optimization. arXiv:2206.05318, 2022

2022 arXiv
[13]

D. C. Liu and J. Nocedal. On the limited memory BFGS method for large scale optimization. Math. Program., 45:503–528, 1989

1989
[14]

J. J. Mor´ e and S. M. Wild. Benchmarking derivative-free optimization algorithms. SIAM J. Optim., 20:172–191, 2009

2009
[15]

M. J. D. Powell. The NEWUOA software for unconstrained optimization without derivatives. In G. Di Pillo and M. Roma, editors, Large-scale Nonlinear Optimization, pages 255–297. Springer, Boston, 2006

2006
[16]

C. W. Royer and S. J. Wright. Complexity analysis of second-order line-search algorithms for smooth nonconvex optimization. SIAM J. Optim., 28:1448–1477, 2018

2018
[17]

Y. Yuan. A review on subspace methods for nonlinear optimization. In S. Y. Jang, Y. R. Kim, Lee D.-W., and I. Yie, editors, Proceedings of the International Congress of Mathematicians (2014), pages 807–827. Kyung Moon Sa Co. Ltd., Seoul, 2014

2014
[18]

Yuan and J

Y. Yuan and J. Stoer. A subspace study on conjugate gradient algorithms. ZAMM Z. Angew. Math. Mech., 75:69–77, 1995. 11

1995
[19]

Zhang, D

C. Zhang, D. Ge, B. Jiang, and Y. Ye. DRSOM: A dimension reduced second-order method and preliminary analyses. arXiv:2208.00208, 2022

2022 arXiv
[20]

Z. Zhang. On Derivative-Free Optimization Methods(in Chinese). PhD thesis, Chinese Academy of Sciences, Beijing, 2012

2012
[21]

Z. Zhang. PRIMA: Reference Implementation for Powell’s Methods with Modernization and Amelioration. https://www.libprima.net, 2023. 12

2023

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

A. S. Bandeira, K. Scheinberg, and L. N. Vicente. Convergence of trust-region methods based on probabilistic models. SIAM J. Optim., 24:1238–1264, 2014

work page 2014

[2] [2]

Cartis and L

C. Cartis and L. Roberts. Scalable subspace methods for derivative-free nonlinear least- squares optimization. Math. Program., 199:461–524, 2023

work page 2023

[3] [3]

A. R. Conn, K. Scheinberg, and L. N. Vicente. Introduction to Derivative-Free Optimization, volume 8 of MOS-SIAM Ser. Optim.SIAM, Philadelphia, 2009

work page 2009

[4] [4]

A. R. Conn, Ph. L. Toint, A. Sartenaer, and N. I. M. Gould. On iterated-subspace minimization methods for nonlinear optimization. In Adams L. and Nazareth J. L., editors, Linear and Nonlinear Conjugate Gradient-Related Methods, pages 50–78. SIAM, Philadelphia, 1996

work page 1996

[5] [5]

E. D. Dolan and J. J. Mor´ e. Benchmarking optimization software with performance profiles. Math. Program., 91:201–213, 2002. 10

work page 2002

[6] [6]

Fasano and S

G. Fasano and S. Lucidi. A nonmonotone truncated Newton-Krylov method exploiting negative curvature directions, for large scale unconstrained optimization. Optim. Lett., 3:521–535, 2009

work page 2009

[7] [7]

N. I. M. Gould, D. Orban, and Ph. L. Toint. CUTEst: a constrained and unconstrained testing environment with safe threads for mathematical optimization. Comput. Optim. Appl., 60:545–557, 2015

work page 2015

[8] [8]

Gratton, C

S. Gratton, C. W. Royer, L. N. Vicente, and Z. Zhang. Complexity and global rates of trust-region methods based on probabilistic models. IMA J. Numer. Anal., 38:1579–1597, 2018

work page 2018