The convergence of the Generalized Lanczos Trust-Region Method for the Trust-Region Subproblem

Fa Wang; Zhongxiao Jia

REVIEW 3 major objections 4 minor 32 references

The convergence of the Generalized Lanczos Trust-Region Method for the Trust-Region Subproblem

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves the first a-priori error bounds for the Lagrange multiplier and residual of the GLTR trust-region solver, with rates governed by the condition number of $A+\lambda_{opt}I$.

desk verdict First a-priori bounds for GLTR multiplier and residual errors, with two localized proof gaps that are likely repairable. read the letter →

arxiv 1908.02094 v1 pith:4GEIRNSI submitted 2019-08-06 math.NA cs.NA

Zhongxiao Jia , Fa Wang This is my paper · ORCID

classification math.NAcs.NA MSC 90C2090C3065K0565F10

keywords trust-regionsubproblemGLTRmethoda-prioriboundLagrangianmultiplierChebyshevpolynomialeigenvalueproblemsymmetricLanczosprocessKrylovsubspace

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 establishes the first a-priori convergence theory for the generalized Lanczos trust-region method (GLTR), the standard Lanczos-based solver for large-scale trust-region subproblems. It proves that the error in the optimal Lagrange multiplier, $\lambda_{opt}-\lambda_k$, and the error in the objective value, $q(s_k)-q(s_{opt})$, are bounded by constants times $\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{2(k+1)}$, while the solution error, the sine of the angle between $s_k$ and $s_{opt}$, and the residual norm $\|(A+\lambda_k I)s_k+g\|$ are bounded by constants times $\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{k+1}$, with $\kappa$ the condition number of $A+\lambda_{opt}I$. These are the first bounds of their kind for the multiplier and the residual. If correct, they make the number of Lanczos iterations needed to reach a given accuracy predictable from the spectrum alone, and they explain why objective and multiplier errors reach a given level in roughly half the iterations needed for the solution and residual errors.

What carries the argument

The load-bearing identity is that the projected TRS at iteration $k$ is equivalent to the Rayleigh–Ritz eigenproblem of the augmented matrix $M$ restricted to $\widetilde{\mathcal{S}}_k=\mathcal{S}_k\oplus\mathcal{S}_k$, so that $\lambda_k$ is exactly the rightmost eigenvalue of the projected matrix and $s_k$ is a scaled block of the corresponding Ritz vector. The rate is produced by the generating function $\sum_{j=0}^{\infty}(j+1)t^j U_j(x)=\frac{1-t^2}{(1+t^2-2tx)^2}$ for Chebyshev polynomials of the second kind $U_j$, which turns the best-uniform-approximation error of $\frac{1}{(x-\eta)^2}$ on $[-1,1]$ into the tail sum $\sum_{j=k+1}^{\infty}(j+1)t^j$, of order $t^{k+1}$. With $t=\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}$, this supplies both the $(k+1)$-rate for the Ritz-vector angle and, through the linear-system identity in Theorem 4.7, the squared $2(k+1)$-rate for the multiplier and objective errors.

What would settle it

Take a small TRS with exactly representable eigenvalues, for example $n=4$ with $A$ having two eigenvalues close to $-\lambda_{opt}$ and a tiny but nonzero component of $g$ in that near-degenerate subspace, and compute $\lambda_k$, $q(s_k)$, $\sin\angle(s_k,s_{opt})$, and the residual norm in exact arithmetic for $k=1,2,\dots$. If any of these quantities decreases at a slower geometric rate than the claimed $t^{k+1}$ or $t^{2(k+1)}$, or if $\mathrm{sep}(\mu_1,C_k)>0$ yet the bound (5.9) fails to decrease, the central claim would be refuted.

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Extended reading notes

Core claim

The paper's central claim is that, once the TRS is viewed through the equivalence between its solution and the rightmost eigenpair of the augmented matrix $M$, the GLTR iteration is a Rayleigh–Ritz projection of $M$ onto the subspace built from two copies of the Krylov subspace $\mathcal{K}_k(g,A)$. Under this view $\lambda_k$ is the rightmost Ritz value of $M$, and the convergence of $\lambda_k$ to $\lambda_{opt}$, of $s_k$ to $s_{opt}$, and of the residual to zero all reduce to the convergence of Ritz pairs of $M$ plus polynomial approximation bounds for the eigenvector components. The paper proves those polynomial bounds using the generating function of Chebyshev polynomials of the second kind, obtaining the rate $t^{k+1}$ with $t=\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}$ for the Ritz-vector angle and its square for $\lambda_{opt}-\lambda_k$ and $q(s_k)-q(s_{opt})$. The resulting a-priori bounds are stated in Theorem 4.11 for the multiplier error, Theorem 4.10 for the objective value, Theorem 5.4 for the angle, Theorem 5.6 for the solution error, and Theorem 5.7 for the residual norm.

Load-bearing premise

The bounds require the easy case of the TRS with $\|s_{opt}\|=\|s_k\|=\Delta$ and no Lanczos breakdown, and the angle bound also requires that the separation between the rightmost projected eigenvalue of the augmented matrix and all other projected eigenvalues stays positive for large $k$; the paper does not say how large $k$ must be for this separation condition to hold.

Editorial extensions

If this is right

A user can predict, before running GLTR, that the multiplier and objective value reach accuracy $\varepsilon$ in $O(\ln(1/\varepsilon)/\ln(1/t))$ iterations, with $t=(\sqrt{\kappa}-1)/(\sqrt{\kappa}+1)$, and that the solution and residual need about twice that many.
The residual norm $\|(A+\lambda_k I)s_k+g\|$, the standard computable stopping criterion, is now proven to decrease at the same geometric rate as the solution error, validating its use as a convergence measure.
The factor-of-two relation between the squared rate and the linear rate explains the observed behavior that $\lambda_k$ and $q(s_k)$ converge in roughly half the iterations needed for $s_k$ and the residual.
The bounds have prefactors built from the extreme eigenvalues of $A+\lambda_{opt}I$, $\|g\|$, and $\Delta$, and rates governed only by $\kappa$, not by the internal Lanczos coefficients or the dimension $n$, so they give a spectrum-level prediction that carries across problem instances.
Combined with the known monotonic increase of $\lambda_k$, the new bound shows $\lambda_k$ approaches $\lambda_{opt}$ from below at a guaranteed geometric rate, closing the gap left by earlier mixed bounds that did not decrease with $k$.

Reading between the lines

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

The same Rayleigh–Ritz viewpoint should extend to the hard case of the TRS, which this paper explicitly excludes; a hard-case variant would likely yield rates depending on the gap between the smallest eigenvalue of $A$ and the next eigenvalue, and on the angle of $g$ with the smallest eigenspace.
Because the rate depends only on $\kappa$, the bounds imply that preconditioning the TRS through the weighted-norm formulation should accelerate GLTR exactly as it accelerates conjugate gradients; this is a concrete, testable prediction not made in the paper.
The Chebyshev generating-function trick used for $\frac{1}{(x-\eta)^2}$ should apply to other rational functions appearing in eigenvector error estimates, such as higher powers or shifted resolvents, potentially giving rates for other projection methods.
The constants in the bounds are worst-case and the paper's own experiments show they can be much larger than the observed errors; whether the $t$-rates are sharp, in the sense that some spectrum attains them, is left open and could be settled by constructing eigenvalue distributions that saturate the polynomial approximation error.

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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

Two things to know. First, this is the first paper to give a-priori convergence rates for the Lagrange multiplier and the residual norm in the GLTR method. That is a real gap in the literature. Second, the rates are almost certainly right, but two steps in the written proofs do not hold, so the paper as submitted is not rigorous.

What is genuinely new and good: the route through the Adachi et al. 2n x 2n eigenvalue equivalence turns lambda_k convergence into a Ritz-value convergence problem, which is the right tool. Theorem 4.7, where the multiplier error is bounded via CG error on a linear system, is elegant. The numerical section is honest: the authors report the bounds themselves, not just the rates, and they correctly note when a bound is an overestimate. They also flag explicitly that (5.10) is asymptotic, which is good practice.

Now the soft spots. The stress-test note is correct: the tail bound in (4.31) is not a valid upper bound. For f(z) = (z+1)t^z, 0 < t < 1, the series tail is larger than the integral, not smaller. At t=0.5, k=1, the printed right-hand side is about 1.78 while the exact tail is about 2.67, so the displayed inequality is simply false. Since Theorem 4.6, the lambda_k-to-lambda_opt step in Theorem 4.8, and Theorems 5.4 and 5.7 all build on (4.31), the stated a-priori bounds are not proved as written. The fix is easy: the exact tail sum has the same t^{k+1} exponential factor, so the central rates are likely repairable, but the correction has to be written.

The second gap is Theorem 5.5: inequality (5.10) drops the positive term (lambda_opt - lambda_k)||s_opt - s_k|| and is not rigorous. The authors admit this in the preamble to the theorem, but then Theorem 5.7 uses (5.10) to present (5.12) as a rigorous a-priori bound. That is a genuine gap. The authors should either bound the dropped term and absorb it into the constant, or recast (5.12) as an asymptotic bound.

The sep(mu_1, C_k) > 0 condition in Theorem 5.4 is standard for Ritz-vector bounds, but there is no guidance on how large k must be, so the constant in (5.9) can blow up for small k. Minor, but worth a remark.

Bottom line: this paper deserves a serious referee. It fills a genuine gap, the rates are almost certainly correct, and the errors are localized with plausible fixes. I would send it to review with a request to fix (4.31) and to either justify or reclassify (5.12). I would not cite it in its current form.

Referee Report

3 major / 4 minor

Summary. The paper develops a convergence theory for the generalized Lanczos trust-region (GLTR) method applied to the trust-region subproblem (1.1). It uses the equivalence, due to Adachi et al., between the TRS and the rightmost eigenvalue problem of the augmented matrix M in (2.1). At iteration k, the GLTR projection is identified with a Rayleigh-Ritz projection of M onto a 2(k+1)-dimensional subspace, and the paper derives a-priori bounds for the multiplier error λ_opt−λ_k, the objective error q(s_k)−q(s_opt), sin ∠(s_k,s_opt), the solution error ‖s_k−s_opt‖, and the residual norm ‖(A+λ_k I)s_k+g‖. The claimed convergence rates are t^{k+1} for the angle, solution error, and residual, and t^{2(k+1)} for λ_opt−λ_k and q(s_k)−q(s_opt), where t=(√κ−1)/(√κ+1) with κ the condition number of A+λ_opt I.

Significance. The paper targets a genuinely open point: no previous a-priori bound existed for λ_opt−λ_k or for the residual norm of the GLTR approximate solution. The eigenvalue-equivalence viewpoint is a natural and potentially powerful framework, and the paper correctly identifies that the difficult part is bounding the component ‖(I−π_k)y_2‖ of the eigenvector of M. The numerical experiments in Section 6 support the plausibility of the claimed rates and show the bounds tracking the actual errors. If the proof gaps identified below are repaired, the result would be a solid contribution to the understanding of Krylov-subspace methods for the TRS. The paper is also to be credited for clearly describing the relationship between its bounds and earlier work in [31] and [32], and for isolating the Chebyshev-polynomial approximation problem that controls the convergence.

major comments (3)

[Section 4, Theorem 4.5 and Eq. (4.31)] The tail estimate in (4.31) is not a valid upper bound. For f(z)=(z+1)t^z with 0<t<1, the series tail ∑_{j=m}^{∞} f(j) is greater than, not less than, the integral ∫_m^{∞} f(z)dz. Moreover, the integration by parts evaluation omits the positive term 1/(ln t)^2. Concretely, for t=0.5 and k=1 the right-hand side of the displayed bound is approximately 1.78, while the exact tail ∑_{j=2}^{∞}(j+1)t^j equals 2.0, so the inequality is false. Since the bounds (4.27), (4.28), (4.34)–(4.35), and consequently Theorems 4.6, 4.8, 4.11, 5.4, and 5.7 all rely on this tail estimate, the stated a-priori bounds are not established as written. The exact formula ∑_{j=m}^{∞}(j+1)t^j = t^m(m(1−t)+1)/(1−t)^2 still contains the same factor t^{k+1}, so the exponential rates are likely repairable, but the current proof needs correction.
[Section 5, discussion after Eq. (5.7)] Theorem 5.4 assumes sep(μ_1, C_k)>0 for k suitably large, but the paper does not prove that this separation is bounded below uniformly or even eventually positive in a way compatible with the stated a-priori bound. The argument sep(μ_1,C_k)≥sep(μ_1^{(k)},C_k)−|μ_1−μ_1^{(k)}| is insufficient because sep(μ_1^{(k)},C_k) itself may shrink as k grows; this is not merely a technicality, since the constant 1/sep(μ_1,C_k) multiplies the rate factor in (5.9). Without a quantitative lower bound on sep(μ_1,C_k), the displayed bound is conditional on an unquantified spectral gap and cannot be called a complete a-priori result for sin ∠(s_k,s_opt). A proof that the projected matrix C_k converges to the compression L of (5.5) in the relevant sense, so that sep(μ_1,C_k)→sep(μ_1,L)>0, would close this gap.
[Section 5, Theorems 5.5 and 5.7, Eqs. (5.10) and (5.12)] The inequality (5.10) is not established because the positive term (λ_opt−λ_k)‖s_opt−s_k‖ is dropped while claiming the result is an upper bound. The triangle inequality gives ‖r_k‖ ≤ (λ_opt−λ_k)Δ + (α_1+λ_opt)‖s_opt−s_k‖ + (λ_opt−λ_k)‖s_opt−s_k‖. Dropping the last nonnegative term makes the inequality invalid in general, so Theorem 5.7 as stated is not a rigorous bound. Since the dropped term is of higher order in the rate t^{k+1}, the result is repairable either by keeping the term and bounding it by the product of (4.52) and (5.11), or by stating (5.10) as an asymptotic inequality with explicit quantified error. As written, the claimed 'rigorous a-priori bound' in Theorem 5.7 is not proved.

minor comments (4)

[Theorem 4.7, proof] There is a typographical error: 'the the squared initial error' should read 'the squared initial error'.
[Lemma 4.2] In the displayed text after (4.18), the chain α_1 ≥ α_{n−1} ≥ ⋯ ≥ α_n appears to be a typo; it should be α_1 ≥ α_2 ≥ ⋯ ≥ α_n.
[Section 4, notation after Lemma 4.2] The notation ¯P_k is defined as polynomials of degree at most k+1, and the paper later writes p_k ∈ ¯P_{k−1} for a degree-k polynomial. This is consistent but potentially confusing; a brief reminder would improve readability.
[Section 6, paragraph before Example 1] The text states that the authors 'use the Matlab functions eigs and svds with the stopping tolerance 10^{-14} to compute λ_opt, s_opt and ‖M‖'; presumably svds is used for ‖M‖, but the sentence could be clarified to avoid the impression that svds computes λ_opt.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and the self-citations are general perturbation lemmas, not assumed conclusions.

full rationale

The paper's derivation chain is not circular. The central reductions are (i) the equivalence lambda_opt - lambda_k = mu_1 - mu_1^(k) via the Adachi-Nakatsukasa-Takeda eigenproblem formulation, and (ii) standard Ritz-value and Ritz-vector perturbation inequalities from Jia (1995), Jia-Stewart (1999), and Huang-Jia-Lin (2013) that convert sin angle terms into eigenvalue and eigenvector errors. These self-citations are not load-bearing in a circular way: the cited theorems are general, parameter-free matrix perturbation results whose stated assumptions do not include the trust-region convergence rates being proved. The genuinely new content is the polynomial approximation analysis of 1/(x-eta)^2 in Theorems 4.4-4.5 and the algebraic identities in Theorems 4.7-4.8 and 5.5-5.7 that propagate the derived rates to lambda_k and the residual norm. No parameter is fitted to the target errors, and no target rate is assumed as an input. The assumptions 'k suitably large' and sep(mu_1, C_k) > 0 limit the validity of some bounds but do not create circularity. A separate reviewer objection that the tail integral estimate in (4.31) is numerically false is a correctness issue with a plausible repair, not a definitional reduction to the paper's own inputs.

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

No free parameters or invented entities. All bounds are in terms of problem data (alpha_1, alpha_n, lambda_opt, beta_0, Delta, kappa). The axioms are the standard TRS assumptions and the external perturbation lemmas.

assumptions (6)

domain assumption A is symmetric nonsingular and g != 0, Delta > 0
Standard TRS setting, stated in (1.1).
domain assumption The TRS is in the easy case: g is not orthogonal to the eigenspace of the smallest eigenvalue of A, and ||s_opt|| = Delta
Section 2.1 restricts to the easy case; the hard case is excluded from the analysis.
domain assumption The projected TRS satisfies ||s_k|| = Delta, so lambda_k > 0 for all k
Assumed in Section 3 after equation (3.14) to ensure lambda_k > 0 and the eigenvalue equivalence applies.
domain assumption The symmetric Lanczos process does not break down before iteration k, and at k_max it breaks down so T_kmax has the exact eigenvalues of A
Used in Theorem 4.7 and Theorem 4.8.
domain assumption sep(mu_1, C_k) > 0 for k suitably large
Assumed in Section 5 for the Ritz vector convergence bound of Lemma 5.3.
standard math Perturbation and convergence lemmas of Jia [16], Jia-Stewart [17], Huang-Jia-Lin [12], and Zhang et al. [31] are correct
These external results are used as black boxes for Ritz value/vector bounds and CG error bounds.

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Pith. "Pith review of The convergence of the Generalized Lanczos Trust-Region Method for the Trust-Region Subproblem." pith.science (2026). https://pith.science/paper/4GEIRNSI

@misc{pith2026190802094,
  author       = {Pith},
  title        = {Pith review of: The convergence of the Generalized Lanczos Trust-Region Method for the Trust-Region Subproblem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GEIRNSI}},
  note         = {Machine review of arXiv:1908.02094}
}

abstract

Solving the trust-region subproblem (TRS) plays a key role in numerical optimization and many other applications. The generalized Lanczos trust-region (GLTR) method is a well-known Lanczos type approach for solving a large-scale TRS. The method projects the original large-scale TRS onto a $k$ dimensional Krylov subspace, whose orthonormal basis is generated by the symmetric Lanczos process, and computes an approximate solution from the underlying subspace. There have been some a-priori error bounds for the optimal solution and the optimal objective value in the literature, but no a-priori result exists on the convergence of Lagrangian multipliers involved in projected TRS's and the residual norm of approximate solution. In this paper, a general convergence theory of the GLTR method is established, and a-priori bounds are derived for the errors of the optimal Lagrangian multiplier, the optimal solution, the optimal objective value and the residual norm of approximate solution. Numerical experiments demonstrate that our bounds are realistic and predict the convergence rates of the three errors and residual norms accurately.

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Works this paper leans on

32 extracted references · 31 canonical work pages

[31]

L. H. Zhang, C. G. Shen, and R. C. Li , On the generalized Lanczos trust-region method , SIAM J. Optim., 27 (2017), pp. 2110–2142

work page 2017
[32]

L. H. Zhang, W. H. Yang, C. Shen, and J. Feng , Error bounds of the Lanczos approach for the trust-region subproblem , Front. Math. China, 13 (2018), pp. 459–481

work page 2018
[1]

Adachi, S

S. Adachi, S. Iw ata, Y. Nakatsukasa, and A. Takeda , Solving the trust-region subproblem by a generalized eigenvalue problem , SIAM J. Optim., 27 (2017), pp. 269–291

work page 2017
[2]

R. H. Byrd, R. B. Schnabel, and G. A. Shultz , Approximate solution of the trust region problem by minimization over two-dimensional subspaces , Math. Prog., 40 (1988), pp. 247– 263

work page 1988
[3]

A. R. Conn, N. I. M. Gould, and P. L. Toint , Trust-region Methods, SIAM, Philadelphia, 2000

work page 2000
[4]

Ei Attar , Special Functions and Orthogonal Polynomials , Lulu Press, USA, 2006

R. Ei Attar , Special Functions and Orthogonal Polynomials , Lulu Press, USA, 2006

work page 2006
[5]

J. B. Erw ay, and P. E. Gill , A subspace minimization method for the trust-region step , SIAM J. Optim., 20 (2010), pp. 1439–1461

work page 2010
[6]

J. B. Erw ay, P. E. Gill, and J. D. Griffin , Iterative methods for ﬁnding a trust-region step , SIAM J. Optim., 20 (2009), pp. 1110–1131

work page 2009

Show all 32 references

[7]

Fortin, and H

C. Fortin, and H. Wolkowicz , The trust region subproblem and semideﬁnite programming , Optim. Methods Softw., 19 (2004), pp. 41–67

2004
[8]

N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint , Solving the trust-region subproblem using the Lanczos method , SIAM J. Optim., 9 (1999), pp. 504–525

1999
[9]

N. I. M. Gould, D. P. Robinson, and H. S. Thorne , On solving trust-region and other regularised subproblems in optimization , Math. Prog. Comput., 2 (2010), pp. 21–57

2010
[10]

Gander, C

W. Gander, C. H. Golub, and U. von Matt , A constrained eigenvalue problem , Linear Algebra Appl., 114 (1989), pp. 815–839

1989
[11]

Greenbaum , Iterative Methods for Solving Linear Systems , SIAM, Philadephia, 1997

A. Greenbaum , Iterative Methods for Solving Linear Systems , SIAM, Philadephia, 1997

1997
[12]

T. M. Huang, Z. Jia, and W. W. Lin , On the convergence of Ritz pairs and reﬁned Ritz vectors for quadratic eigenvalue problems , BIT Numer Math, 53 (2013), pp. 941–958

2013
[13]

W. W. Hager , Minimizing a quadratic over a sphere , SIAM J. Optim., 12 (2001), pp. 188–208

2001
[14]

W. W. Hager, and Y. Krylyuk , Graph partitioning and continuous quadratic programming , SIAM J. Alg. Discrete Methods, 12( 1999), pp. 500–523

1999
[15]

N. J. Higham , Functions of Matrices: Theory and Computation , SIAM, Philadelphia, PA, 2008

2008
[16]

Jia , The convergence of generalized Lanczos methods for large un symmetric eigenproblems , SIAM J

Z. Jia , The convergence of generalized Lanczos methods for large un symmetric eigenproblems , SIAM J. Matrix Anal. Appl., 16 (1995), pp. 843–862

1995
[17]

Jia, and G

Z. Jia, and G. W. Stew art , On the convergence of Ritz values, Ritz vectors and reﬁned Ri tz vectors, Tech. Rep. TR-3986, Department of Computer Science, Unive rsity of Maryland, College Park, MD, 1999

1999
[18]

Lukˇsan, C

L. Lukˇsan, C. Matonoha, and J. Vl ˇcek, On Lagrange multipliers of trust-region subproblems , BIT, 48 (2008), pp. 763–768

2008
[19]

Meurant , The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Pre- cision Computations , SIAM, Philadelphia, 2006

G. Meurant , The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Pre- cision Computations , SIAM, Philadelphia, 2006

2006
[20]

J. J. Mor ´e, and D. C. Sorensen , Computing a trust region step , SIAM J. Sci. Statist. Com- put., 4 (1983), pp. 553–572

1983
[21]

Nocedal and S

J. Nocedal and S. J. Wright , Numerical Optimization , Second Edition, Springer, 2006

2006
[22]

Rendl, and H

F. Rendl, and H. Wolkowicz , A semideﬁnite framework for trust region subproblems with applications to large scale minimization , Math. Prog., 77 (1997), pp. 273–299

1997
[23]

Rojas, S

M. Rojas, S. A. Santos, and D. C. Sorensen , A new matrix-free algorithm for the large-scale trust-region subproblem, SIAM J. Optim., 11 (2001), pp. 611–646

2001
[24]

Rojas, S

M. Rojas, S. A. Santos, and D. C. Sorensen , Algorithm 873: LSTRS: MATLAB software for large-scale trust-region subproblems and regularizat ion, ACM Trans. Math. Softw., 34 (2008), pp. 1–28

2008
[25]

Saad , Numerical Methods for Large Eigenvalue Problems , revised Edition, SIAM, Philade- phia, 2011

Y. Saad , Numerical Methods for Large Eigenvalue Problems , revised Edition, SIAM, Philade- phia, 2011

2011
[26]

D. C. Sorensen , Minimization of a large-scale quadratic function subject t o a spherical con- straint, SIAM J. Optim., 7 (1997), pp. 141–161

1997
[27]

Steihaug , The conjugate gradient method and trust regions in large sca le optimization , SIAM J

T. Steihaug , The conjugate gradient method and trust regions in large sca le optimization , SIAM J. Numer. Anal., 20 (1983), pp. 626–637

1983
[28]

G. W. Stew art and J.-G. Sun , Matrix Perturbation Theory , Academic Press, INC., Boston, 1990

1990
[29]

P. L. Toint , Towards an eﬃcient sparsity exploiting Newton method for mi nimization, Sparse Matrices and Their Uses, Academic Press, 1981, pp. 57–88

1981
[30]

Yuan , On the truncated conjugate gradient method , Math

Y. Yuan , On the truncated conjugate gradient method , Math. Prog., 87 (2000), pp. 561–573

2000

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [31]

L. H. Zhang, C. G. Shen, and R. C. Li , On the generalized Lanczos trust-region method , SIAM J. Optim., 27 (2017), pp. 2110–2142

work page 2017

[2] [32]

L. H. Zhang, W. H. Yang, C. Shen, and J. Feng , Error bounds of the Lanczos approach for the trust-region subproblem , Front. Math. China, 13 (2018), pp. 459–481

work page 2018

[3] [1]

Adachi, S

S. Adachi, S. Iw ata, Y. Nakatsukasa, and A. Takeda , Solving the trust-region subproblem by a generalized eigenvalue problem , SIAM J. Optim., 27 (2017), pp. 269–291

work page 2017

[4] [2]

R. H. Byrd, R. B. Schnabel, and G. A. Shultz , Approximate solution of the trust region problem by minimization over two-dimensional subspaces , Math. Prog., 40 (1988), pp. 247– 263

work page 1988

[5] [3]

A. R. Conn, N. I. M. Gould, and P. L. Toint , Trust-region Methods, SIAM, Philadelphia, 2000

work page 2000

[6] [4]

Ei Attar , Special Functions and Orthogonal Polynomials , Lulu Press, USA, 2006

R. Ei Attar , Special Functions and Orthogonal Polynomials , Lulu Press, USA, 2006

work page 2006

[7] [5]

J. B. Erw ay, and P. E. Gill , A subspace minimization method for the trust-region step , SIAM J. Optim., 20 (2010), pp. 1439–1461

work page 2010

[8] [6]

J. B. Erw ay, P. E. Gill, and J. D. Griffin , Iterative methods for ﬁnding a trust-region step , SIAM J. Optim., 20 (2009), pp. 1110–1131

work page 2009