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Solving a linear program via a single unconstrained minimization

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Pith's one-line read The paper proves that a primal-dual linear program is equivalent to a single unconstrained minimization of a smooth convex merit function, with a regularized Newton method reaching ε-accuracy in O(ε^{−3/2}) iterations.

desk verdict The central equivalence and the global-rate theorem hold up, but a noncentral Hessian claim in Section 4.2 is false, and the practical claims outrun the evidence. read the letter →

arxiv 2505.21232 v2 pith:PTBFKEUU submitted 2025-05-27 math.OC

Adilet Otemissov , Alina Abdikarimova This is my paper
classification math.OC MSC 90C0590C25
keywords linearprogrammingprimal-dualoptimalitymeritfunctionunconstrainedminimizationLevenberg-Marquardtregularizationglobalconvergencezeroresidualproblemhomotopymethod
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 that solving the primal-dual pair of linear programs is equivalent to one unconstrained minimization problem. The proposed merit function $f_q(x,\lambda,s)$ is convex, twice continuously differentiable, nonnegative, and reaches zero exactly at primal-dual optimal solutions, so LP solving becomes a zero-residual unconstrained optimization over the whole space. The authors prove that a Newton method with Levenberg-Marquardt regularization attains primal and dual residuals below $\epsilon$ in $O(\epsilon^{-3/2})$ iterations whenever the LP optimal set is nonempty and bounded, and that the per-iteration linear system can be solved in $O(m^3+n^3)$ time. A modified merit function with a tunable parameter $\nu>0$ keeps the Hessian nonsingular and yields a homotopy-style heuristic that converges faster in experiments. This gives an exact reformulation that needs no penalty parameter to be driven to zero and no barrier to keep iterates feasible.

What carries the argument

The load-bearing object is the merit function $f_q$ in (MF): a sum of squared residual terms for the primal-dual optimality conditions, with complementary slackness replaced by the duality gap $c^Tx-b^T\lambda$ and with nonnegativity enforced by $\max\{-x_j,0\}^q$ and $\max\{-s_j,0\}^q$ for $q>2$. Its convexity and twice continuous differentiability make unconstrained second-order minimization possible, while its zero-level set encodes exactly the LP solutions. The singular Hessian at optimal points is handled by adding $\mu_k I$ (Levenberg-Marquardt regularization), and the Lipschitz continuity of the Hessian at $q=3$ plugs into an existing regularized-Newton convergence theorem to yield the global rate. A second mechanism is the modified function $h_{q,\nu}=f_q+\nu\|\lambda\|_2^2+\frac{\nu}{q(q-1)}\sum_j(\max\{x_j,0\}^q+\max\{s_j,0\}^q)$, whose Hessian is positive definite for $\nu>0$, so the heuristic can take Newton steps while $\nu$ decays to zero.

What would settle it

Run Algorithm 1 on a linear program whose optimal set is provably nonempty and bounded, starting from a point with large norm, and record how many iterations are needed to push the residuals below a decreasing sequence of tolerances $\epsilon$. If the count grows faster than a constant times $\epsilon^{-3/2}$, or if the iterates escape to infinity, Theorem 3.2's claim fails for that instance. A direct check of the proof mechanism is to test whether the sublevel set $\{(x,\lambda,s): f_3(x,\lambda,s)\le f_3(x_0,\lambda_0,s_0)\}$ is bounded for a family of LPs with bounded optimal sets.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.1: for any fixed $q>2$, $(x^*,\lambda^*,s^*)$ is optimal for the primal-dual LP pair if and only if it minimizes $f_q$ defined in (MF) and $f_q(x^*,\lambda^*,s^*)=0$, where $f_q$ sums half the squared duality gap $(c^Tx-b^T\lambda)^2$, half the squared primal and dual feasibility violations $\|Ax-b\|_2^2$ and $\|A^T\lambda+s-c\|_2^2$, and order-$q$ penalty terms $\max\{-x_j,0\}^q + \max\{-s_j,0\}^q$ scaled by $1/(q(q-1))$. Since $f_q\ge0$, the LP optimal set is exactly the zero-level set of $f_q$, making (MF) a zero-residual problem. Theorem 3.2 then states that Algorithm 1, i.e. Newton's method with Levenberg-Marquardt regularization using $\mu_k=\sqrt{\|\nabla f(x_k,\lambda_k,s_k)\|/2}$ and $\alpha_k=1$, drives the residuals $\|c^Tx_k-b^T\lambda_k\|$, $\|Ax_k-b\|$, $\|A^T\lambda_k+s_k-c\|$ and the violations $x_k,s_k\ge-\epsilon$ all below $\epsilon$ within $O(\epsilon^{-3/2})$ iterations, under the assumption that the primal-dual optimal set is nonempty and bounded. The paper also introduces a modified merit function $h_{q,\nu}$ whose Hessian is nonsingular for every $\nu>0$ and which equals $f_q$ at $\nu=0$; Algorithm 2 decreases $\nu$ geometrically while taking Newton steps and is offered as a heuristic that performed faster in the reported experiments, especially in higher dimensions.

Load-bearing premise

The proof of the fast global rate assumes that the set of all points where the merit function is no higher than at the starting point is bounded, and it takes as given that boundedness of the LP optimal set guarantees this; that implication is asserted but not demonstrated, so if it fails the $O(\epsilon^{-3/2})$ bound has no support.

Editorial extensions

If this is right

  • Any linear program with a nonempty bounded optimal set can be solved by driving one smooth unconstrained merit function to zero, starting from any point in $\mathbb{R}^{2n+m}$.
  • The value of $\min f_q$ acts as a certificate: if the computed minimum stays strictly above zero, the primal-dual pair has no optimal solution (it is infeasible or unbounded), and the simplified homogeneous formulation can then identify which case holds.
  • The iteration complexity $O(\epsilon^{-3/2})$ does not depend on the problem dimension, and with the factored linear-system solve each iteration costs $O(m^3+n^3)$ rather than $O((2n+m)^3)$.
  • For $q=3$ the Hessian is 1-Lipschitz, giving a concrete regularization choice $\mu_k=\sqrt{\|\nabla f_k\|/2}$ that satisfies the conditions of the convergence theorem.

Reading between the lines

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

  • The same template could be applied to other KKT systems: replacing complementarity by a squared bilinear term such as $(x^T\lambda)^2$ yields an exact unconstrained reformulation, though the resulting merit function is nonconvex for nonlinear programs.
  • Adaptive schedules for $\nu$ rather than the fixed geometric factor $\theta$ might remove the stalling observed for large-dimensional problems; the paper lists adaptive $\nu$ updates as future work.
  • The zero-residual structure suggests that local error bound conditions, if they hold for $f_q$, would upgrade the observed fast practical convergence into a proof of local quadratic convergence, an avenue the paper mentions but does not carry out.
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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 proposes an exact unconstrained reformulation of a primal-dual linear program in standard form, using a convex twice continuously differentiable merit function f_q (Eq. (MF)) that combines a squared duality gap, squared constraint residuals, and smoothed penalties for nonnegativity. Theorem 2.1 establishes that, when the LP has an optimal solution, the minimizers of f_q coincide with the LP optimal solutions and f_q attains zero there; Corollary 2.3 characterizes infeasible or unbounded cases by min f_q > 0. For q = 3 the Hessian is shown to be 1-Lipschitz (Lemma 2.7) but singular at optimal solutions. The main theoretical result (Theorem 3.2) applies Mishchenko's regularized Newton theorem to obtain an O(epsilon^{-3/2}) global iteration bound for reaching approximate primal-dual optimality conditions under the assumption that the LP optimal set is bounded. The paper further introduces a modified merit function h_{q,nu} (Eq. (HMF)) and a homotopy heuristic (Algorithm 2), supported by numerical experiments on random LPs.

Significance. The reformulation is elegant, and the main convergence theorem is a plausible and nontrivial corollary of existing regularized Newton theory, giving a smooth exact unconstrained formulation of LP with a provable global rate for a second-order method. The paper is careful in stating the zero-residual structure and provides a reproducible numerical study with available code. If the missing level-set lemma is supplied and the HMF Hessian claim is corrected, the contribution would be a useful addition to the literature on penalty and smoothing reformulations of LP. The worst-case rate O(epsilon^{-3/2}) is nonstandard and worth highlighting; the numerical experiments honestly show that the adaptive-mu variant is slow while the heuristic variants are faster, which is a useful empirical observation.

major comments (2)
  1. [Section 3.2, proof of Theorem 3.2] The proof asserts without proof that the boundedness of the LP optimal set implies boundedness of the initial sublevel set {f_3 <= f_3(x0, lambda0, s0)} and cites [39]. This implication is load-bearing because it is exactly Theorem 3.1's Assumption 2, and the manuscript does not state or prove the underlying lemma. Please add an explicit lemma: if f is convex and its argmin is nonempty and bounded, then every sublevel set {f <= a} is bounded; provide the proof (e.g., by showing that an unbounded sublevel set contains a nonzero recession direction along which convexity forces f to be constant, contradicting boundedness of the argmin). This will make the proof of Theorem 3.2 self-contained and verify the applicability of [39].
  2. [Section 4.2, Eq. (HMF)] The claim that the Hessian of h_{q,nu} is positive definite for all (x, lambda, s) when nu > 0 is false. At x = s = 0, the added terms nu ||lambda||^2 and nu/(q(q-1)) sum(max{x_j,0}^q + max{s_j,0}^q) contribute zero curvature in the x and s blocks. For example, with n = 2, m = 1, A = [1 1], c = [1 1]^T, b = 1, the direction d = (1, -1, 0, 0) at (0,0,0) satisfies A d = 0 and c^T d = 0, so d^T nabla^2 h_{q,nu}(0,0,0) d = 0. Thus the Hessian is singular for every nu > 0. This affects the abstract's claim of nonsingular Hessians and the motivation for Algorithm 2. Please correct the claim (for example, by adding a strong convexity term involving x and s, or by stating the actual positive semidefiniteness and singularity properties) and adjust the discussion of Algorithm 2 accordingly.
minor comments (4)
  1. [Section 3.2, Theorem 3.2] The proof uses the same symbol epsilon for the tolerance on f_3 (3.7) and for the tolerances on the optimality constraints (3.2)-(3.6). The iteration counts O(epsilon^{-1}) and O(epsilon^{-3/2}) are derived correctly, but the statement would be clearer if the target accuracy in the merit function were written as a separate parameter, say delta = O(epsilon^2) or O(epsilon^3), to avoid the appearance that O(epsilon^{-1/2}) directly yields all bounds.
  2. [Table 1 and Section 4.3] Algorithms 1b and 2 use q = 2.1, but Lemma 2.7 and Theorem 3.2 are stated only for q = 3. The text should state explicitly that the experiments with q = 2.1 are heuristic and not covered by the theoretical rate.
  3. [Section 4.4] The conclusion that the problem is unbounded from the observed behavior (gradient approaching zero, f_q bounded away from zero, and ||Ax_k - b|| approaching zero) is a heuristic inference from the limit behavior of the iterates; a rigorous certificate would require the homogeneous formulation described in Section 2.1. Please soften the wording or add a remark that this is an empirical observation.
  4. [Section 2.2, proof of Lemma 2.7] The display in the proof of Lemma 2.7 has a small typographical error with double parentheses and a missing closing brace; please clean up the typesetting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity is found; Theorem 3.2 rests on an external convergence theorem and no fitted or self-cited inputs.

full rationale

The central equivalence in Theorem 2.1 is built directly into the definition of the merit function fq in (MF), which sums squares of the residual expressions from (OC); calling this an equivalence is the intended reformulation, not a hidden circular derivation. The main convergence result, Theorem 3.2, is obtained by importing Mishchenko's Theorem 3.1 as an external result, verifying the L-Lipschitz Hessian in Lemma 2.7, and using boundedness of the LP optimal set to obtain bounded sublevel sets; the bounded-sublevel-set inference is cited to [39] and does not depend on the authors' own work. No parameter is fitted and then relabeled a prediction: Algorithm 1's only free design choice is the standard Levenberg-Marquardt regularization, and Algorithm 2 is explicitly described as heuristic, with its theta acknowledged in Section 6 as tuned by trial and error. There are no self-citations at all in the reference list, so no load-bearing self-citation chain exists. The separate mathematical error in Section 4.2 concerning positive definiteness of the Hessian of h_{q,nu} is a correctness issue, not circularity, and does not enter the proof of Theorem 3.2.

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

No new physical or ontological entities are introduced; fq and h_{q,nu} are new mathematical constructions, not entities requiring independent empirical evidence. The main proof rests on external theorems, and the two listed free parameters are the merit exponent and the heuristic homotopy schedule.

free parameters (2)
  • Merit exponent q = 3 in theory, 2.1 in experiments
    Chosen by hand; q > 2 is needed for twice differentiability, q = 3 gives a 1-Lipschitz Hessian used in Theorem 3.2, while the numerical tests switch to q = 2.1.
  • Homotopy schedule nu0, theta = theta = 0.8, nu0 not stated in text
    Tuned by trial and error as admitted in Section 6 for Algorithm 2. The faster observed convergence is not covered by a theorem and depends on this schedule.
assumptions (6)
  • standard math Optimality conditions (OC) are necessary and sufficient for optimality of a primal-dual LP triple.
    Used to define (MF) and to prove Theorem 2.1.
  • standard math max{-x,0}^q is convex and twice continuously differentiable for q > 2.
    Required for convexity and smoothness of fq; cited to [8].
  • standard math For q = 3, the Hessian of fq is 1-Lipschitz.
    Proved in Lemma 2.7 and needed to apply Mishchenko's theorem with L = 1.
  • standard math Mishchenko's Theorem 3.1 gives O(epsilon^{-1/2}) global convergence for regularized Newton under Lipschitz Hessian and bounded level set.
    Imported from [32] and used as the engine of Theorem 3.2.
  • domain assumption Boundedness of the LP optimal set implies boundedness of every sublevel set of f3.
    Invoked in the proof of Theorem 3.2 with citation [39], but not derived in the paper for this specific merit function.
  • standard math The simplified homogeneous model has a strictly complementary solution whose tau/kappa certificates identify primal and dual infeasibility.
    Used in Section 2.1 to detect infeasibility; cited to [1, 47, 49].

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Pith. "Pith review of Solving a linear program via a single unconstrained minimization." pith.science (2026). https://pith.science/paper/PTBFKEUU

@misc{pith2026250521232,
  author       = {Pith},
  title        = {Pith review of: Solving a linear program via a single unconstrained minimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTBFKEUU}},
  note         = {Machine review of arXiv:2505.21232}
}
abstract

This paper proposes a novel approach for solving linear programs. We reformulate a primal-dual linear program as an unconstrained minimization of a convex and twice continuously differentiable merit function. When the optimal set of the primal-dual pair is nonempty, its optimal set is equal to the optimal set of the proposed merit function. Minimizing this merit function poses some challenges due to its Hessian being singular at some points in the domain, including the optimal solutions. We handle singular Hessians using the Newton method with Levenberg-Marquardt regularization. We show that the Newton method with Levenberg-Marquardt regularization yields global convergence to a solution of the primal-dual linear program in at most $O(\epsilon^{-3/2})$ iterations requiring only the assumption that the optimal set of the primal-dual linear program is bounded. Testing on random synthetic problems demonstrates convergence to optimal solutions to very high accuracy significantly faster than the derived worst-case bound. We further introduce a modified merit function that depends on a scalar parameter $\nu > 0$, whose Hessian is nonsingular for all $\nu > 0$ and which reduces exactly to the original merit function when $\nu = 0$. Based on this formulation, we propose a heuristic scheme that performs Newton steps while gradually decreasing $\nu$ toward zero. Numerical experiments indicate that this approach achieves faster convergence, particularly on higher-dimensional problems.

Figures

Figures reproduced from arXiv: 2505.21232 by the authors.

Figure 1
Figure 1. Left: A contour plot of (4.1). Right: Illustration of max [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Left: A contour plot of (4.3). Right: Illustration of max [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. This figure shows results for Algorithm 1a (leftmost plots), Algorithm 1b (middle [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: This figure shows results for Algorithm 1a (leftmost plot), Algorithm 1b (middle plot) [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

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Works this paper leans on

52 extracted references · 51 canonical work pages

  1. [39]

    Roman A. Polyak. Regularized newton method for unconstrained convex optimization. Mathematical Programming, 120(1):125–145, 2009

    work page 2009

  2. [1]

    Andersen and Knud D

    Erling D. Andersen and Knud D. Andersen. The Mosek Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm , pages 197–232. Springer US, Boston, MA, 2000

    work page 2000

  3. [2]

    Optimization iii: Convex optimization, 2023

    Aharon Ben-Tal and Arkadi Nemirovski. Optimization iii: Convex optimization, 2023. Lecture notes, Georgia Institute of Technology

    work page 2023

  4. [3]

    Bertsekas

    Dimitri P. Bertsekas. Necessary and sufficient conditions for a penalty method to be exact. Mathematical Programming, 9(1):87–99, December 1975

    work page 1975

  5. [4]

    Convex Optimization

    Stephen Boyd and Lieven Vandenberghe. Convex Optimization . Cambridge University Press, 2004

    2004

  6. [5]

    Cartis, N

    C. Cartis, N. I. M. Gould, and Ph. L. Toint. On the complexity of steepest descent, newton’s and regularized newton’s methods for nonconvex unconstrained optimization problems. SIAM Journal on Optimization , 20(6):2833–2852, 2010

    work page 2010

  7. [6]

    Coralia Cartis, Nicholas I. M. Gould, and Philippe L. Toint. Adaptive cubic regularisation methods for unconstrained optimization. part i: motivation, convergence and numerical results. Mathematical Programming, 127(2):245–295, 2011

    work page 2011

  8. [7]

    Coralia Cartis, Nicholas I. M. Gould, and Philippe L. Toint. Adaptive cubic regularisa- tion methods for unconstrained optimization. part ii: worst-case function- and derivative- evaluation complexity. Mathematical Programming, 130(2):295–319, 2011

    work page 2011

Show all 52 references
  1. [8]

    Edwin K. P. Chong and Stanislaw H. Zak. An Introduction to Optimization . Wiley- Interscience, New York, 2 edition, 2001

    2001

  2. [9]

    Faster randomized interior point methods for tall/wide linear programs.Journal of Machine Learning Research, 23:336:1–336:48, 2022

    Agniva Chowdhury, Gregory Dexter, Palma London, Haim Avron, and Petros Drineas. Faster randomized interior point methods for tall/wide linear programs.Journal of Machine Learning Research, 23:336:1–336:48, 2022

    2022

  3. [10]

    Conn, Nicholas I

    Andrew R. Conn, Nicholas I. M. Gould, and Philippe L. Toint. Trust Region Methods. MPS- SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadel- phia, PA, 2000

    2000

  4. [11]

    Convergence properties of the inexact levenberg-marquardt method under local error bound conditions

    Hiroshige Dan, Nobuo Yamashita, and Masao Fukushima. Convergence properties of the inexact levenberg-marquardt method under local error bound conditions. Optimization Methods and Software , 17(4):605–626, 2002

    2002

  5. [12]

    George B. Dantzig. Application of the simplex method to a transportation problem. Activity Analysis of Production and Allocation , pages 359–373, 1951

    1951

  6. [13]

    George B. Dantzig. A proof of the equivalence of the programming problem and the game problem. Activity Analysis of Production and Allocation , pages 330–335, 1951

    1951

  7. [14]

    Dolgopolik

    M.V. Dolgopolik. A unifying theory of exactness of linear penalty functions. Optimization, 65(6):1167–1202, 2016

    2016

  8. [15]

    Y. G. Evtushenko, A. I. Golikov, and N. Mollaverdy. Augmented lagrangian method for large-scale linear programming problems. Optimization Methods and Software, 20(4–5):515– 524, 2005. 20

    2005

  9. [16]

    A regularized newton method for monotone nonlinear equations and its application

    Jinyan Fan and Yaxiang Yuan. A regularized newton method for monotone nonlinear equations and its application. Optimization Methods and Software , 29(1):102–119, 2014

    2014

  10. [17]

    coin-or/clp: Release releases/1.17.10, August 2024

    John Forrest, Stefan Vigerske, Ted Ralphs, John Forrest, Lou Hafer, jpfasano, Haroldo Gambini Santos, Jan-Willem, Matthew Saltzman, a andre, Bjarni Kristjansson, h-i gassmann, Alan King, Arevall, Bohdan Mart, Pierre Bonami, Ruan Luies, Samuel Brito, and to st. coin-or/clp: Rel...

    2024

  11. [18]

    I. L. Galabova and J. A. J. Hall. The ‘idiot’ crash quadratic penalty algorithm for lin- ear programming and its application to linearizations of quadratic assignment problems. Optimization Methods and Software , 35(3):488–501, 2020

    2020

  12. [19]

    P. E. Gill, G. H. Golub, W. Murray, and M. A. Saunders. Methods for modifying matrix factorizations. Mathematics of Computation , 28(126):505–535, 1974

    1974

  13. [20]

    O. G¨ uler. Augmented lagrangian algorithms for linear programming. Journal of Optimiza- tion Theory and Applications , 75(3):445–470, 1992

    1992

  14. [21]

    Stochastic subspace cubic newton method

    Filip Hanzely, Nikita Doikov, Peter Richt´ arik, and Yurii Nesterov. Stochastic subspace cubic newton method. In Proceedings of the 37th International Conference on Machine Learning, ICML’20. JMLR.org, 2020

    2020

  15. [22]

    Hillier and Gerald J

    Frederick S. Hillier and Gerald J. Lieberman. Introduction to Operations Research. McGraw- Hill Education, 10 edition, 2014

    2014

  16. [23]

    Qi Huangfu, Irina Galabova, Matthias Feldmeier, and Julian A. J. Hall. HiGHS – high performance software for linear optimization. INFORMS Journal on Computing , 35(1):1– 20, 2023

    2023

  17. [24]

    A new polynomial-time algorithm for linear programming

    Narendra Karmarkar. A new polynomial-time algorithm for linear programming. Combi- natorica, 4(4):373–395, 1984

    1984

  18. [25]

    L. G. Khachiyan. A polynomial algorithm in linear programming. Soviet Mathematics Doklady, 20:191–194, 1979. Originally published in Russian in Doklady Akademii Nauk SSSR, 244 (1979), 1093–1096

    1979

  19. [26]

    Victor Klee and George J. Minty. How good is the simplex algorithm? Inequalities III , pages 159–175, 1972

    1972

  20. [27]

    Levenberg

    K. Levenberg. A method for the solution of certain nonlinear problems in least squares. Quarterly of Applied Mathematics , 2:164–168, 1944

    1944

  21. [28]

    Regularized newton methods for convex minimization problems with singular solutions

    Dong-Hui Li, Masao Fukushima, Liqun Qi, and Nobuo Yamashita. Regularized newton methods for convex minimization problems with singular solutions. Computational Opti- mization and Applications , 28(2):131–147, 2004

    2004

  22. [29]

    Truncated regularized newton method for convex minimiza- tions

    Ying-Jie Li and Dong-Hui Li. Truncated regularized newton method for convex minimiza- tions. Computational Optimization and Applications , 43(1):119–131, 2009

    2009

  23. [30]

    O. L. Mangasarian. A newton method for linear programming. Journal of Optimization Theory and Applications , 121(1):1–18, 2004

    2004

  24. [31]

    Marquardt

    D. Marquardt. An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics , 11:431–441, 1963. 21

    1963

  25. [32]

    Regularized newton method with global ≀(1/k2) convergence

    Konstantin Mishchenko. Regularized newton method with global ≀(1/k2) convergence. SIAM Journal on Optimization , 33(3):1440–1462, 2023

    2023

  26. [33]

    Mor´ e, D

    Jorge J. Mor´ e, D. C. Sorensen, and G. H. Golub. Newton’s method. In Studies in Nu- merical Analysis, volume 24 of MAA Studies in Mathematics , pages 29–82. Mathematical Association of America, Washington, DC, 1984

    1984

  27. [34]

    J. L. Nazareth. Homotopy techniques in linear programming. Algorithmica, 1(1):529–535, 1986

    1986

  28. [35]

    J. L. Nazareth. The homotopy principle and algorithms for linear programming. SIAM Journal on Optimization , 1(3):316–332, 1991

    1991

  29. [36]

    Interior-Point Polynomial Algorithms in Convex Programming

    Yurii Nesterov and Arkadii Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994

    1994

  30. [37]

    Yurii Nesterov and B. T. Polyak. Cubic regularization of newton method and its global performance. Mathematical Programming, 108(1):177–205, 2006

    2006

  31. [38]

    Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer Series in Opera- tions Research and Financial Engineering. Springer New York, NY, 2 edition, 2006. eBook ISBN: 978-0-387-40065-5; Softcover ISBN: 978-1-4939-3711-0

    2006

  32. [40]

    H. H. Rosenbrock. An automatic method for finding the greatest or least value of a function. The Computer Journal , 3(3):175–184, 01 1960

    1960

  33. [41]

    Convergence properties of the regularized newton method for the unconstrained nonconvex optimization

    Kenji Ueda and Nobuo Yamashita. Convergence properties of the regularized newton method for the unconstrained nonconvex optimization. Applied Mathematics and Opti- mization, 62(1):27–46, 2010

    2010

  34. [42]

    A regularized newton method without line search for unconstrained optimization

    Kenji Ueda and Nobuo Yamashita. A regularized newton method without line search for unconstrained optimization. Computational Optimization and Applications , 59(1):321–351, 2014

    2014

  35. [43]

    Vanderbei

    Robert J. Vanderbei. Linear Programming: Foundations and Extensions . International Series in Operations Research & Management Science. Springer, 5 edition, 2020

    2020

  36. [44]

    Random projections for linear programming

    Ky Vu, Pierre-Louis Poirion, and Leo Liberti. Random projections for linear programming. Mathematics of Operations Research, 43(4):1051–1071, 2018

    2018

  37. [45]

    A new newton method for convex optimization problems with singular hessian matrices

    Tianji Wang and Qingdao Huang. A new newton method for convex optimization problems with singular hessian matrices. AIMS Mathematics , 8(9):21161–21175, 2023

    2023

  38. [46]

    Laurence A. Wolsey. Integer Programming. John Wiley & Sons, 2 edition, 2020

    2020

  39. [47]

    Stephen J. Wright. Primal-dual interior-point methods. Society for Industrial and Applied Mathematics, USA, 1997

    1997

  40. [48]

    A deep learning approach for solving linear programming problems

    Dawen Wu and Abdel Lisser. A deep learning approach for solving linear programming problems. Neurocomputing, 520:15–24, 2023. 22

    2023

  41. [49]

    A simplified homogeneous and self-dual linear programming algorithm and its implementation

    Xiaojie Xu, Pi-Fang Hung, and Yinyu Ye. A simplified homogeneous and self-dual linear programming algorithm and its implementation. Annals of Operations Research, 62(1):151– 171, 1996

    1996

  42. [50]

    Yamashita and M

    N. Yamashita and M. Fukushima. On the rate of convergence of the levenberg-marquardt method. In Goetz Alefeld and Xiaojun Chen, editors, Topics in Numerical Analysis , pages 239–249, Vienna, 2001. Springer Vienna

    2001

  43. [51]

    Todd, and Shinji Mizuno

    Yinyu Ye, Michael J. Todd, and Shinji Mizuno. An o(√nl)-iteration homogeneous and self-dual linear programming algorithm. Mathematics of Operations Research, 19(1):53–67, 1994

    1994

  44. [52]

    On the quadratic convergence of the cubic regularization method under a local error bound condition

    Man-Chung Yue, Zirui Zhou, and Anthony Man-Cho So. On the quadratic convergence of the cubic regularization method under a local error bound condition. SIAM Journal on Optimization, 29(1):904–932, 2019. 23

    2019

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.