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2026
An objective-function-free algorithm for general smooth constrained optimization
Bellavia, S., Gratton, S., Morini, B. & Toint, P., 2 Feb 2026, Arxiv.Research output: Working paper › Preprint
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A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad, Shampoo and Muon
Gratton, S. & Toint, P., 17 Apr 2026, Arxiv.Research output: Working paper › Preprint
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Iteration complexity of the Difference-of-Convex Algorithm for unconstrained optimization: a simple proof
Gratton, S. & Toint, P., 22 Jan 2026, Arxiv.Research output: Working paper › Preprint
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2025
A Fast Newton Method Under Local Lipschitz Smoothness
Gratton, S., Jerad, S. & Toint, P., 2025, Arxiv.Research output: Working paper
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A Simple First-Order Algorithm for Full-Rank Equality Constrained Optimization
Gratton, S. & Toint, P., Oct 2025, 2510-16390: Arxiv.Research output: Working paper
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A Stochastic Objective-Function-Free Adaptive Regularization Method with Optimal Complexity
Gratton, S., Jerad, S. & Toint, P., Mar 2025, In: Open Journal of Mathematical Optimization. 6, 5, 32 p., 5.Research output: Contribution to journal › Article › peer-review
Open AccessFile36 Downloads (Pure) -
Complexity and performance for two classes of noise-tolerant first-order algorithms
Gratton, S., Jerad, S. & Toint, P., 2025, (Accepted/In press) In: Optimization Methods and Software.Research output: Contribution to journal › Article › peer-review
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Fast Stochastic Second-Order Adagrad for Nonconvex Bound-Constrained Optimization
Bellavia, S., Gratton, S., Morini, B. & Toint, P., 2025, Arxiv.Research output: Working paper
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2024
Complexity of Adagrad and other first-order methods for nonconvex optimization problems with bounds constraints
Toint, P., 10 Jul 2024, Arxiv.Research output: Working paper
Open AccessFile34 Downloads (Pure) -
2023
Divergence of the ADAM algorithm with fixed-stepsize: a (very) simple example
TOINT, P., Aug 2023, Arxiv, 3 p.Research output: Working paper › Discussion paper
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OFFO minimization algorithms for second-order optimality and their complexity
Gratton, S. & TOINT, P., 15 Feb 2023, In: Computational Optimization and Applications. 84, 2, p. 573-607 35 p.Research output: Contribution to journal › Article › peer-review
File42 Downloads (Pure) -
The Impact of Noise on Evaluation Complexity: The Deterministic Trust-Region Case
Bellavia, S., Gurioli, G., Morini, B. & TOINT, P., Feb 2023, In: Journal of Optimization Theory and Applications. 196, 2, p. 700-729 30 p.Research output: Contribution to journal › Article › peer-review
Open Access -
2022
Evaluation complexity of algorithms for nonconvex optimization
Cartis, C., Gould, N. I. M. & TOINT, P., Jul 2022, SIAM. 600 p. (SIAM-MOS Series on Optimization)Research output: Book/Report/Journal › Book
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First-Order Objective-Function-Free Optimization Algorithms and Their Complexity
Gratton, S., Jerad, S. & TOINT, P., 7 Mar 2022, Arxiv.Research output: Working paper
File81 Downloads (Pure) -
2021
High-Order Evaluation Complexity for Convexly-Constrained Optimization with Non-Lipschitzian Group Sparsity Terms
Chen, X. & Toint, P., May 2021, In: Mathematical Programming. 187, 1-2, p. 47-78 32 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile60 Downloads (Pure) -
2020
A concise second-order complexity analysis for unconstrained optimization using high-order regularized models
Cartis, C., Gould, N. I. M. & Toint, P. L., 3 Mar 2020, In: Optimization Methods and Software. 35, 2, p. 243-256 14 p.Research output: Contribution to journal › Article › peer-review
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Adaptive regularization algorithms with inexact evaluations for nonconvex optimization
Bellavia, S., Gurioli, G., Morini, B. & Toint, P., 2 Jan 2020, In: SIAM Journal on Optimization. 29, 4, p. 2881-2915 35 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile84 Downloads (Pure) -
An algorithm for the minimization of nonsmooth nonconvex functions using inexact evaluations and its worst-case complexity
Gratton, S., Simon, E. & Toint, P., 21 Jan 2020, In: Mathematical Programming. 187, 1-2, p. 1-24 19 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile88 Downloads (Pure) -
A stochastic cubic regularisation method with inexact function evaluations and random derivatives for finite sum minimisation
Bellavia, S., Gurioli, G., Morini, B. & Toint, P. L., 2020, (Accepted/In press).Research output: Contribution to conference › Paper › peer-review
Open AccessFile124 Downloads (Pure) -
Sharp worst-case evaluation complexity bounds for arbitrary-order nonconvex optimization with inexpensive constraints
Toint, P., Cartis, C. & Gould, N. I. M., Jan 2020, In: SIAM Journal on Optimization. 30, 1, p. 513-541 29 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile98 Downloads (Pure) -
Strong Evaluation Complexity Bounds for Arbitrary-Order Optimization of Nonconvex Nonsmooth Composite Functions
Cartis, C., Gould, N. & Toint, P., 30 Jan 2020, Arxiv.Research output: Working paper
File81 Downloads (Pure) -
2019
Complexity of partially separable convexly constrained optimization with non-Lipschitzian singularities
Chen, X., Toint, P. & Wang, H., 15 Apr 2019, In: SIAM Journal on Optimization. 29, 1, p. 874-903 30 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile65 Downloads (Pure) -
Evaluation complexity bounds for smooth constrained nonlinear optimization using scaled KKT conditions and high-order models
Cartis, C., Gould, N. I. M. & Toint, P., Jun 2019, Springer Optimization and Its Applications: Algorithms, Complexity and Applications. Demetriou, I. & Pardalos, P. (eds.). Springer Heidelberg, p. 5-26 22 p. (Springer Optimization and Its Applications; vol. 145).Research output: Contribution in Book/Catalog/Report/Conference proceeding › Chapter
Open Access -
Optimality of orders one to three and beyond: Characterization and evaluation complexity in constrained nonconvex optimization
Cartis, C., Gould, N. I. M. & Toint, P., 10 Aug 2019, In: Journal of Complexity. 53, p. 68-94 27 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile67 Downloads (Pure) -
2018
Second-order optimality and beyond: Characterization and Evaluation Complexity in Convexly Constrained Nonlinear Optimization
Cartis, C., Gould, N. I. M. & Toint, P., 1 Oct 2018, In: Foundations of Computational Mathematics. 18, 5, p. 1073-1107 35 p.Research output: Contribution to journal › Article › peer-review
Open Access -
Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization
Cartis, C., Gould, N. I. M. & Toint, P., 1 Jan 2018, Invited Lectures. Sirakov, B., de Souza, P. N. & Viana, M. (eds.). World Scientific Publishing Co Pte Ltd, p. 3729-3768 40 p. (Proceedings of the International Congress of Mathematicians, ICM 2018; vol. 4).Research output: Contribution in Book/Catalog/Report/Conference proceeding › Conference contribution
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2017
Worst-case evaluation complexity of regularization methods for smooth unconstrained optimization using Hölder continuous gradients
Toint, P., 2 Nov 2017, In: Optimization Methods and Software. 32, 6, p. 1273-1298 26 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile55 Downloads (Pure) -
2016
Evaluation complexity for nonlinear constrained optimization using unscaled KKT conditions and high-order models
Birgin, E., Gardenghi, J., Martinez, J.-M., Santos, S. & Toint, P., 2016, In: SIAM Journal on Optimization. 26, 2, 20 p.Research output: Contribution to journal › Article › peer-review
File72 Downloads (Pure) -
2015
Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models
Cartis, C., Gould, N. I. M. & Toint, P., 18 Nov 2015, 18 p. (naXys technical report).Research output: Working paper
File91 Downloads (Pure) -
On the evaluation complexity of constrained nonlinear least-squares and general constrained nonlinear optimization using second-order methods
Cartis, C., Gould, N. I. M. & Toint, P. L., 2015, In: SIAM Journal on Numerical Analysis. 53, 2, p. 836-851 16 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile211 Downloads (Pure) -
Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models
Birgin, E., Gardenghi, J., Martinez, J.-M., Santos, S. A. & Toint, P., Jun 2015, 2015: Namur center for complex systems, 8 p. (naXys Technical Reports; vol. 05-2015).Research output: Working paper
File107 Downloads (Pure) -
Worst-case evaluation complexity of non-monotone gradient-related algorithms for unconstrained optimization
Cartis, C., Rodrigues Sampaio, P. & Toint, P. L., 4 May 2015, In: Optimization. 64, 5, p. 1349-1361 13 p.Research output: Contribution to journal › Article › peer-review
File164 Downloads (Pure) -
2014
On the complexity of finding first-order critical points in constrained nonlinear optimization
Cartis, C., Gould, N. I. M. & Toint, P. L., 2014, In: Mathematical Programming. 144, 1-2, p. 93-106 14 p.Research output: Contribution to journal › Article › peer-review
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2013
An example of slow convergence for Newton's method on a function with globally Lipschitz continuous Hessian
Cartis, C., Gould, N. I. M. & Toint, P., 5 May 2013, Namur center for complex systems, 9 p.Research output: Working paper
File137 Downloads (Pure) -
A note about the complexity of minimizing Nesterov's smooth Chebyshev-Rosenbrock function
Cartis, C., Gould, N. I. M. & Toint, P. L., 1 Jun 2013, In: Optimization Methods and Software. 28, 3, p. 451-457 7 p.Research output: Contribution to journal › Article › peer-review
File60 Downloads (Pure) -
On the evaluation complexity of cubic regularization methods for potentially rank-deficient nonlinear least-squares problems and its relevance to constrained nonlinear optimization
Cartis, C., Gould, N. & Toint, P., 29 Oct 2013, In: SIAM Journal on Optimization. 23, 3, p. 1553-1574 22 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile191 Downloads (Pure) -
2012
An adaptive cubic regularization algorithm for nonconvex optimization with convex constraints and its function-evaluation complexity
Cartis, C., Gould, N. & Toint, P., 1 Oct 2012, In: IMA Journal of Numerical Analysis. 32, 4, p. 1662-1695 34 p.Research output: Contribution to journal › Article › peer-review
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Complexity bounds for second-order optimality in unconstrained optimization
Cartis, C., Gould, N. I. M. & Toint, P., 1 Feb 2012, In: Journal of Complexity. 28, p. 93-108 16 p.Research output: Contribution to journal › Article › peer-review
File61 Downloads (Pure) -
Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization
Toint, P., Cartis, C. & Gould, N., 1 Apr 2012, In: Optimization Methods and Software. 27, 2, p. 197-219 23 p.Research output: Contribution to journal › Article › peer-review
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On the complexity of the steepest-descent with exact linesearches
Cartis, C., Gould, N. I. M. & Toint, P., 2012, FUNDP. Namur center for complex systems. (NAXYS Technical Report; vol. 16-2012)Research output: Book/Report/Journal › Other report
Open AccessFile121 Downloads (Pure) -
On the oracle complexity of first-order and derivative-free algorithms for smooth nonconvex minimization
Cartis, C., Gould, N. I. M. & Toint, P., 1 Jan 2012, In: SIAM Journal on Optimization. 22, 1, p. 66-86 21 p.Research output: Contribution to journal › Article › peer-review
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Updating the regularization parameter in the adaptive cubic regularization algorithm
Gould, N., Porcelli, M. & Toint, P., 1 Sept 2012, In: Computational Optimization and Applications. 53, 1, p. 1-22 22 p.Research output: Contribution to journal › Article › peer-review
File173 Downloads (Pure) -
2011
Adaptive cubic regularisation methods for unconstrained optimization. Part II: Worst-case function- and derivative-evaluation complexity
Cartis, C., Gould, N. & Toint, P., 1 Dec 2011, In: Mathematical Programming. 130, 2, p. 295-319 25 p.Research output: Contribution to journal › Article › peer-review
File260 Downloads (Pure) -
On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming
Cartis, C., Gould, N. & Toint, P., 1 Jan 2011, In: SIAM Journal on Optimization. 21, 4, p. 1721-1739 19 p.Research output: Contribution to journal › Article › peer-review
File170 Downloads (Pure) -
Optimal Newton-type methods for nonconvex smooth optimization problems
Cartis, C., Gould, N. & Toint, P., 2011, Namur: Namur center for complex systems.Research output: Book/Report/Journal › Other report
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2010
On the complexity of steepest descent, Newton's and regularized Newton's methods for nonconvex unconstrained optimization problems
Cartis, C., Gould, N. & Toint, P., 1 Jan 2010, In: SIAM Journal on Optimization. 20, 6, p. 2833-2852 20 p.Research output: Contribution to journal › Article › peer-review
Open AccessFile212 Downloads (Pure) -
2009
Trust-region and other regularisations of linear least-squares problems
Cartis, C., Gould, N. & Toint, P., 1 Mar 2009, In: BIT Numerical Mathematics. 49, 1, p. 21-53 33 p.Research output: Contribution to journal › Article › peer-review
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