Papers

Google scholar
ResearchGate 

Preprints

    1. Hong T.M. Chu, Meixia Lin, K.C. Toh, Wasserstein distributionally robust optimization and its tractable regularization formulations, arXiv:2402.03942
    2. D. Hou, L. Liang, and K.C. Toh, A sparse smoothing Newton method for solving discrete optimal transport problems, arXiv:2311.06448
    3. K.Y. Ding, N.C. Xiao, and K.C. Toh, Adam-family methods with decoupled weight decay in deep learning, arXiv:2310.08858
    4. W.J. Li, W. Bian, K.C. Toh, On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models, arXiv:2308.16690
    5. S.L. Hu, D.F. Sun, and K.C. Toh, Quantifying low rank approximations of third order symmetric tensors, arXiv:2307.10855
    6. N.C. Xiao, X.Y. Hu, and K.C. Toh, Convergence guarantees for stochastic subgradient methods in nonsmooth nonconvex optimization, arXiv:2307.10053
    7. Y.C. Yuan, M.X. Lin, D.F. Sun, and K.C. Toh, Adaptive sieving: A dimension reduction technique for sparse optimization problems, arXiv:2306:17369.
    8. K.Y. Ding, J.Y. Li, and K.C. Toh, Nonconvex stochastic Bregman proximal gradient method with application to deep learning, arXiv:2306.14522 
    9. N.C. Xiao, X. Liu, and K.C. Toh, A partial exact penalty function approach for constrained optimization, arXiv:2304.01467
    10. L. Liang, D.F. Sun, and K.C. Toh, A squared smoothing Newton method for semidefinite programming, arXiv:2303.05825
    11. N.C. Xiao, X.Y. Hu, X. Liu, and K.C. Toh, CDOpt: A python package for a class of Riemannian optimization, arXiv:2212.02698
    12. N. Hoang Anh Mai, V. Magron, J.B. Lasserre, K.C. Toh, Tractable hierarchies of convex relaxations for polynomial optimization on the nonnegative orthant, arXiv:2209.06175 
    13. C.P. Lee, L. Liang, T.Y. Tang, and K.C. Toh, Accelerating nuclear-norm regularized low-rank matrix optimization through Burer-Monteiro decomposition, arXiv:2204.14067
      The above is updated from the version under the title “Escaping Spurious Local Minima of Low-Rank Matrix Factorization Through Convex Lifting”
    14. L. Yang, and K.C. Toh, An inexact Bregman proximal gradient method and its inertial variants with absolute and relative stopping criteria, arXiv:2109.05690
    15. M.X. Lin,  D.F. Sun, K.C. Toh and C.J. Wang, Estimation of sparse Gaussian graphical models with hidden clustering structure, arXiv:2004:08115
    16. K. Fujii, N. Ito, S. Kim, M. Kojima, Y. Shinano, and K.C. Toh,  Solving challenging large scale QAPs, arXiv:2101.09629
    17. X.L. Song, D.F. Sun, and K.C. Toh,
      Mesh independence of a majorized ABCD method for sparse PDE-constrained optimization problems, arXiv:2001.02118

2021–present

  1. L. Yang, L. Liang, H.T.M.  Chu, and K.C. Toh, A corrected inexact proximal augmented Lagrangian method with a relative error criterion for a class of group-quadratic regularized optimal transport problems,
    J. Scientific Computing, arXiv:2311.01976
  2. T.Y. Tang and K.C. Toh, A feasible method for general convex low-rank SDP problems,
    SIAM J. Optimization, in print.     arXiv:2312.07908
  3. P. Zhou, X.Y. Xie, Z.C. Lin, K.C. Toh, and S.C. Yan, Win: Weight-Decay-Integrated Nesterov Acceleration for Faster Network Training, 
    J. Machine Learning Research, 25 (2024), Article 83.
  4. T.Y. Tang, K.C. Toh, N.C. Xiao, and Y.Y. Ye, A Riemannian dimension-reduced second order method with application in sensor network localization,
    SIAM J. Scientific Computing,     arXiv:2304.10092 
  5. N.C. Xiao, X.Y. Hu, X. Liu, and K.C. Toh, Adam-family methods for nonsmooth optimization with convergence guarantees,
    J. Machine Learning Research, 25 (2024), Article 48. arXiv:2305.03938 
  6. Y.J. Zhang, Y. Cui, B. Sen, and K.C. Toh, On efficient and scalable computation of the nonparametric maximum likelihood estimator in mixture models,
    J. Machine Learning Research, 25 (2024), Article 8.     arXiv:2208.07514 
  7. X.Y. Hu, N.C. Xiao, X. Liu, and K.C. Toh, A constraint dissolving approach for nonsmooth optimization over the Stiefel manifold,
    IMA Journal of Numerical Analysis, arXiv:2205.10500 
  8. Y. J. Zhang, K.C. Toh, and D.F. Sun, Learning graph Laplacian with MCP,
    Optimization Methods and Software, in print.      arXiv:2010.11559
  9. T.Y. Tang, and K.C. Toh, Self-adaptive ADMM for semi-strongly convex problems,
    Mathematical Programming Computation, 16 (2024), pp. 113-150. arXiv:2310.00376
  10. M.X. Lin,  Y.C. Yuan, D.F. Sun, and K.C. Toh,
    A highly efficient algorithm for solving exclusive Lasso problems,
    Optimization Methods and Software, in print. arXiv:2306:14196.
    The above is a revised version of “Adaptive sieving with PPDNA: Generating solution paths of exclusive Lasso models”, arXiv:2009:08719
  11. T.Y. Tang, and K.C. Toh, A feasible method for solving an SDP relaxation of the quadratic knapsack problem, Mathematics of Operations Research, 49 (2024), pp. 19-39. arXiv:2303.06599
  12. T.Y. Tang, and K.C. Toh, Solving graph equipartition SDPs on an algebraic variety,
    Mathematical Programming, 204 (2024), pp. 299-347.     arXiv:2112.04256
  13. N.C. Xiao, X. Liu, and K.C. Toh, Constraint dissolving approaches for Riemannian optimization,
    Mathematics of Operations Research, 49 (2024), pp. 366-397.     arXiv:2203.10319 
  14. K.Y. Ding, X.Y. Lam, and K.C. Toh, On proximal augmented Lagrangian based decomposition methods for dual block-angular convex composite programming problems,
    Computational Optimization and Applications, 86 (2023), pp. 117–161.     arXiv:2303.06893
  15. X.Y. Hu, N.C. Xiao, X. Liu, and K.C. Toh, An improved unconstrained approach for bilevel optimization,
    SIAM J. Optimization, 33 (2023), pp. 2801-2829.     arXiv:2208.00732 
  16. H.T. Chu, L. Liang, K.C. Toh, and L. Yang, An efficient implementable inexact entropic proximal point algorithm for a class of linear programming problems,
    Computational Optimization and Applications, 85 (2023), pp. 107–146. arXiv:2011.14312
  17. H. Yang, L. Liang, L. Carlone, and K.C. Toh, An inexact projected gradient method with rounding and lifting by nonlinear programming for solving rank-one semidefinite relaxation of polynomial optimization,
    Mathematical Programming, 201 (2023), pp. 409–472.     arXiv:2105.14033
    Solver available at github 
  18. H.T. Chu, K.C. Toh, and Y.J. Zhang, On regularized square-root regression problems: distributionally robust interpretation and fast computations,
    J. Machine Learning Research, 23 (2022), article 308. arXiv:2109.03632
  19. Y.C. Yuan, T.H. Chang, D.F. Sun, and K.C. Toh, A dimension reduction technique for large-scale structured sparse optimization problems with application to convex clustering,
    SIAM J. Optimization, 32 (2022), pp. 2294-2318. arXiv:2108.07462
  20. L. Liang, X.D. Li, D.F. Sun, and K.C. Toh, QPPAL: A two-phase proximal augmented Lagrangian method for high dimensional convex quadratic programming problems,
    ACM Transactions on Mathematical Software, 48 (2022), Article 33.     arXiv:2103.13108
  21. L. Yang and K.C. Toh, Bregman proximal point algorithm revisited: A new inexact version and its inertial variant,
    SIAM J. Optimization, 32 (2022), pp. 1523-1554.     arXiv:2105.10370
  22. W.J. Li, W. Bian, K.C. Toh, DC algorithms for a class of sparse group L0 regularized optimization problems,
    SIAM J. Optimization, 32 (2022), pp. 1614-1641. arXiv:2109.05251
  23. M.X. Lin,  D.F. Sun, and K.C. Toh, An augmented Lagrangian method with constraint generations for shape-constrained convex regression problems,
    Mathematical Programming Computation, 14 (2022), pp. 223–270. Springer Nature ShareIt
    arXiv:2012.04862, old version: arXiv:2002.11410
  24. Y. Cui, L. Liang, D.F. Sun, and K.C. Toh,
    On degenerate doubly nonnegative projection problems,
    Mathematics of Operations Research, 47 (2022), pp. 2219-2239. arXiv:2009.11272, DOI
  25. S.Y. Kim, M. Kojima, and K.C. Toh,
    Doubly nonnegative relaxations for quadratic and polynomial optimization problems with binary and box constraints,
    Mathematical Programming, 193 (2022), pp. 761–787. Optimization Online, DOI
  26. T.-D. Quoc, L. Liang, K.C. Toh,
    A new homotopy proximal variable-metric framework for composite convex minimization,
    Mathematics of Operations Research, 47 (2022), pp. 508–539. arXiv:1812.05243, DOI
  27. R. Wang, N.H. Xiu, and K.C. Toh,
    Subspace quadratic regularization method for group sparse m    ultinomial logistic regression,
    Computational Optimization and Applications, 79 (2021), pp. 531–559.
  28. L. Liang, D.F. Sun, and K.C. Toh,
    An inexact augmented Lagrangian method for second-order cone programming with applications,
    SIAM J. Optimization, 31 (2021), pp. 1748–1773. arXiv:2010.08772
  29. N. Zhang, Y.J. Zhang, D.F. Sun, and K.C. Toh,
    An efficient linearly convergent regularized proximal point algorithm for fused multiple graphical Lasso problems,
    SIAM J. Mathematics of Data Science, 3 (2021), pp. 524–543.     arXiv:1902.06952
  30. L. Yang, J. Li, D.F. Sun, and K.C. Toh,
    A fast globally linearly convergent algorithm for the computation of Wasserstein barycenters,
    J. Machine Learning Research, 22 (2021), article 21. arXiv:1809.04249
  31. X.Y. Lam, D.F Sun, and K.C. Toh,
    A semi-proximal augmented Lagrangian based decomposition method for primal block angular convex composite quadratic conic programming problems,
    INFORMS J. Optimization, 3 (2021), pp. 254–277. arXiv:1812.04941
  32. S.Y. Kim, M. Kojima, and K.C. Toh,
    A Newton-bracketing method for a simple conic optimization problem, 
    Optimization Methods and Software, 36 (2021), pp. 371–388. arXiv:1905.12840.
  33. L. Chen, X.D. Li, D.F. Sun, and K.C. Toh,
    On the equivalence of inexact proximal ALM and ADMM for a class of convex composite programming,
    Mathematical Programming, 185 (2021), pp. 111–161. arXiv:1803.10803
  34. D.F. Sun, K.C. Toh, and Y.C. Yuan,
    Convex clustering: model, theoretical guarantee and efficient algorithm,
    J. Machine Learning Research, 22 (2021), Article 9.     arXiv:1810.0267

    2017–2020
  35. P.P. Tang, C.J. Wang, D.F. Sun, and K.C. Toh,
    A sparse semismooth Newton based proximal majorization-minimization algorithm for nonconvex square-root-loss regression problems,
    J. Machine Learning Research, 21 (2020), Article 226. arXiv:1903.11460
  36. X.D. Li, D.F. Sun, and K.C. Toh,
    An asymptotically superlinearly convergent semismooth Newton augmented Lagrangian method for Linear Programming,
    SIAM J. Optimization,  30 (2020), pp. 2410–2440.     arXiv:1903.09546.
  37. Y.J. Zhang, N. Zhang, D.F. Sun, and K.C. Toh,
    A proximal point dual Newton algorithm for solving group graphical Lasso problems,
    SIAM J. Optimization, 30 (2020), pp. 2197–2220. arXiv:1906.04647.
  38. S.Y. Kim, M. Kojima, and K.C. Toh,
    A geometrical analysis of a class of nonconvex conic programs for convex conic reformulations of quadratic and polynomial optimization problems,
    SIAM J. Optimization, 30 (2020), pp. 1251–1273.     arXiv:1901.02179.
  39. S.Y. Kim, M. Kojima, and K.C. Toh,
    Doubly nonnegative relaxations are equivalent to completely positive reformulations of quadratic optimization problems with block-clique graph structures, 
    J. Global Optimization, 77 (2020), pp. 513–541. arXiv:1903.07325.
  40. C. Ding, D.F. Sun, J. Sun, and K.C. Toh,
    Spectral operators of matrices: semismoothness and characterizations of the generalized Jacobian,
    SIAM J. Optimization, 30 (2020), pp. 630–659.     arXiv:1810.09856.
  41.  X.D. Li, D.F. Sun and K.C. Toh,
    On the efficient computation of a generalized Jacobian of the projector over the Birkhoff polytope,
    Mathematical Programming, 179 (2020),  pp. 419–446.     arXiv:1702.05934. Springer Nature ShareIt.
  42. Y.J. Zhang, N. Zhang, D.F. Sun and K.C. Toh,
    An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems,
    Mathematical Programming, 179 (2020), pp. 223–263. arXiv:1712.05910. Springer Nature ShareIt.
  43. D.F. Sun, K.C. Toh, Y.C. Yuan, and X.Y. Zhao,
    SDPNAL+: A Matlab software for semidefinite programming with bound constraints (version 1.0),
    Optimization Methods and Software, 35 (2020), 87–115. arXiv:1710.10604.
  44. S.L. Hu, D.F. Sun, and K.C. Toh,
    Best nonnegative rank-one approximations of tensors,
    SIAM J. Matrix Analysis and Applications, 40 (2019), pp. 1527–1554.  arXiv:1810.13372.
  45. Y. Cui, D.F. Sun, and K.C. Toh,
    Computing the best approximation over the intersection of a polyhedral set and the doubly nonnegative cone,
    SIAM J. Optimization, 29 (2019), pp. 2785–2813. arXiv:1803.06566.
  46. Z.Y. Lou, D.F. Sun, K.C. Toh, and N.H. Xiu,
    Solving the OSCAR and SLOPE models using a semismooth Newton-based augmented Lagrangian method,
    J. Machine Learning Research, 20 (2019), Article 106. arXiv:1803.10740.
  47. M.X. Lin, Y.J. Liu, D.F. Sun, and K.C. Toh,
    Efficient sparse semismooth Newton methods for the clustered Lasso problem,
    SIAM J. Optimization, 29 (2019), pp. 2026–2052. arXiv:1808.07181.
  48. L. Chen, D.F. Sun, K.C. Toh, and N. Zhang,
    A unified algorithmic framework of symmetric Gauss-Seidel decomposition based proximal ADMMs for convex composite programming,
    J. Computational Mathematics, 37 (2019), pp. 739–757. arXiv:1812.06579.
  49. N. Ito, S. Kim, M. Kojima, A. Takeda, and K.C. Toh,
    BBCPOP: A sparse doubly nonnegative relaxation of polynomial optimization problems with binary, box and complementarity constraints,
    ACM Transactions on Mathematical Software, 45 (2019), Article 34.
    arXiv:1804.00761. BBCPOP Matlab Software.
    Valid lower bounds for large QAPs computed by Hans Mittelmann using BBCPOP.
  50. N. Arima, S.Y. Kim, M. Kojima, and K.C. Toh,
    Lagrangian-conic relaxations, Part II: Applications to polynomial optimization problems,
    Pacific J. Optimization, 15 (2019), pp. 415–439. Optimization Online.
  51. L. Chen, D.F. Sun and K.C. Toh,
    Some problems on the Gauss-Seidel iteration method in degenerate cases (in Chinese)
    Journal On Numerical Methods and Computer Applications, 40 (2019), pp. 98–110. 
  52. Y. Cui, D.F. Sun and K.C. Toh,
    On the R-superlinear convergence of the KKT residuals generated by the augmented Lagrangian method for convex composite conic programming,
    Mathematical Programming, 178 (2019), pp. 381–415.  arXiv:1706.08800. Springer Nature SharedIt.
  53. X.D. Li, D.F. Sun and K.C. Toh,
    A block symmetric Gauss-Seidel decomposition theorem for convex composite quadratic programming and its applications,
    Mathematical Programming, 175 (2019), pp. 395–418. arXiv:1703.06629. Springer Nature SharedIt.
  54. N. Ito, S. Kim, M. Kojima, A. Takeda, and K.C. Toh,
    Equivalences and differences in conic relaxations of combinatorial quadratic optimization problems,
    J. Global Optimization, 72 (2018), pp. 619–653. Optimization Online. Springer Nature SharedIt.
  55. X.D. Li, D.F. Sun and K.C. Toh,
    On efficiently solving the subproblems of a level-set method for fused lasso problems,
    SIAM J. Optimization, 28 (2018), pp. 1842–1866.
    arXiv:1706.08732. Detailed computational results in the paepr.
  56. X.D. Li, D.F. Sun, and K.C. Toh,
    QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming,
    Mathematical Programming Computation, 10 (2018), pp. 703–743. arXiv:1512.08872. Springer Nature SharedIt.
  57. K. Natarajan, D.J. Shi, and K.C. Toh,
    Bounds for random binary quadratic programs,
    SIAM J. Optimization, 28 (2018), pp. 671–692.
  58. X.D. Li, D.F. Sun, and K.C. Toh,
    A highly efficient semismooth Netwon augmented Lagrangian method for solving Lasso problems,
    SIAM J. Optimization, 28 (2018), pp. 433–458.     arXiv:1607.05428.
  59. Z.W. Li, L.F. Cheong, S.G. Yang, and K.C. Toh,
    Simultaneous clustering and model selection: algorithm, theory and applications,
    IEEE Transactions on Pattern Analysis and Machine Intelligence, 40 (2018), pp. 1964–1978.
  60. X.Y. Lam, J.S. Marron, D.F. Sun, and K.C. Toh,
    Fast algorithms for large scale generalized distance weighted discrimination,
    J. Computational and Graphical Statistics, 27 (2018), pp. 368–379. arXiv:1604.05473.
    R package. Matlab package
  61. T. Weisser, J.B. Lasserre, and K.C. Toh,
    A bounded degree SOS hierarchy for large scale polynomial optimization with sparsity,
    Mathematical Programming Computation, 10 (2018), pp. 1–32. arXiv:1607.01151. Springer Nature SharedIt.
  62. C. Ding, D.F. Sun, J. Sun, and K.C. Toh,
    Spectral operators of matrices,
    Mathematical Programming, 168 (2018), pp. 509–531. arXiv:1401.2269.
  63. Ethan Fang, H. Liu, K.C. Toh, W.-X. Zhou,
    Max-norm optimization for robust matrix recovery,
    Mathematical Programming, 167 (2018), pp. 5–35. Optimization Online. Springer Nature SharedIt.
  64. N. Arima, S.Y. Kim, M. Kojima, and K.C. Toh,
    Lagrangian-conic relaxations, Part I: A unified framework and its applications to quadratic optimization problems,
    Pacific J. Optimization, 14 (2018), pp.161–192. Optimization Online.
  65. N. Ito, A. Takeda, and K.C. Toh,
    A unified formulation and fast accelerated proximal gradient method for classification,
    J. Machine Learning Research, 18 (2017), Article 16.
  66. N. Arima, S.Y. Kim, M. Kojima, and K.C. Toh,
    A robust Lagrangian-DNN method for a class of quadratic optimizaiton problems,
    Computational Optimization and Applications, 66 (2017), pp. 453–479. Optimization Online.
  67. L. Chen, D.F. Sun, and K.C. Toh,
    A note on the convergence of ADMM for linearly constrained convex optimization problems,
    Computational Optimization and Applications, 66 (2017), pp. 327—343.  arXiv:1507.02051
  68. J.B. Lasserre, K.C. Toh, and S.G. Yang,
    A bounded-SOS-hierarchy for polynomial optimization,
    EURO J. Computational Optimization, 5 (2017), pp. 87–117. arXiv:1501.06126.
  69. L. Chen, D.F. Sun, and K.C. Toh,
    An efficient inexact symmetric Gauss-Seidel based majorized ADMM for high-dimensional convex composite conic programming,
    Mathematical Programming, 161 (2017), pp. 237–270. arXiv:1506.00741. Springer Nature SharedIt.

    2014–2016

  70. D.F. Sun, K.C. Toh, and L.Q. Yang,
    An efficient inexact ABCD method for least squares semidefinite programming,
    SIAM J. Optimization, 26 (2016), pp. 1072–1100. arXiv:1505.04278.
    Detailed computational results for over 600 problems tested in the paper.
  71. Y. Cui, X.D. Li, D.F. Sun and K.C. Toh,
    On the convergence properties of a majorized ADMM for linearly constrained convex optimization problems with coupled objective functions,
    J. Optimization Theory and Applications, 169 (2016), pp. 1013–1041. arXiv:1502.00098. Springer Nature SharedIt
  72. M. Li, D.F. Sun, and K.C. Toh,
    A majorized ADMM with indefinite proximal terms for linearly constrained convex composite optimization,
    SIAM J. Optimization, 26 (2016), pp. 922–950. arXiv:1412.1911.
  73. S.Y. Kim, M. Kojima, and K.C. Toh,
    A Lagrangian-DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems,
    Mathematical Programming, 156 (2016), pp. 161–187.
  74. C.H. Chen, Y.J. Liu, D.F. Sun, and K.C. Toh,
    A semismooth Newton-CG dual proximal point algorithm for spectral norm approximation problems,
    Mathematical Programming, 155 (2016), pp. 435–470.
  75. X.D. Li, D.F. Sun and K.C. Toh,
    A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions,
    Mathematical Programming, 155 (2016), pp. 333–373. arXiv:1409.2679.
  76. L.Q. Yang, D.F. Sun, and K.C. Toh,
    SDPNAL+: a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints,
    Mathematical Programming Computation, 7 (2015), pp. 331-366. arXiv:1406.0942.
    More recent computational results (computed in Dec 2017).
    Numerical experiments on a variety of large scale SDPs with the matrix dimension n up to 9,261 and the number of equality constraints m up to 12,326,390 show that the proposed method is very efficient on certain large SDPs. We are also able to solve the SDP problem fap36 (with n=4,110 and m=1,154,467) in the Seventh DIMACS Implementation Challenge much more efficiently (in 23 hours in 2015) and accurately than previous attempts. The approximate optimal objective value we obtained for fap36 is 69.85, with the corresponding solution having relative primal and dual infeasibilities, and complementarity gap ⟨X,S⟩ all less than 1e-6.
  77. D.F. Sun, K.C. Toh and L.Q. Yang,
    A convergent 3-block semi-proximal alternating direction method of multipliers for conic programming with 4-type constraints,
    SIAM J. Optimization, 25 (2015), pp. 882–915. arXiv:1404.5378.
    Detailed computational results for over 400 problems tested in the paper.
    Supplementary note: more detailed comparison between the performance of our algorithm and various variants of ADMMs.
  78. M. Li, D.F. Sun, and K.C. Toh,
    A convergent 3-block semi-proximal ADMM for convex minimization with one strongly convex block,
    Asia Pacific J. Operational Research, 32 (2015), 1550024. arXiv:1410.7933.
  79. Y.X. Wang, C.M. Lee, L.F. Cheong, and K.C. Toh,
    Practical matrix completion and corruption recovery using proximal alternating robust subspace minimization,
    International J. of Computer Vision, 111 (2015), pp. 315–344. arXiv:1309.1539.
  80. C. Tang, K.K. Phoon, and K.C. Toh,
    Effect of footing width on Ny and failure envelope of eccentrically and obliquely loaded strip footings on sand,
    Canadian Geotechnical Journal, 52 (2015), pp. 694–707.
  81. J. Peng, T. Zhu, H. Luo, and K.C. Toh,
    Semidefinite relaxation of quadratic assignment problems based on nonredundant matrix splitting,
    Computational Optimization and Applications, 60 (2015), pp. 171–198.
  82. K.F. Jiang, D.F. Sun, and K.C. Toh,
    A partial proximal point algorithm for nuclear norm regularized matrix least squares problems,
    Mathematical Programming Computation, 6 (2014), pp. 281–325.
  83. Z. Gong, Z.W. Shen, and K.C. Toh,
    Image restoration with mixed or unknown noises,
    Multiscale Modeling and Simulation, 12 (2014), pp. 458–487.
  84. B. Wu, C. Ding, D.F. Sun, and K.C. Toh,
    On the Moreau-Yoshida regularization of the vector k-norm related functions,
    SIAM J. Optimization, 24 (2014), pp. 766–794.
  85. K. Natarajan, D.J. Shi, and K.C. Toh,
    A probabilistic model for minimax regret in combinatorial optimization,
    Operations Research, 62 (2014), pp. 160–181.
  86. C. Ding, D.F Sun and K.C. Toh,
    An introduction to a class of matrix cone programming,
    Mathematical Programming, 144 (2014), pp. 141–179.
  87. C. Tang, K.K. Phoon, and K.C. Toh,
    Lower bound limit analysis for seismic passive earth pressure on rigid walls,
    International J. of Geomechanics, 14 (2014), 04014022.
  88. C. Tang, K.C. Toh, and K.K. Phoon,
    Axisymmetric lower bound limit analysis using finite elements and second-order cone programming,
    J. of Engineering Mechanics, 140 (2014), pp. 268–278.

    2011–2013

  89. Z.Z. Zhang, G.L. Li, K.C. Toh, and W.K. Sung,
    3D chromosome modeling with semi-definite programming and Hi-C data,
    J. Computational Biology, 20 (2013), pp. 831–846.
  90. J.F. Yang, D.F. Sun, and K.C. Toh,
    A proximal point algorithm for log-determinant optimization with group Lasso regularization,
    SIAM J. Optimization, 23 (2013), pp. 857–893.
  91. X.V. Doan, K.C. Toh, and S. Vavasis,
    A proximal point algorithm for sequential feature extraction applications,
    SIAM J. Scientific Computing, 35 (2013), pp. 517–540.
  92. T.H.H. Tran, K.C. Toh, and K.K. Phoon,
    Preconditioned IDR(s) iterative solver for non-symmetric linear system associated with FEM analysis of shallow foundation,
    International J. for Numerical and Analytical Methods in Geomechanics, 37 (2013), pp. 2972–2986.
  93. K. B. Chaudhary, K.K. Phoon, and K.C. Toh,
    Inexact block diagonal preconditioners to mitigate the effects of relative differences in material stiffnesses,
    International J. Geomechanics, 13 (2013), pp. 273–291.
  94. K. B. Chaudhary, K.K. Phoon, and K.C. Toh,
    Effective block diagonal preconditioners for Biot’s consolidation equations in piled-raft foundations,
    International J. Numerical and Analytical Methods in Geomechanics, 37 (2013), pp. 871–892.
  95. K.F. Jiang, D.F. Sun, and K.C. Toh,
    An inexact accelerated proximal gradient method for large scale linearly constrained convex SDP    ,
    SIAM J. Optimization, 22 (2012), pp. 1042–1064.
  96. Y.J. Liu, D.F. Sun, and K.C. Toh,
    An implementable proximal point algorithmic framework for nuclear norm minimization,
    Mathematical Programming, 133 (2012), pp. 399–436.
  97. X.Y. Zhao, and K.C. Toh,
    Infeasible potential reduction algorithms for semidefinite programming,
    Pacific J. Optimization, 8 (2012), pp. 725–753.
  98. X. Chen, K.K. Phoon, and K.C. Toh,
    Performance of zero-level fill-in preconditioning techniques for iterative solutions in geotechnical applications,
    International J. Geomechanics, 12 (2012), pp. 596–605.
  99. Z. Shen, K.C. Toh, and S. Yun,
    An accelerated proximal gradient algorithm for frame based image restoration via the balanced approach,
    SIAM J. Imaging Sciences, 4 (2011), pp. 573–596.
  100. S. Yun, P. Tseng, and K.C. Toh,
    A block coordinate gradient descent method for regularized convex separable optimization and covariance selection,
    Mathematical Programming, 129 (2011), pp. 331–355.
  101. L. Li, and K.C. Toh,
    A polynomial-time inexact primal-dual infeasible path-following algorithm for convex quadratic SDP,
    Pacific J. Optimization, 7 (2011), pp. 43–61.
  102. S. Yun, and K.C. Toh,
    A coordinate gradient descent method for L1-regularized convex minimization,
    Computational Optimization and Applications, 48 (2011), pp. 273–307.
    Erratum: In Lemma 3.4, add Assumption 2 so that equation (22) is valid.

    Before 2011

  103. K.C. Toh, and S.W. Yun
    An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems,
    Pacific J. Optimization, 6 (2010), pp. 615–640.
    Numerical results suggest that our algorithm is efficient and robust in solving large-scale random matrix completion problems. In particular, we are able to solve random matrix completion problems with matrix dimensions up to $10^5$ each in less than 10 minutes on a modest PC.
  104. Lu Li and K.C. Toh
    An inexact interior point method for L1-regularized sparse covariance selection,
    Mathematical Programming Computation, 2 (2010), pp. 291–315.
  105. L. Li, and K.C. Toh,
    A polynomial-time inexact interior-point method for convex quadratic symmetric cone programming,
    J. Math-for-industry, 2 (2010), pp. 199–212.
  106. X.-W. Liu, G.Y. Zhao, and K.C. Toh,
    On the implementation of a log-barrier progressive hedging method for multistage stochastic programs,
    J. of Computational and Applied Mathematics, 234 (2010), pp. 579–592.
  107. C.J. Wang, D.F. Sun, and K.C. Toh,
    Solving log-determinant optimization problems by a Newton-CG primal proximal point algorithm,
    SIAM J. Optimization, 20 (2010), pp. 2994–3013.
  108. X.Y. Zhao, D.F. Sun, and K.C. Toh,
    A Newton-CG augmented Lagrangian method for semidefinite programming,
    SIAM J. Optimization, 20 (2010), pp. 1737–1765.
    Numerical experiments on a variety of large scale SDPs with the matrix dimension n up to 4,110 and the number of equality constraints m up to 2,156,544 show that the proposed method is very efficient on certain large SDPs. We are also able to solve the SDP problem fap36 (with n = 4,110 and m = 1,154,467) in the Seventh DIMACS Implementation Challenge much more accurately than previous attempts. The approximate optimal objective value we obtained for fap36 is 69.85, with the corresponding solution having relative primal and dual infeasibilities, and complementarity gap (Tr(XS)) all less than 1e-6.
  109. N.-H. Z. Leung and K.-C. Toh,
    An SDP-based divide-and-conquer algorithm for large scale noisy anchor-free graph realization,
    SIAM J. Scientific Computing, 31 (2010), pp. 4351–4372.
    A movie showing how the divide-and-conquer algorithm computes the conformation of a protein molecule.
  110. P. Biswas, K.C. Toh, and Y. Ye,
    A distributed SDP approach for large scale noisy anchor-free graph realization with applications to molecular conformation,
    SIAM J. Scientific Computing, 30 (2008), pp. 1251–1277.
  111. K.C. Toh,
    An inexact primal-dual path-following algorithm for convex quadratic SDP,
    Mathematical Programming, 112 (2008), pp. 221–254.
  112. K.C. Toh, and K.K. Phoon,
    Comparison between iterative solution of symmetric and non-symmetric forms of Biot’s FEM equations using the generalized Jacobi preconditioner,
    International J. for Numerical and Analytical Methods in Geomechanics, 32 (2008), pp. 1131–1146.
  113. X. Chen, K.K. Phoon, and K.C. Toh,
    Partitioned versus global Krylov subspace iterative methods for FE solution of 3-D Biot’s problem,
    Computer Methods in Applied Mechanics and Engineering, 196 (2007), pp. 2737–2750.
  114. J.S. Chai, and K.C. Toh,
    Preconditioning and iterative solution of symmetric indefinite linear systems arising from interior point methods for linear programming,
    Computational Optimization and Applications, 36 (2007), pp. 221–247.
  115. K.C. Toh, R.H. Tutuncu, and M.J. Todd,
    Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems,
    Pacific J. Optimization (special issue dedicated to Masakazu Kojima’s 60th birthday), 3 (2007), pp. 135–164.
  116. R.M. Freund, F. Ordonez, and K.C. Toh,
    Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems,
    Mathematical Programming, 109 (2007), pp. 445–475.
  117. Z. Cai and K.C. Toh,
    Solving second order cone programming via the augmented systems,
    SIAM J. Optimization, 17 (2006), pp. 711–737.
  118. P. Biswas, T.C. Liang, K.C. Toh, T.C. Wang, and Y. Ye,
    Semidefinite programming approaches for sensor network localization with noisy distance measurements,
    IEEE Transactions on Automation Science and Engineering, regular paper, 3 (2006), pp. 360–371.
  119. X. Chen, K.C. Toh, and K.K. Phoon,
    A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations,
    International J. Numerical Methods in Engineering, 65 (2006), pp. 785–807.
  120. J.S. Chai and K.C. Toh,
    Computation of condition numbers for linear programming problems using Pena’s method,
    Optimization Methods and Software, 21 (2006), pp. 419–443.
  121. G.L. Zhou, and K.C. Toh,
    Superlinear convergence of a Newton-type algorithm for monotone equations,
    J. Optimization Theory and Applications, 125 (2005), pp. 205–221.
  122. G.L. Zhou, K.C. Toh, and J. Sun,
    Efficient algorithms for the smallest enclosing ball problem,
    Computational Optimization and Applications, 30 (2005), pp. 147–160.
  123. K.K. Phoon, K.C. Toh, and X. Chen,
    Block constrained versus generalized Jacobi preconditioners iterative solution of large-scale Biot’s FEM equations,
    Computers and Structures, 82 (2004), pp. 2401–2411.
  124. K.C. Toh, K.K. Phoon, and S.H. Chan,
    Block preconditioners for symmetric indefinite linear systems,
    International J. Numerical Methods in Engineering, 60 (2004), pp. 1361–1381.
  125. S. K. Chua, K. C. Toh and G. Y. Zhao,
    An analytic center cutting plane method with deep cuts for semidefinite feasibility problems,
    J. Optimization Theory and Applications, 123 (2004), pp. 291–318.
  126. G.L. Zhou, K.C. Toh, and G.Y. Zhao,
    Convergence analysis of an infeasible interior point algorithm based on a regularized central path for linear complementarity problems,
    Computational Optimization and Applications, 27 (2004), pp. 269–283.
  127. K. C. Toh,
    Solving large scale semidefinite programs via an iterative solver on the augmented systems,
    SIAM J. Optimization, 14 (2004), pp. 670–698.
  128. G.L. Zhou, and K.C. Toh,
    Polynomiality of an inexact infeasible interior point algorithm for semidefinite programming,
    Mathematical Programming, 99 (2004), pp. 261–282.
  129. K.K. Phoon, K.C.Toh, S.H. Chan, and F.H. Lee,
    Fast iterative solution of large undrained soil-structure interaction problems,
    International J. for Numerical and Analytical Methods in Geomechanics, 27 (2003), pp. 159–181.
  130. G.L. Zhou, K.C. Toh, and D.F. Sun,
    A globally and quadratically convergent algorithm for minimizing a sum of Euclidean norms,
    J. Optimization Theory and Applications, 119 (2003), pp. 357–377.
  131. R.H Tutuncu, K.C. Toh, and M.J. Todd,
    Solving semidefinite-quadratic-linear programs using SDPT3,
    Mathematical Programming, 95 (2003), pp. 189–217.
  132. K.C. Toh, G.Y Zhao, and J. Sun,
    A multiple-cut analytic center cutting plane method for semidefinite feasibility problems,
    SIAM J. Optimizaton, 12 (2002), pp. 1126–1146.
  133. J. Sun, K.C. Toh, and G.Y Zhao,
    An analytic center cutting plane method for semidefinite feasibility problems,
    Mathematics of Operations Research, 27 (2002), pp. 332–346.
  134. K.C. Toh, and M. Kojima,
    Solving some large scale semidefinite programs via the conjugate residual method,
    SIAM J. Optimization, 12 (2002), pp. 669–691.
  135. K.C. Toh,
    A note on the calculation of step-lengths in interior-point methods for semidefinite programming,
    Computational Optimization and Applications, 21 (2002), pp. 301–310.
  136. K.K. Phoon, K.C. Toh, S.H. Chan, and F.H. Lee
    An efficient diagonal preconditioner for finite element solution of Biot’s consolidation equations,
    International J. Numerical Methods in Engineering, 55 (2002), pp. 377–400.
  137. A. Ron, Z.W. Shen, and K.C. Toh,
    Computing the Sobolev regularity of refinable functions by the the Arnoldi Method,
    SIAM J. Matrix Analysis and Applications, 23 (2001), pp. 57–76.
  138. K.C. Toh,
    Some new search directions for primal-dual interior point methods in semidefinite programming,
    SIAM J. Optimization, 11 (2000), pp. 223–242.
  139. K.C. Toh, and L.N. Trefethen,
    The Kreiss Matrix Theorem on a general complex domain,
    SIAM J. Matrix Analysis and Applications, 21 (1999), pp. 145–165.
  140. K.C. Toh, M.J. Todd, and R.H. Tutuncu,
    SDPT3 — a Matlab software package for semidefinite programming,
    Optimization Methods and Software, 11 (1999), pp. 545–581.
  141. K.C. Toh,
    Primal-dual path-following algorithms for determinant maximization problems with linear matrix inequalities,
    Computational Optimization and Applications, 14 (1999), pp. 309–330.
  142. T.A. Driscoll, K.C. Toh and L.N. Trefethen,
    From potential theory to matrix iterations in six steps,
    SIAM Review, 40 (1998), pp. 547-578.
  143. M.J. Todd, K.C. Toh, and R.H. Tutuncu,
    On the Nesterov-Todd direction in semidefinite programming,
    SIAM J. of Optimization, 8 (1998), pp. 769–796.
  144. K.C. Toh and L.N. Trefethen,
    The Chebyshev Polynomials of a Matrix,
    SIAM J. Matrix Analysis and Applications, 20 (1998), pp. 400-419.
  145. K.C. Toh,
    GMRES vs. ideal GMRES,
    SIAM J. of Matrix Analysis and Applications, 18 (1997), pp. 30–36.
  146. K.C. Toh and L.N. Trefethen,
    Calculation of pseudospectra by the Arnoldi iteration,
    SIAM J. of Scientific Computing, 17 (1996), pp. 1–15.
  147. K.C. Toh and L.N. Trefethen,
    Pseudozeros of polynomials and pseudospectra of companion matrices,
    Numerische Mathematik, 68 (1994), pp. 403–425.
  148. K.C. Toh and S. Mukherjee,
    Hypersingular and finite part integrals in the boundary element method,
    International J. of Solids and Structures, 31 (1994), pp. 2299–2312.

Refereed Conference Papers

  1. Anh Duc Nguyen, Tuan Dung Nguyen, Quang Minh Nguyen, Hoang H Nguyen, Lam M. Nguyen, Kim-Chuan Toh, On Partial Optimal Transport: Revising the Infeasibility of Sinkhorn and Efficient Gradient Method, Oral presentation, 38th AAAI Conference on Artificial Intelligence (AAAI-24), 2024. arXiv:2312.13970
  2. Y.C. Yuan, D.F. Sun, and K.C. Toh, An efficient semismooth Newton based algorithm for convex clustering,
    Oral presentation, International Conference on Machine Learning (ICML) 2018. arXiv:1802.07091.
  3. Z.W. Li, S.G. Yang, L.-F. Cheong, and K.C. Toh, Simultaneous Clustering and Model Selection for Tensor Affinities,
    Spotlight presentation, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
  4. Z.Z. Zhang, G.L. Li, K.C. Toh, and W. Sung, Inference of spatial organizations of chromosomes using semidefinite embedding approach and Hi-C data,
    RECOMB 2013, The 17th Annual International Conference on Research in Computational Molecular Biology, Beijing, China, April 7-10, 2013.
     In “Research in Computational Molecular Biology”, Lecture Notes in Computer Science, Volume 7821, 2013, Springer, pp. 317–332.
  5. Krishna B. Chaudhary, K.K. Phoon, and K.C. Toh, Fast iterative solution of large soil-structure interaction problems in varied ground conditions,
    Proceedings of 14th Asian Regional Conference on Soil Mechanics and Geotechnical Engineering, Hong Kong, China, 23-27 May 2011.
  6. K. B. Chaudhary, K.K. Phoon, and K.C. Toh, Comparison of MSSOR versus ILU(0) Preconditioners for Biot’s FEM Consolidation Equations,
    The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG), 1-6 October 2008, Goa, India.
  7. X. Chen, K.K. Phoon, and K.C. Toh, Symmetric indefinite preconditioners for FE solution of Biot’s consolidation problem,
    Geotechnical Engineering in the Information Technology Age (2006): CDROM. Reston: ASCE. (GeoCongress2006, 26 Feb – 1 Mar 2006, Atlanta, United States).
  8. K.C. Toh, R.H. Tutuncu, and M.J. Todd, On the implementation of SDPT3 (version 3.1) — a Matlab software package for semidefinite-quadratic-linear programming,
    IEEE Conference on Computer-Aided Control System Design, Taipei, Taiwan, 2-4 September 2004.
  9. F. Ting, W.J. Heng, and K.C. Toh, Question classification for e-learning by artificial neural network,
    Fourth International Conference on Information, Communications & Signal Processing and Fourth IEEE Pacific-Rim Conference On Multimedia, 15-18 December 2003, Singapore.
  10. K.K. Phoon, K.C. Toh, S.H. Chan, and F.H. Lee, A generalized Jacobi preconditioner for finite element solution of large-scale consolidation problems,
    in Second MIT Conference on Computational Fluid and Solid Mechanics, 17–20 June 2003, Massachusetts Institute of Technology, Cambridge, United States, Vol.1, pp. 573–577, 2003.
  11. G.L. Zhou, K.C. Toh, and J. Sun, Efficient algorithms for the smallest enclosing ball problem in high dimensional space,
    Novel Approaches to Hard Discrete Optimization, Proceedings of Fields Institute of Mathematics, P. Pardalos and H. Wolkowicz eds., Canadian Mathematical Society, 2002.

Book chapters and others

  1. K.C. Toh, Some numerical issues in the development of SDP algorithms, INFORMS OS Today, Volume 8 Number 2 (2008), pp. 7–20.
  2. K.C. Toh, M.J. Todd, and R.H. Tutuncu, On the implementation and usage of SDPT3 — a Matlab software package for semidefinite-quadratic-linear programming, version 4.0, in Handbook on semidefinite, cone and polynomial optimization: theory, algorithms, software and applications, M. Anjos and J.B. Lasserre eds., Springer, 2012, pp. 715–754.
    Here is the complete performance results obtained by SDPT3-4.0 on over 400 problems.
  3. X.Y. Fang and K.C. Toh, Using a distributed SDP approach to solve simulated protein molecular conformation problems,
    in Distance Geometry: Theory, Methods, and Applications, A. Mucherino, C. Lavor, L. Liberti, and N. Maculan eds., Springer, 2013, pp. 351–376.
  4. K.F. Jiang, D.F. Sun, and K.C. Toh, Solving nuclear norm regularized and semidefinite matrix least squares problems with linear equality constraints,
    Fields Institute Communications Volume 69, Discrete Geometry and Optimization, K. Bezdek, Y. Ye, and A. Deza eds., Springer, 2013, pp. 133–162.