Preprints

 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
 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, arXiv:2208.07514
 X.Y. Hu, N.C. Xiao, X. Liu, and K.C. Toh, An improved unconstrained approach for bilevel
optimization, arXiv:2208.00732  X.Y. Hu, N.C. Xiao, X. Liu, and K.C. Toh, A constraint dissolving approach for nonsmooth optimization over the Stiefel manifold, arXiv:2205.10500
 C.P. Lee, L. Liang, T.Y. Tang, and K.C. Toh, Escaping spurious local minima of lowrank matrix factorization through convex lifting, arXiv:2204.14067
 N.C. Xiao, X. Liu, and K.C. Toh, Constraint dissolving approaches for Riemannian optimization, arXiv:2203.10319
 T.Y. Tang, and K.C. Toh, Solving graph equipartition SDPs on an algebraic variety,
arXiv:2112.04256  L. Yang, and K.C. Toh, An inexact Bregman proximal gradient method and its inertial variants, arXiv:2109.05690
 H.T. Chu, K.C. Toh, and Y.J. Zhang, On regularized squareroot regression problems: distributionally robust interpretation and fast computations, arXiv:2109.03632
 H. Yang, L. Liang, L. Carlone, and K.C. Toh, An inexact projected gradient method with rounding and lifting by nonlinear programming for solving rankone semidefinite relaxation of polynomial optimization,
arXiv:2105.14033
Solver available at github  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, arXiv:2011.14312
 M.X. Lin, Y.C. Yuan, D.F. Sun, and K.C. Toh,
Adaptive sieving with PPDNA: Generating solution paths of exclusive Lasso models, arXiv:2009:08719  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
 Y. J. Zhang, K.C. Toh, and D.F. Sun, Learning Graph Laplacian with MCP, arXiv:2010.11559
 K. Fujii, N. Ito, S. Kim, M. Kojima, Y. Shinano, and K.C. Toh, Solving challenging large scale QAPs, arXiv:2101.09629
 X.L. Song, D.F. Sun, and K.C. Toh,
Mesh independence of a majorized ABCD method for sparse PDEconstrained optimization problems, arXiv:2001.02118
2021–present
 Y.C. Yuan, T.H. Chang, D.F. Sun, and K.C. Toh, A dimension reduction technique for largescale structured sparse optimization problems with application to convex clustering,
SIAM J. Optimization, in print, 2022. arXiv:2108.07462  L. Liang, X.D. Li, D.F. Sun, and K.C. Toh, QPPAL: A twophase proximal augmented Lagrangian method for high dimensional convex quadratic programming problems,
ACM Transactions on Mathematical Software, in print, 2022. arXiv:2103.13108  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. 15231554. arXiv:2105.10370  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. 16141641. arXiv:2109.05251  M.X. Lin, D.F. Sun, and K.C. Toh, An augmented Lagrangian method with constraint generations for shapeconstrained convex regression problems,
Mathematical Programming Computation, 14 (2022), pp. 223–270. Springer Nature ShareIt
arXiv:2012.04862, old version: arXiv:2002.11410  Y. Cui, L. Liang, D.F. Sun, and K.C. Toh,
On degenerate doubly nonnegative projection problems,
Mathematics of Operations Research, 47 (2022), pp. 22192239. arXiv:2009.11272, DOI  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  T.D. Quoc, L. Liang, K.C. Toh,
A new homotopy proximal variablemetric framework for composite convex minimization,
Mathematics of Operations Research, 47 (2022), pp. 508–539. arXiv:1812.05243, DOI  R. Wang, N.H. Xiu, and K.C. Toh,
Subspace quadratic regularization method for group sparse multinomial logistic regression,
Computational Optimization and Applications, 79 (2021), pp. 531–559.
 L. Liang, D.F. Sun, and K.C. Toh,
An inexact augmented Lagrangian method for secondorder cone programming with applications,
SIAM J. Optimization, 31 (2021), pp. 1748–1773. arXiv:2010.08772  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  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  X.Y. Lam, D.F Sun, and K.C. Toh,
A semiproximal 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  S.Y. Kim, M. Kojima, and K.C. Toh,
A Newtonbracketing method for a simple conic optimization problem,
Optimization Methods and Software, 36 (2021), pp. 371–388. arXiv:1905.12840.  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  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  P.P. Tang, C.J. Wang, D.F. Sun, and K.C. Toh,
A sparse semismooth Newton based proximal majorizationminimization algorithm for nonconvex squarerootloss regression problems,
J. Machine Learning Research, 21 (2020), Article 226. arXiv:1903.11460  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.  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.  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.  S.Y. Kim, M. Kojima, and K.C. Toh,
Doubly nonnegative relaxations are equivalent to completely positive reformulations of quadratic optimization problems with blockclique graph structures,
J. Global Optimization, 77 (2020), pp. 513–541. arXiv:1903.07325.  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.  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.  Y.J. Zhang, N. Zhang, D.F. Sun and K.C. Toh,
An efficient Hessian based algorithm for solving largescale sparse group Lasso problems,
Mathematical Programming, 179 (2020), pp. 223–263. arXiv:1712.05910. Springer Nature ShareIt.  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.  S.L. Hu, D.F. Sun, and K.C. Toh,
Best nonnegative rankone approximations of tensors,
SIAM J. Matrix Analysis and Applications, 40 (2019), pp. 1527–1554. arXiv:1810.13372.  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.  Z.Y. Lou, D.F. Sun, K.C. Toh, and N.H. Xiu,
Solving the OSCAR and SLOPE models using a semismooth Newtonbased augmented Lagrangian method,
J. Machine Learning Research, 20 (2019), Article 106. arXiv:1803.10740.  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.  L. Chen, D.F. Sun, K.C. Toh, and N. Zhang,
A unified algorithmic framework of symmetric GaussSeidel decomposition based proximal ADMMs for convex composite programming,
J. Computational Mathematics, 37 (2019), pp. 739–757. arXiv:1812.06579.  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.  N. Arima, S.Y. Kim, M. Kojima, and K.C. Toh,
Lagrangianconic relaxations, Part II: Applications to polynomial optimization problems,
Pacific J. Optimization, 15 (2019), pp. 415–439. Optimization Online.  L. Chen, D.F. Sun and K.C. Toh,
Some problems on the GaussSeidel iteration method in degenerate cases (in Chinese)
Journal On Numerical Methods and Computer Applications, 40 (2019), pp. 98–110.  Y. Cui, D.F. Sun and K.C. Toh,
On the Rsuperlinear 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.  X.D. Li, D.F. Sun and K.C. Toh,
A block symmetric GaussSeidel decomposition theorem for convex composite quadratic programming and its applications,
Mathematical Programming, 175 (2019), pp. 395–418. arXiv:1703.06629. Springer Nature SharedIt.  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.  X.D. Li, D.F. Sun and K.C. Toh,
On efficiently solving the subproblems of a levelset method for fused lasso problems,
SIAM J. Optimization, 28 (2018), pp. 1842–1866.
arXiv:1706.08732. Detailed computational results in the paepr.  X.D. Li, D.F. Sun, and K.C. Toh,
QSDPNAL: A twophase augmented Lagrangian method for convex quadratic semidefinite programming,
Mathematical Programming Computation, 10 (2018), pp. 703–743. arXiv:1512.08872. Springer Nature SharedIt.  K. Natarajan, D.J. Shi, and K.C. Toh,
Bounds for random binary quadratic programs,
SIAM J. Optimization, 28 (2018), pp. 671–692.
 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.  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.
 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  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.  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.  Ethan Fang, H. Liu, K.C. Toh, W.X. Zhou,
Maxnorm optimization for robust matrix recovery,
Mathematical Programming, 167 (2018), pp. 5–35. Optimization Online. Springer Nature SharedIt.  N. Arima, S.Y. Kim, M. Kojima, and K.C. Toh,
Lagrangianconic relaxations, Part I: A unified framework and its applications to quadratic optimization problems,
Pacific J. Optimization, 14 (2018), pp.161–192. Optimization Online.  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.  N. Arima, S.Y. Kim, M. Kojima, and K.C. Toh,
A robust LagrangianDNN method for a class of quadratic optimizaiton problems,
Computational Optimization and Applications, 66 (2017), pp. 453–479. Optimization Online.  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  J.B. Lasserre, K.C. Toh, and S.G. Yang,
A boundedSOShierarchy for polynomial optimization,
EURO J. Computational Optimization, 5 (2017), pp. 87–117. arXiv:1501.06126.  L. Chen, D.F. Sun, and K.C. Toh,
An efficient inexact symmetric GaussSeidel based majorized ADMM for highdimensional convex composite conic programming,
Mathematical Programming, 161 (2017), pp. 237–270. arXiv:1506.00741. Springer Nature SharedIt.2014–2016
 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.  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  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.  S.Y. Kim, M. Kojima, and K.C. Toh,
A LagrangianDNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems,
Mathematical Programming, 156 (2016), pp. 161–187.  C.H. Chen, Y.J. Liu, D.F. Sun, and K.C. Toh,
A semismooth NewtonCG dual proximal point algorithm for spectral norm approximation problems,
Mathematical Programming, 155 (2016), pp. 435–470.  X.D. Li, D.F. Sun and K.C. Toh,
A Schur complement based semiproximal ADMM for convex quadratic conic programming and extensions,
Mathematical Programming, 155 (2016), pp. 333–373. arXiv:1409.2679.  L.Q. Yang, D.F. Sun, and K.C. Toh,
SDPNAL+: a majorized semismooth NewtonCG augmented Lagrangian method for semidefinite programming with nonnegative constraints,
Mathematical Programming Computation, 7 (2015), pp. 331366. 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 1e6.  D.F. Sun, K.C. Toh and L.Q. Yang,
A convergent 3block semiproximal alternating direction method of multipliers for conic programming with 4type 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.  M. Li, D.F. Sun, and K.C. Toh,
A convergent 3block semiproximal ADMM for convex minimization with one strongly convex block,
Asia Pacific J. Operational Research, 32 (2015), 1550024. arXiv:1410.7933.  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.  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.  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.  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.  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.  B. Wu, C. Ding, D.F. Sun, and K.C. Toh,
On the MoreauYoshida regularization of the vector knorm related functions,
SIAM J. Optimization, 24 (2014), pp. 766–794.  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.  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.  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.  C. Tang, K.C. Toh, and K.K. Phoon,
Axisymmetric lower bound limit analysis using finite elements and secondorder cone programming,
J. of Engineering Mechanics, 140 (2014), pp. 268–278.2011–2013
 Z.Z. Zhang, G.L. Li, K.C. Toh, and W.K. Sung,
3D chromosome modeling with semidefinite programming and HiC data,
J. Computational Biology, 20 (2013), pp. 831–846.  J.F. Yang, D.F. Sun, and K.C. Toh,
A proximal point algorithm for logdeterminant optimization with group Lasso regularization,
SIAM J. Optimization, 23 (2013), pp. 857–893.  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.  T.H.H. Tran, K.C. Toh, and K.K. Phoon,
Preconditioned IDR(s) iterative solver for nonsymmetric linear system associated with FEM analysis of shallow foundation,
International J. for Numerical and Analytical Methods in Geomechanics, 37 (2013), pp. 2972–2986.  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.  K. B. Chaudhary, K.K. Phoon, and K.C. Toh,
Effective block diagonal preconditioners for Biot’s consolidation equations in piledraft foundations,
International J. Numerical and Analytical Methods in Geomechanics, 37 (2013), pp. 871–892.  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.  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.  X.Y. Zhao, and K.C. Toh,
Infeasible potential reduction algorithms for semidefinite programming,
Pacific J. Optimization, 8 (2012), pp. 725–753.  X. Chen, K.K. Phoon, and K.C. Toh,
Performance of zerolevel fillin preconditioning techniques for iterative solutions in geotechnical applications,
International J. Geomechanics, 12 (2012), pp. 596–605.  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.  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.  L. Li, and K.C. Toh,
A polynomialtime inexact primaldual infeasible pathfollowing algorithm for convex quadratic SDP,
Pacific J. Optimization, 7 (2011), pp. 43–61.  S. Yun, and K.C. Toh,
A coordinate gradient descent method for L1regularized 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
 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 largescale 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.  Lu Li and K.C. Toh
An inexact interior point method for L1regularized sparse covariance selection,
Mathematical Programming Computation, 2 (2010), pp. 291–315.  L. Li, and K.C. Toh,
A polynomialtime inexact interiorpoint method for convex quadratic symmetric cone programming,
J. Mathforindustry, 2 (2010), pp. 199–212.  X.W. Liu, G.Y. Zhao, and K.C. Toh,
On the implementation of a logbarrier progressive hedging method for multistage stochastic programs,
J. of Computational and Applied Mathematics, 234 (2010), pp. 579–592.  C.J. Wang, D.F. Sun, and K.C. Toh,
Solving logdeterminant optimization problems by a NewtonCG primal proximal point algorithm,
SIAM J. Optimization, 20 (2010), pp. 2994–3013.  X.Y. Zhao, D.F. Sun, and K.C. Toh,
A NewtonCG 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 1e6.  N.H. Z. Leung and K.C. Toh,
An SDPbased divideandconquer algorithm for large scale noisy anchorfree graph realization,
SIAM J. Scientific Computing, 31 (2010), pp. 4351–4372.
A movie showing how the divideandconquer algorithm computes the conformation of a protein molecule.
 P. Biswas, K.C. Toh, and Y. Ye,
A distributed SDP approach for large scale noisy anchorfree graph realization with applications to molecular conformation,
SIAM J. Scientific Computing, 30 (2008), pp. 1251–1277.  K.C. Toh,
An inexact primaldual pathfollowing algorithm for convex quadratic SDP,
Mathematical Programming, 112 (2008), pp. 221–254.  K.C. Toh, and K.K. Phoon,
Comparison between iterative solution of symmetric and nonsymmetric 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.  X. Chen, K.K. Phoon, and K.C. Toh,
Partitioned versus global Krylov subspace iterative methods for FE solution of 3D Biot’s problem,
Computer Methods in Applied Mechanics and Engineering, 196 (2007), pp. 2737–2750.  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.  K.C. Toh, R.H. Tutuncu, and M.J. Todd,
Inexact primaldual pathfollowing 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.  R.M. Freund, F. Ordonez, and K.C. Toh,
Behavioral measures and their correlation with IPM iteration counts on semidefinite programming problems,
Mathematical Programming, 109 (2007), pp. 445–475.  Z. Cai and K.C. Toh,
Solving second order cone programming via the augmented systems,
SIAM J. Optimization, 17 (2006), pp. 711–737.  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.  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.  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.  G.L. Zhou, and K.C. Toh,
Superlinear convergence of a Newtontype algorithm for monotone equations,
J. Optimization Theory and Applications, 125 (2005), pp. 205–221.  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.  K.K. Phoon, K.C. Toh, and X. Chen,
Block constrained versus generalized Jacobi preconditioners iterative solution of largescale Biot’s FEM equations,
Computers and Structures, 82 (2004), pp. 2401–2411.  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.  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.  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.  K. C. Toh,
Solving large scale semidefinite programs via an iterative solver on the augmented systems,
SIAM J. Optimization, 14 (2004), pp. 670–698.  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.  K.K. Phoon, K.C.Toh, S.H. Chan, and F.H. Lee,
Fast iterative solution of large undrained soilstructure interaction problems,
International J. for Numerical and Analytical Methods in Geomechanics, 27 (2003), pp. 159–181.  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.  R.H Tutuncu, K.C. Toh, and M.J. Todd,
Solving semidefinitequadraticlinear programs using SDPT3,
Mathematical Programming, 95 (2003), pp. 189–217.  K.C. Toh, G.Y Zhao, and J. Sun,
A multiplecut analytic center cutting plane method for semidefinite feasibility problems,
SIAM J. Optimizaton, 12 (2002), pp. 1126–1146.  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.  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.  K.C. Toh,
A note on the calculation of steplengths in interiorpoint methods for semidefinite programming,
Computational Optimization and Applications, 21 (2002), pp. 301–310.  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.  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.  K.C. Toh,
Some new search directions for primaldual interior point methods in semidefinite programming,
SIAM J. Optimization, 11 (2000), pp. 223–242.  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.  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.  K.C. Toh,
Primaldual pathfollowing algorithms for determinant maximization problems with linear matrix inequalities,
Computational Optimization and Applications, 14 (1999), pp. 309–330.  T.A. Driscoll, K.C. Toh and L.N. Trefethen,
From potential theory to matrix iterations in six steps,
SIAM Review, 40 (1998), pp. 547578.  M.J. Todd, K.C. Toh, and R.H. Tutuncu,
On the NesterovTodd direction in semidefinite programming,
SIAM J. of Optimization, 8 (1998), pp. 769–796.  K.C. Toh and L.N. Trefethen,
The Chebyshev Polynomials of a Matrix,
SIAM J. Matrix Analysis and Applications, 20 (1998), pp. 400419.  K.C. Toh,
GMRES vs. ideal GMRES,
SIAM J. of Matrix Analysis and Applications, 18 (1997), pp. 30–36.  K.C. Toh and L.N. Trefethen,
Calculation of pseudospectra by the Arnoldi iteration,
SIAM J. of Scientific Computing, 17 (1996), pp. 1–15.  K.C. Toh and L.N. Trefethen,
Pseudozeros of polynomials and pseudospectra of companion matrices,
Numerische Mathematik, 68 (1994), pp. 403–425.  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
 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.  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.  Z.Z. Zhang, G.L. Li, K.C. Toh, and W. Sung, Inference of spatial organizations of chromosomes using semidefinite embedding approach and HiC data,
RECOMB 2013, The 17th Annual International Conference on Research in Computational Molecular Biology, Beijing, China, April 710, 2013.
In “Research in Computational Molecular Biology”, Lecture Notes in Computer Science, Volume 7821, 2013, Springer, pp. 317–332.  Krishna B. Chaudhary, K.K. Phoon, and K.C. Toh, Fast iterative solution of large soilstructure interaction problems in varied ground conditions,
Proceedings of 14th Asian Regional Conference on Soil Mechanics and Geotechnical Engineering, Hong Kong, China, 2327 May 2011.  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), 16 October 2008, Goa, India.  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).  K.C. Toh, R.H. Tutuncu, and M.J. Todd, On the implementation of SDPT3 (version 3.1) — a Matlab software package for semidefinitequadraticlinear programming,
IEEE Conference on ComputerAided Control System Design, Taipei, Taiwan, 24 September 2004.  F. Ting, W.J. Heng, and K.C. Toh, Question classification for elearning by artificial neural network,
Fourth International Conference on Information, Communications & Signal Processing and Fourth IEEE PacificRim Conference On Multimedia, 1518 December 2003, Singapore.  K.K. Phoon, K.C. Toh, S.H. Chan, and F.H. Lee, A generalized Jacobi preconditioner for finite element solution of largescale 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.  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
 K.C. Toh, Some numerical issues in the development of SDP algorithms, INFORMS OS Today, Volume 8 Number 2 (2008), pp. 7–20.
 K.C. Toh, M.J. Todd, and R.H. Tutuncu, On the implementation and usage of SDPT3 — a Matlab software package for semidefinitequadraticlinear 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 SDPT34.0 on over 400 problems.  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.  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.