On the solution of large-scale SDP problems by the modified barrier method using iterative solvers The limiting factors of second-order methods for large-scale semidefinite optimization are the storage and factorization of the Newton matrix. For a particular algorithm based on the modified barrier method, we propose to use iterative solvers instead of the routinely used direct factorization techniques. The preconditioned conjugate gradient method proves to be a viable alternative for problems with a large number of variables and modest size of the constrained matrix. We further propose to avoid explicit calculation of the Newton matrix either by an implicit scheme in the matrix-vector product or using a finite-difference formula. This leads to huge savings in memory requirements and, for certain problems, to further speed-up of the algorithm. (Source:

References in zbMATH (referenced in 24 articles , 1 standard article )

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  1. Bellavia, Stefania; Gondzio, Jacek; Porcelli, Margherita: A relaxed interior point method for low-rank semidefinite programming problems with applications to matrix completion (2021)
  2. Huang, Aiqun: A proximal augmented method for semidefinite programming problems (2021)
  3. Yamashita, Hiroshi; Yabe, Hiroshi; Harada, Kouhei: A primal-dual interior point trust-region method for nonlinear semidefinite programming (2021)
  4. Andreani, Roberto; Haeser, Gabriel; Viana, Daiana S.: Optimality conditions and global convergence for nonlinear semidefinite programming (2020)
  5. Brune, Alexander; Kočvara, Michal: On barrier and modified barrier multigrid methods for three-dimensional topology optimization (2020)
  6. Bellavia, Stefania; Gondzio, Jacek; Porcelli, Margherita: An inexact dual logarithmic barrier method for solving sparse semidefinite programs (2019)
  7. Andrievskii, Boris R.; Selivanov, Anton A.: New results on the application of the passification method. A survey (2018)
  8. Diamond, Steven; Boyd, Stephen: Matrix-free convex optimization modeling (2016)
  9. Kočvara, Michal; Mohammed, Sudaba: Primal-dual interior point multigrid method for topology optimization (2016)
  10. Polyak, Roman A.: The Legendre transformation in modern optimization (2016)
  11. Polyak, Roman: Lagrangian transformation and interior ellipsoid methods in convex optimization (2015)
  12. Huang, Aiqun; Xu, Chengxian: A trust region method for solving semidefinite programs (2013)
  13. Huang, Aiqun; Xu, Chengxian: A globally convergent filter-type trust region method for semidefinite programming (2012)
  14. Malick, Jérôme; Roupin, Frédéric: Solving (k)-cluster problems to optimality with semidefinite programming (2012)
  15. Pan, Shaohua; Chiang, Yungyen; Chen, Jein-Shan: SOC-monotone and SOC-convex functions vs. matrix-monotone and matrix-convex functions (2012)
  16. Zhao, Xin-Yuan; Sun, Defeng; Toh, Kim-Chuan: A Newton-CG augmented Lagrangian method for semidefinite programming (2010)
  17. Kočvara, Michal; Stingl, Michael: Erratum to: “On the solution of large-scale SDP problems by the modified barrier method using iterative solvers” (2009)
  18. Stingl, M.; Kočvara, M.; Leugering, G.: A sequential convex semidefinite programming algorithm with an application to multiple-load free material optimization (2009)
  19. Jarre, Florian; Rendl, Franz: An augmented primal-dual method for linear conic programs (2008)
  20. Kočvara, Michal; Stingl, Michael: On the solution of large-scale SDP problems by the modified barrier method using iterative solvers (2007)

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