Global Optimization Algorithms Using Tensor Trains

  • Dmitry A. ZheltkovEmail author
  • Alexander Osinsky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)


Global optimization problem arises in a huge amount of applications including parameter estimation of different models, molecular biology, drug design and many others. There are several types of methods for this problem: deterministic, stochastic, heuristic and metaheuristic. Deterministic methods guarantee that found solution is the global optima, but complexity of such methods allows to use them only for problems of relatively small dimensionality, simple functional and area of optimization.

Non-deterministic methods are based on some simple models of stochastic, physical, biological and other processes. On practice such methods are often much faster then direct methods. But for the most of them there is no proof of such fast convergence even for some simple cases.

In this paper we consider global optimization method based on tensor train decomposition. The method is non-deterministic and exploits tensor structure of functional. Theoretical results proving its fast convergence in some simple cases to global optimum are provided.



The work was supported by the RAS presidium program 1 “Fundamental mathematics and its applications” and by program 26 “Fundamental foundations of design of algorithms and software for prospective and high-performance computing systems”.


  1. 1.
    Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudo-skeleton approximations. Linear Algebra Appl. 261(1–3), 1–21 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Tyrtyshnikov, E.E.: Incomplete cross approximation in the mosaic-skeleton method. Computing 64(4), 367–380 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Goreinov, S.A., Tyrtyshnikov, E.E.: The maximal-volume concept in approximation by low-rank matrices. Contemp. Math. 208, 47–51 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Goreinov, S.A., Oseledets, I.V., Savostyanov, D.V., Tyrtyshnikov, E.E., Zamarashkin, N.L.: How to find a good submatrix. In: Olshevsky, V., Tyrtyshnikov, E. (eds.) Matrix Methods: Theory, Algorithms, Applications, pp. 247–256. World Scientific, Hackensack (2010)CrossRefGoogle Scholar
  5. 5.
    Oseledets, I.V., Tyrtyshnikov, E.E.: Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31(5), 3744–3759 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Oseledets, I.V., Tyrtyshnikov, E.E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70–88 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Osinsky A.I.: Probabilistic estimation of the rank-1 cross approximation accuracy. arXiv preprint arXiv:1706.10285 (2017)
  9. 9.
    Zheltkova, V.V., Zheltkov, D.A., Grossman, Z., Bocharov, G.A., Tyrtyshnikov, E.E.: Tensor based approach to the numerical treatment of the parameter estimation problems in mathematical immunology. J. Inverse Ill-posed Prob. 26(1), 51–66 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sulimov, A.V., et al.: Evaluation of the novel algorithm of flexible ligand docking with moveable target-protein atoms. Comput. Struct. Biotechnol. J. 15, 275–285 (2017)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Marchuk Institute of Numerical MathematicsMoscowRussia

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