Adaptive Descent Splitting Method for Decomposable Optimization Problems

  • Igor Konnov
  • Olga PinyaginaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12095)


We suggest a modified descent splitting method for optimization problems having a special decomposable structure. The proposed modification maintains the basic convergence properties but enables one to reduce computational efforts per iteration and to provide computations in a distributed manner. On the one hand, it consists in component-wise choice of descent directions together with a special threshold control. On the other hand, it involves a simple adaptive step-size choice, which takes into account the problem behavior along the iteration sequence. Preliminary computational tests confirm the efficiency of the proposed modification.


Descent splitting method Adaptive step-size choice Decomposable optimization problem Threshold control Coordinate-wise step 


  1. 1.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two monotone operators. SIAM. J. Num. Anal. 16(6), 964–979 (1979)CrossRefGoogle Scholar
  2. 2.
    Gabay, D.: Application of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-value Problems, pp. 299–331. North-Holland, Amsterdam (1983)CrossRefGoogle Scholar
  3. 3.
    Fukushima, M., Mine, H.: A generalized proximal point algorithm for certain non-convex minimization problems. Int. J. Syst. Sci. 12, 989–1000 (1981)CrossRefGoogle Scholar
  4. 4.
    Patriksson, M.: Cost approximations: a unified framework of descent algorithms for nonlinear programs. SIAM J. Optim. 8(2), 561–582 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Patriksson, M.: Nonlinear Programming and Variational Inequality Problems: A Unified Approach. Kluwer, Dordrecht (1999)CrossRefGoogle Scholar
  6. 6.
    Konnov, I.V., Kum, S.: Descent methods for mixed variational inequalities in a Hilbert space. Nonlinear Anal. Theory Methods Appl. 47(1), 561–572 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Konnov, I.V.: Iterative solution methods for mixed equilibrium problems and variational inequalities with non-smooth functions. In: Haugen, I.N., Nilsen, A.S. (eds.) Game Theory: Strategies, Equilibria, and Theorems, pp. 117–160. NOVA, Hauppauge (2008)Google Scholar
  8. 8.
    Konnov, I.V.: Descent methods for mixed variational inequalities with non-smooth mappings. In: Reich, S., Zaslavski, A.J. (eds.) Optimization Theory and Related Topics. Contemporary Mathematics, vol. 568, pp. 121–138 (2012). Amer. Math. Soc., ProvidenceGoogle Scholar
  9. 9.
    Konnov, I.V.: Salahuddin: two-level iterative method for non-stationary mixed variational inequalities. Russ. Math. (Iz. VUZ) 61(10), 44–53 (2017)CrossRefGoogle Scholar
  10. 10.
    Konnov, I.V.: Sequential threshold control in descent splitting methods for decomposable optimization problems. Optim. Methods Softw. 30(6), 1238–1254 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Konnov, I.: Conditional gradient method without line-search. Russ. Math. 62(1), 82–85 (2018)CrossRefGoogle Scholar
  12. 12.
    Konnov, I.: A simple adaptive step-size choice for iterative optimization methods. Adv. Model. Optim. 20(2), 353–369 (2018)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Konnov, I., Pinyagina, O.: Splitting method with adaptive step-size. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds.) MOTOR 2019. LNCS, vol. 11548, pp. 46–58. Springer, Cham (2019). Scholar

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Authors and Affiliations

  1. 1.Department of System Analysis and Information Technologies, Institute of Computational Mathematics and Information TechnologiesKazan Federal UniversityKazanRussia
  2. 2.Department of Data Mining and Operations Research, Institute of Computational Mathematics and Information TechnologiesKazan Federal UniversityKazanRussia

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