Primal-Dual Methods for Solving Infinite-Dimensional Games

  • Pavel Dvurechensky
  • Yurii NesterovEmail author
  • Vladimir Spokoiny


In this paper, we show that the infinite-dimensional differential games with simple objective functional can be solved in a finite-dimensional dual form in the space of dual multipliers for the constraints related to the end points of the trajectories. The primal solutions can be easily reconstructed by the appropriate dual subgradient schemes. The suggested schemes are justified by the worst-case complexity analysis.


Convex optimization Primal-dual optimization methods  Saddle-point problems Differential games 

Mathematics Subject Classification

90C06 90C25 90C60 91A23 49N70 



The research presented in this paper was partially supported by the Laboratory of Structural Methods of Data Analysis in Predictive Modeling, MIPT, through the RF government Grant, ag.11.G34.31.0073 and by RFBR, Research Project No. 13-01-12007 ofi_m.


  1. 1.
    Nesterov, Y.: Primal-dual subgradient methods for convex problems. Math. Program. 120(1), 221–259 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Devolder, O., Glineur, F., Nesterov, Y.: Double smoothing technique for large-scale linearly constrained convex optimization. SIAM J. Optim. 22(2), 702–727 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Antsaklis, P.J., Michel, A.N.: Linear Systems. Birkhauser Book, Boston (2006)zbMATHGoogle Scholar
  4. 4.
    Maurice, S.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)zbMATHCrossRefGoogle Scholar
  5. 5.
    Nesterov, Y.: Dual extrapolation and its applications for solving variational inequalities and related problems. J. Math. Program. Ser. B 109(2), 319–344 (2007)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Pavel Dvurechensky
    • 1
  • Yurii Nesterov
    • 2
    Email author
  • Vladimir Spokoiny
    • 3
  1. 1.PreMoLab, Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.COREUniversite catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Weierstrass Institute and Humboldt UniversityBerlinGermany

Personalised recommendations